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Номер 4, страница 79, часть 1 - гдз по физике 9 класс рабочая тетрадь Грачев, Погожев
Авторы: Грачев А. В., Погожев В. А., Боков П. Ю., Вишнякова Е. А.
Тип: рабочая тетрадь
Издательство: Просвещение
Год издания: 2021 - 2026
Часть: 1
Цвет обложки: синий
ISBN: 978-5-09-098826-1 (ч.1) 978-5-09-098880-3 (ч.2) 978-5-09-098881-0 (ч.3)
Популярные ГДЗ в 9 классе
Часть 1. Глава 2. Динамика. Параграф 16. Динамика равномерного движения материальной точки по окружности - номер 4, страница 79.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1435001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => "80" "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/16-4" "field_display_title" => "4" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1434806 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 1203 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2021" "field_publication_year_until" => null "field_part" => "1, 2, 3" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "594" "field_priority" => null "field_default_folder" => "/fizika_09/grachev-rt/" "field_isbn" => "978-5-09-098826-1 (ч.1) 978-5-09-098880-3 (ч.2) 978-5-09-098881-0 (ч.3)" "field_cover" => array:3 [ …3] "field_cover_alts" => array:3 [ …3] "field_covers" => array:3 [ …3] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 1203 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2021" "field_publication_year_until" => null "field_part" => "1, 2, 3" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "594" "field_priority" => null "field_default_folder" => "/fizika_09/grachev-rt/" "field_isbn" => "978-5-09-098826-1 (ч.1) 978-5-09-098880-3 (ч.2) 978-5-09-098881-0 (ч.3)" "field_cover" => array:3 [ …3] "field_cover_alts" => array:3 [ …3] "field_covers" => array:3 [ …3] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1434806 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1073} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:3 [ 0 => App\Models\Branch {#1165 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1434806 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\Term {#1166 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Book {#1168 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 1203 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Term {#1167 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #original: array:10 [ "id" => 6493 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Физика" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "fizika" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1171 #items: array:1 [ 0 => App\Models\Term {#1172 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\Term {#1174 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:4 [ 0 => App\Models\Term {#1176 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4180 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Грачев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Васильевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4180 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Грачев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Васильевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1177 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Погожев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Погожев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Владимир" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1178 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3888 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Боков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Павел" "field_patronymic" => "Юрьевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3888 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Боков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Павел" "field_patronymic" => "Юрьевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Term {#1179 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Вишнякова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Екатерина" "field_patronymic" => "Анатольевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 4007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Вишнякова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Екатерина" "field_patronymic" => "Анатольевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Term {#1182 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #original: array:10 [ "id" => 6736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Term {#1184 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:1 [ 0 => App\Models\Term {#1186 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Term {#1191 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Term {#1193 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1194 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2021" "field_publication_year_until" => null "field_part" => "1, 2, 3" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "594" "field_priority" => null "field_default_folder" => "/fizika_09/grachev-rt/" "field_isbn" => "978-5-09-098826-1 (ч.1) 978-5-09-098880-3 (ч.2) 978-5-09-098881-0 (ч.3)" "field_cover" => array:3 [ 0 => "/media/fizika_09/grachev-rt/covers/cover1.webp?ts=1759003248" 1 => "/media/fizika_09/grachev-rt/covers/cover2.webp?ts=1759003248" 2 => "/media/fizika_09/grachev-rt/covers/cover3.webp?ts=1759003248" ] "field_cover_alts" => array:3 [ 0 => "" 1 => "" 2 => "" ] "field_covers" => array:3 [ 0 => array:4 [ "path" => "/media/fizika_09/grachev-rt/covers/cover1.webp?ts=1759003248" "alt" => "" "width" => "2293" "height" => "2989" ] 1 => array:4 [ "path" => "/media/fizika_09/grachev-rt/covers/cover2.webp?ts=1759003248" "alt" => "" "width" => "2265" "height" => "3003" ] 2 => array:4 [ "path" => "/media/fizika_09/grachev-rt/covers/cover3.webp?ts=1759003248" "alt" => "" "width" => "2241" "height" => "2999" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1195 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1196 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1197 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Term {#1199 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "field_cases" => array:6 [ …6] "field_translit" => "siniy" ] #original: array:6 [ "id" => 49 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "синий" "field_cases" => array:6 [ …6] "field_translit" => "siniy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #original: array:50 [ "id" => 1203 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1170} "field_class" => Illuminate\Database\Eloquent\Collection {#1171} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173} "field_author" => Illuminate\Database\Eloquent\Collection {#1175} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1180} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1181} "field_country" => Illuminate\Database\Eloquent\Collection {#1183} "field_city" => Illuminate\Database\Eloquent\Collection {#1185} "field_series" => Illuminate\Database\Eloquent\Collection {#1187} "field_umk" => Illuminate\Database\Eloquent\Collection {#1188} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1189} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1190} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1192} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1194} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2021" "field_publication_year_until" => null "field_part" => "1, 2, 3" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "594" "field_priority" => null "field_default_folder" => "/fizika_09/grachev-rt/" "field_isbn" => "978-5-09-098826-1 (ч.1) 978-5-09-098880-3 (ч.2) 978-5-09-098881-0 (ч.3)" "field_cover" => array:3 [ 0 => "/media/fizika_09/grachev-rt/covers/cover1.webp?ts=1759003248" 1 => "/media/fizika_09/grachev-rt/covers/cover2.webp?ts=1759003248" 2 => "/media/fizika_09/grachev-rt/covers/cover3.webp?ts=1759003248" ] "field_cover_alts" => array:3 [ 0 => "" 1 => "" 2 => "" ] "field_covers" => array:3 [ 0 => array:4 [ "path" => "/media/fizika_09/grachev-rt/covers/cover1.webp?ts=1759003248" "alt" => "" "width" => "2293" "height" => "2989" ] 1 => array:4 [ "path" => "/media/fizika_09/grachev-rt/covers/cover2.webp?ts=1759003248" "alt" => "" "width" => "2265" "height" => "3003" ] 2 => array:4 [ "path" => "/media/fizika_09/grachev-rt/covers/cover3.webp?ts=1759003248" "alt" => "" "width" => "2241" "height" => "2999" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1195} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1196} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1197} "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1198} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1200 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1434806 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1169} "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1202} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1200} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1201 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1434820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:1 [ 0 => App\Models\Term {#1203 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "53" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ 0 => App\Models\Book {#1168} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1434820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1206} "field_page_start" => "53" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1205} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1204} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1207 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1434825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика равномерного движения материальной точки по окружности" "field_branch_order" => "16" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Term {#1208 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "78" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Book {#1168} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:1 [ 0 => App\Models\Branch {#1209 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1434820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_page_start" => "53" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1247 …2} ] #original: array:24 [ "id" => 1434820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_page_start" => "53" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1247 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1434825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика равномерного движения материальной точки по окружности" "field_branch_order" => "16" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1211} "field_page_start" => "78" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1210} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1255} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:2 [ 0 => App\Models\Element {#1144 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1698389 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Edition {#1143 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6173 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_content_source" => null ] #original: array:21 [ "id" => 6173 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1435001" "type" => "task" ] "text" => "<p><strong>4.</strong> На равномерно вращающейся вокруг вертикальной оси O карусели стоят отец (1), сын (2) и дочь (3). Масса отца $m_1 = 70$ кг, масса сына $m_2 = 40$ кг, масса дочери $m_3 = 30$ кг. На рис. 54 схематически показан вид на карусель сверху. Дочь (3) находится на расстоянии $R_3 = 4$ м от оси вращения. Скорость сына $\vec{v}_2$ направлена так, как показано на рисунке, а её модуль равен 3 м/с. Серыми стрелками показаны единичный вектор ускорения и вектор силы, модуль которого равен 100 Н.</p><p>Используя линейку, определите удаления отца ($R_1$) и сына ($R_2$) от оси вращения. Вычислите модули векторов скоростей отца ($\vec{v}_1$) и дочери ($\vec{v}_3$), модули центростремительных ускорений ($a_{цс}$) отца, сына и дочери, а также модули сил трения ($F_{тр}$), действующих на них. Заполните таблицу и изобразите (разным цветом), соблюдая масштабы, векторы скоростей, ускорений и сил трения, обозначив их.</p><p><strong>R, м</strong><strong>$|\vec{v}|$, м/с</strong><strong>$|\vec{a}_{цс}|$, м/с$^2$</strong><strong>$|\vec{F}_{тр}|$, Н</strong></p><p>Отец: (пусто) (пусто) (пусто) (пусто)</p><p>Сын: (пусто) 3 (пусто) (пусто)</p><p>Дочь: 4 (пусто) (пусто) (пусто)</p><p>$|\vec{a}|=1$ м/с$^2$</p><p>$|\vec{F}|=100$ Н</p><p>Рис. 54</p>" "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1924" "height" => 464 "path" => "/media/fizika_09/grachev-rt/0-16/4-1.webp?ts=1759402684" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1999" "height" => 1127 "path" => "/media/fizika_09/grachev-rt/0-16/4-2.webp?ts=1759402684" ] ] ] #original: array:7 [ "id" => 1698389 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135} "task" => array:2 [ "refs" => "1435001" "type" => "task" ] "text" => "<p><strong>4.</strong> На равномерно вращающейся вокруг вертикальной оси O карусели стоят отец (1), сын (2) и дочь (3). Масса отца $m_1 = 70$ кг, масса сына $m_2 = 40$ кг, масса дочери $m_3 = 30$ кг. На рис. 54 схематически показан вид на карусель сверху. Дочь (3) находится на расстоянии $R_3 = 4$ м от оси вращения. Скорость сына $\vec{v}_2$ направлена так, как показано на рисунке, а её модуль равен 3 м/с. Серыми стрелками показаны единичный вектор ускорения и вектор силы, модуль которого равен 100 Н.</p><p>Используя линейку, определите удаления отца ($R_1$) и сына ($R_2$) от оси вращения. Вычислите модули векторов скоростей отца ($\vec{v}_1$) и дочери ($\vec{v}_3$), модули центростремительных ускорений ($a_{цс}$) отца, сына и дочери, а также модули сил трения ($F_{тр}$), действующих на них. Заполните таблицу и изобразите (разным цветом), соблюдая масштабы, векторы скоростей, ускорений и сил трения, обозначив их.</p><p><strong>R, м</strong><strong>$|\vec{v}|$, м/с</strong><strong>$|\vec{a}_{цс}|$, м/с$^2$</strong><strong>$|\vec{F}_{тр}|$, Н</strong></p><p>Отец: (пусто) (пусто) (пусто) (пусто)</p><p>Сын: (пусто) 3 (пусто) (пусто)</p><p>Дочь: 4 (пусто) (пусто) (пусто)</p><p>$|\vec{a}|=1$ м/с$^2$</p><p>$|\vec{F}|=100$ Н</p><p>Рис. 54</p>" "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1924" "height" => 464 "path" => "/media/fizika_09/grachev-rt/0-16/4-1.webp?ts=1759402684" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1999" "height" => 1127 "path" => "/media/fizika_09/grachev-rt/0-16/4-2.webp?ts=1759402684" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1137 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1734375 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Edition {#1136 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6327 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_content_source" => null ] #original: array:21 [ "id" => 6327 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1435001" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса отца (1): $m_1 = 70$ кг</p><p>Масса сына (2): $m_2 = 40$ кг</p><p>Масса дочери (3): $m_3 = 30$ кг</p><p>Расстояние дочери от оси вращения: $R_3 = 4$ м</p><p>Модуль скорости сына: $v_2 = 3$ м/с</p><p>Все данные уже представлены в системе СИ.</p><p><strong>Найти:</strong></p><p>$R_1, R_2$ – расстояния отца и сына от оси вращения;</p><p>$v_1, v_3$ – модули скоростей отца и дочери;</p><p>$a_{цс1}, a_{цс2}, a_{цс3}$ – модули центростремительных ускорений отца, сына и дочери;</p><p>$F_{тр1}, F_{тр2}, F_{тр3}$ – модули сил трения, действующих на отца, сына и дочь.</p><p><strong>Решение:</strong></p><p><strong>1. Определение расстояний $R_1$ и $R_2$</strong></p><p>Для определения расстояний отца ($R_1$) и сына ($R_2$) воспользуемся методом масштабирования, приняв за основу известное расстояние дочери ($R_3 = 4$ м). Измерив на рисунке расстояния от центра O до точек 1, 2 и 3, можно установить, что расстояние до сына составляет примерно 3/4 от расстояния до дочери, а расстояние до отца - примерно 1/2 от расстояния до дочери.</p><p>$R_2 \approx \frac{3}{4} R_3 = \frac{3}{4} \cdot 4 \text{ м} = 3 \text{ м}$</p><p>$R_1 \approx \frac{1}{2} R_3 = \frac{1}{2} \cdot 4 \text{ м} = 2 \text{ м}$</p><p><strong>Ответ:</strong> $R_1 = 2$ м, $R_2 = 3$ м.</p><p><strong>2. Вычисление угловой скорости и линейных скоростей $v_1$ и $v_3$</strong></p><p>Карусель вращается равномерно, следовательно, все точки на ней имеют одинаковую угловую скорость $\omega$. Найдем ее, используя данные для сына:</p><p>$v_2 = \omega R_2 \implies \omega = \frac{v_2}{R_2} = \frac{3 \text{ м/с}}{3 \text{ м}} = 1 \text{ рад/с}$</p><p>Теперь можем найти модули скоростей отца и дочери:</p><p>$v_1 = \omega R_1 = 1 \text{ рад/с} \cdot 2 \text{ м} = 2 \text{ м/с}$</p><p>$v_3 = \omega R_3 = 1 \text{ рад/с} \cdot 4 \text{ м} = 4 \text{ м/с}$</p><p><strong>Ответ:</strong> $v_1 = 2$ м/с, $v_3 = 4$ м/с.</p><p><strong>3. Вычисление модулей центростремительных ускорений ($a_{цс}$)</strong></p><p>Центростремительное ускорение вычисляется по формуле $a_{цс} = \omega^2 R$ или $a_{цс} = v^2/R$.</p><p>Для отца: $a_{цс1} = \omega^2 R_1 = (1 \text{ рад/с})^2 \cdot 2 \text{ м} = 2 \text{ м/с}^2$</p><p>Для сына: $a_{цс2} = \omega^2 R_2 = (1 \text{ рад/с})^2 \cdot 3 \text{ м} = 3 \text{ м/с}^2$</p><p>Для дочери: $a_{цс3} = \omega^2 R_3 = (1 \text{ рад/с})^2 \cdot 4 \text{ м} = 4 \text{ м/с}^2$</p><p><strong>Ответ:</strong> $a_{цс1} = 2$ м/с², $a_{цс2} = 3$ м/с², $a_{цс3} = 4$ м/с².</p><p><strong>4. Вычисление модулей сил трения ($F_{тр}$)</strong></p><p>Сила трения покоя сообщает телам центростремительное ускорение, поэтому, согласно второму закону Ньютона, $F_{тр} = m \cdot a_{цс}$.</p><p>Для отца: $F_{тр1} = m_1 \cdot a_{цс1} = 70 \text{ кг} \cdot 2 \text{ м/с}^2 = 140 \text{ Н}$</p><p>Для сына: $F_{тр2} = m_2 \cdot a_{цс2} = 40 \text{ кг} \cdot 3 \text{ м/с}^2 = 120 \text{ Н}$</p><p>Для дочери: $F_{тр3} = m_3 \cdot a_{цс3} = 30 \text{ кг} \cdot 4 \text{ м/с}^2 = 120 \text{ Н}$</p><p><strong>Ответ:</strong> $F_{тр1} = 140$ Н, $F_{тр2} = 120$ Н, $F_{тр3} = 120$ Н.</p><p><strong>5. Заполнение таблицы и изображение векторов</strong></p><p>Сведем все вычисленные значения в таблицу.</p><div class="table-wrap"><table border="1" cellpadding="5" cellspacing="0" style="border: 1px solid rgb(0, 0, 0);"> <thead> <tr> <th style="border: 1px solid rgb(0, 0, 0);"></th> <th style="border: 1px solid rgb(0, 0, 0);">$R$, м</th> <th style="border: 1px solid rgb(0, 0, 0);">$|\vec{v}|$, м/с</th> <th style="border: 1px solid rgb(0, 0, 0);">$|\vec{a}_{цс}|$, м/с²</th> <th style="border: 1px solid rgb(0, 0, 0);">$|\vec{F}_{тр}|$, Н</th> </tr> </thead> <tbody> <tr> <td style="border: 1px solid rgb(0, 0, 0);"><strong>Отец</strong></td> <td style="border: 1px solid rgb(0, 0, 0);">2</td> <td style="border: 1px solid rgb(0, 0, 0);">2</td> <td style="border: 1px solid rgb(0, 0, 0);">2</td> <td style="border: 1px solid rgb(0, 0, 0);">140</td> </tr> <tr> <td style="border: 1px solid rgb(0, 0, 0);"><strong>Сын</strong></td> <td style="border: 1px solid rgb(0, 0, 0);">3</td> <td style="border: 1px solid rgb(0, 0, 0);">3</td> <td style="border: 1px solid rgb(0, 0, 0);">3</td> <td style="border: 1px solid rgb(0, 0, 0);">120</td> </tr> <tr> <td style="border: 1px solid rgb(0, 0, 0);"><strong>Дочь</strong></td> <td style="border: 1px solid rgb(0, 0, 0);">4</td> <td style="border: 1px solid rgb(0, 0, 0);">4</td> <td style="border: 1px solid rgb(0, 0, 0);">4</td> <td style="border: 1px solid rgb(0, 0, 0);">120</td> </tr> </tbody></table></div><p>На рисунке ниже изображены векторы скоростей, ускорений и сил трения с соблюдением масштаба и направлений. Направление вращения карусели — по часовой стрелке, что следует из направления вектора скорости сына $\vec{v}_2$.</p><ul> <li>Векторы скорости ($\vec{v}$) показаны <strong style="color:blue;">синим</strong> цветом. Они направлены по касательной к траектории движения.</li> <li>Векторы центростремительного ускорения ($\vec{a}_{цс}$) показаны <strong style="color:red;">красным</strong> цветом. Они направлены к центру вращения О.</li> <li>Векторы силы трения ($\vec{F}_{тр}$) показаны <strong style="color:green;">зеленым</strong> цветом. Они также направлены к центру вращения О.</li></ul><svg width="400" height="400" viewBox="-120 -120 240 240" xmlns="http://www.w3.org/2000/svg"> <defs> <marker id="arrow-blue" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="4" markerHeight="4" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="blue"></path> </marker> <marker id="arrow-red" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="4" markerHeight="4" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="red"></path> </marker> <marker id="arrow-green" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="4" markerHeight="4" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="green"></path> </marker> </defs> <!-- Carousel outline --> <circle cx="0" cy="0" r="80" fill="none" stroke="black" stroke-width="1"></circle> <!-- Center O --> <circle cx="0" cy="0" r="2" fill="black"></circle> <text x="5" y="-5" font-family="Times New Roman" font-size="12">O</text> <!-- Point 1: Father (R=2m -> 40px at -30 deg) --> <g transform="translate(34.6, -20)"> <circle cx="0" cy="0" r="3" fill="black"></circle> <text x="5" y="5" font-family="Times New Roman" font-size="12">1</text> <!-- v1 (2 m/s -> 20px) tangent clockwise --> <line x1="0" y1="0" x2="-10" y2="-17.3" stroke="blue" stroke-width="2" marker-end="url(#arrow-blue)"></line> <text x="-25" y="-15" fill="blue" font-size="12">$\vec{v}_1$</text> <!-- a1 (2 m/s^2 -> 20px) towards center --> <line x1="0" y1="0" x2="-34.6" y2="20" stroke="red" stroke-width="2" marker-end="url(#arrow-red)"></line> <text x="-25" y="10" fill="red" font-size="12">$\vec{a}_1$</text> <!-- F1 (140 N -> 28px) towards center --> <line x1="2" y1="-1.15" x2="-25.8" y2="15.0" stroke="green" stroke-width="2" marker-end="url(#arrow-green)"></line> <text x="-2" y="18" fill="green" font-size="12">$\vec{F}_1$</text> </g> <!-- Point 2: Son (R=3m -> 60px at 90 deg) --> <g transform="translate(0, 60)"> <circle cx="0" cy="0" r="3" fill="black"></circle> <text x="5" y="5" font-family="Times New Roman" font-size="12">2</text> <!-- v2 (3 m/s -> 30px) left --> <line x1="0" y1="0" x2="-30" y2="0" stroke="blue" stroke-width="2" marker-end="url(#arrow-blue)"></line> <text x="-20" y="-8" fill="blue" font-size="12">$\vec{v}_2$</text> <!-- a2 (3 m/s^2 -> 30px) towards center --> <line x1="0" y1="0" x2="0" y2="-60" stroke="red" stroke-width="2" marker-end="url(#arrow-red)"></line> <text x="5" y="-30" fill="red" font-size="12">$\vec{a}_2$</text> <!-- F2 (120 N -> 24px) towards center --> <line x1="3" y1="0" x2="3" y2="-24" stroke="green" stroke-width="2" marker-end="url(#arrow-green)"></line> <text x="8" y="-10" fill="green" font-size="12">$\vec{F}_2$</text> </g> <!-- Point 3: Daughter (R=4m -> 80px at 180 deg) --> <g transform="translate(-80, 0)"> <circle cx="0" cy="0" r="3" fill="black"></circle> <text x="5" y="5" font-family="Times New Roman" font-size="12">3</text> <!-- v3 (4 m/s -> 40px) down --> <line x1="0" y1="0" x2="0" y2="40" stroke="blue" stroke-width="2" marker-end="url(#arrow-blue)"></line> <text x="8" y="25" fill="blue" font-size="12">$\vec{v}_3$</text> <!-- a3 (4 m/s^2 -> 40px) towards center --> <line x1="0" y1="0" x2="80" y2="0" stroke="red" stroke-width="2" marker-end="url(#arrow-red)"></line> <text x="35" y="-5" fill="red" font-size="12">$\vec{a}_3$</text> <!-- F3 (120 N -> 24px) towards center --> <line x1="0" y1="3" x2="24" y2="3" stroke="green" stroke-width="2" marker-end="url(#arrow-green)"></line> <text x="5" y="18" fill="green" font-size="12">$\vec{F}_3$</text> </g></svg><p><strong>Ответ:</strong> Все требуемые величины были вычислены, таблица заполнена, а векторы скоростей, ускорений и сил изображены на схеме выше.</p>" ] #original: array:6 [ "id" => 1734375 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123} "task" => array:2 [ "refs" => "1435001" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса отца (1): $m_1 = 70$ кг</p><p>Масса сына (2): $m_2 = 40$ кг</p><p>Масса дочери (3): $m_3 = 30$ кг</p><p>Расстояние дочери от оси вращения: $R_3 = 4$ м</p><p>Модуль скорости сына: $v_2 = 3$ м/с</p><p>Все данные уже представлены в системе СИ.</p><p><strong>Найти:</strong></p><p>$R_1, R_2$ – расстояния отца и сына от оси вращения;</p><p>$v_1, v_3$ – модули скоростей отца и дочери;</p><p>$a_{цс1}, a_{цс2}, a_{цс3}$ – модули центростремительных ускорений отца, сына и дочери;</p><p>$F_{тр1}, F_{тр2}, F_{тр3}$ – модули сил трения, действующих на отца, сына и дочь.</p><p><strong>Решение:</strong></p><p><strong>1. Определение расстояний $R_1$ и $R_2$</strong></p><p>Для определения расстояний отца ($R_1$) и сына ($R_2$) воспользуемся методом масштабирования, приняв за основу известное расстояние дочери ($R_3 = 4$ м). Измерив на рисунке расстояния от центра O до точек 1, 2 и 3, можно установить, что расстояние до сына составляет примерно 3/4 от расстояния до дочери, а расстояние до отца - примерно 1/2 от расстояния до дочери.</p><p>$R_2 \approx \frac{3}{4} R_3 = \frac{3}{4} \cdot 4 \text{ м} = 3 \text{ м}$</p><p>$R_1 \approx \frac{1}{2} R_3 = \frac{1}{2} \cdot 4 \text{ м} = 2 \text{ м}$</p><p><strong>Ответ:</strong> $R_1 = 2$ м, $R_2 = 3$ м.</p><p><strong>2. Вычисление угловой скорости и линейных скоростей $v_1$ и $v_3$</strong></p><p>Карусель вращается равномерно, следовательно, все точки на ней имеют одинаковую угловую скорость $\omega$. Найдем ее, используя данные для сына:</p><p>$v_2 = \omega R_2 \implies \omega = \frac{v_2}{R_2} = \frac{3 \text{ м/с}}{3 \text{ м}} = 1 \text{ рад/с}$</p><p>Теперь можем найти модули скоростей отца и дочери:</p><p>$v_1 = \omega R_1 = 1 \text{ рад/с} \cdot 2 \text{ м} = 2 \text{ м/с}$</p><p>$v_3 = \omega R_3 = 1 \text{ рад/с} \cdot 4 \text{ м} = 4 \text{ м/с}$</p><p><strong>Ответ:</strong> $v_1 = 2$ м/с, $v_3 = 4$ м/с.</p><p><strong>3. Вычисление модулей центростремительных ускорений ($a_{цс}$)</strong></p><p>Центростремительное ускорение вычисляется по формуле $a_{цс} = \omega^2 R$ или $a_{цс} = v^2/R$.</p><p>Для отца: $a_{цс1} = \omega^2 R_1 = (1 \text{ рад/с})^2 \cdot 2 \text{ м} = 2 \text{ м/с}^2$</p><p>Для сына: $a_{цс2} = \omega^2 R_2 = (1 \text{ рад/с})^2 \cdot 3 \text{ м} = 3 \text{ м/с}^2$</p><p>Для дочери: $a_{цс3} = \omega^2 R_3 = (1 \text{ рад/с})^2 \cdot 4 \text{ м} = 4 \text{ м/с}^2$</p><p><strong>Ответ:</strong> $a_{цс1} = 2$ м/с², $a_{цс2} = 3$ м/с², $a_{цс3} = 4$ м/с².</p><p><strong>4. Вычисление модулей сил трения ($F_{тр}$)</strong></p><p>Сила трения покоя сообщает телам центростремительное ускорение, поэтому, согласно второму закону Ньютона, $F_{тр} = m \cdot a_{цс}$.</p><p>Для отца: $F_{тр1} = m_1 \cdot a_{цс1} = 70 \text{ кг} \cdot 2 \text{ м/с}^2 = 140 \text{ Н}$</p><p>Для сына: $F_{тр2} = m_2 \cdot a_{цс2} = 40 \text{ кг} \cdot 3 \text{ м/с}^2 = 120 \text{ Н}$</p><p>Для дочери: $F_{тр3} = m_3 \cdot a_{цс3} = 30 \text{ кг} \cdot 4 \text{ м/с}^2 = 120 \text{ Н}$</p><p><strong>Ответ:</strong> $F_{тр1} = 140$ Н, $F_{тр2} = 120$ Н, $F_{тр3} = 120$ Н.</p><p><strong>5. Заполнение таблицы и изображение векторов</strong></p><p>Сведем все вычисленные значения в таблицу.</p><div class="table-wrap"><table border="1" cellpadding="5" cellspacing="0" style="border: 1px solid rgb(0, 0, 0);"> <thead> <tr> <th style="border: 1px solid rgb(0, 0, 0);"></th> <th style="border: 1px solid rgb(0, 0, 0);">$R$, м</th> <th style="border: 1px solid rgb(0, 0, 0);">$|\vec{v}|$, м/с</th> <th style="border: 1px solid rgb(0, 0, 0);">$|\vec{a}_{цс}|$, м/с²</th> <th style="border: 1px solid rgb(0, 0, 0);">$|\vec{F}_{тр}|$, Н</th> </tr> </thead> <tbody> <tr> <td style="border: 1px solid rgb(0, 0, 0);"><strong>Отец</strong></td> <td style="border: 1px solid rgb(0, 0, 0);">2</td> <td style="border: 1px solid rgb(0, 0, 0);">2</td> <td style="border: 1px solid rgb(0, 0, 0);">2</td> <td style="border: 1px solid rgb(0, 0, 0);">140</td> </tr> <tr> <td style="border: 1px solid rgb(0, 0, 0);"><strong>Сын</strong></td> <td style="border: 1px solid rgb(0, 0, 0);">3</td> <td style="border: 1px solid rgb(0, 0, 0);">3</td> <td style="border: 1px solid rgb(0, 0, 0);">3</td> <td style="border: 1px solid rgb(0, 0, 0);">120</td> </tr> <tr> <td style="border: 1px solid rgb(0, 0, 0);"><strong>Дочь</strong></td> <td style="border: 1px solid rgb(0, 0, 0);">4</td> <td style="border: 1px solid rgb(0, 0, 0);">4</td> <td style="border: 1px solid rgb(0, 0, 0);">4</td> <td style="border: 1px solid rgb(0, 0, 0);">120</td> </tr> </tbody></table></div><p>На рисунке ниже изображены векторы скоростей, ускорений и сил трения с соблюдением масштаба и направлений. Направление вращения карусели — по часовой стрелке, что следует из направления вектора скорости сына $\vec{v}_2$.</p><ul> <li>Векторы скорости ($\vec{v}$) показаны <strong style="color:blue;">синим</strong> цветом. Они направлены по касательной к траектории движения.</li> <li>Векторы центростремительного ускорения ($\vec{a}_{цс}$) показаны <strong style="color:red;">красным</strong> цветом. Они направлены к центру вращения О.</li> <li>Векторы силы трения ($\vec{F}_{тр}$) показаны <strong style="color:green;">зеленым</strong> цветом. Они также направлены к центру вращения О.</li></ul><svg width="400" height="400" viewBox="-120 -120 240 240" xmlns="http://www.w3.org/2000/svg"> <defs> <marker id="arrow-blue" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="4" markerHeight="4" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="blue"></path> </marker> <marker id="arrow-red" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="4" markerHeight="4" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="red"></path> </marker> <marker id="arrow-green" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="4" markerHeight="4" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="green"></path> </marker> </defs> <!-- Carousel outline --> <circle cx="0" cy="0" r="80" fill="none" stroke="black" stroke-width="1"></circle> <!-- Center O --> <circle cx="0" cy="0" r="2" fill="black"></circle> <text x="5" y="-5" font-family="Times New Roman" font-size="12">O</text> <!-- Point 1: Father (R=2m -> 40px at -30 deg) --> <g transform="translate(34.6, -20)"> <circle cx="0" cy="0" r="3" fill="black"></circle> <text x="5" y="5" font-family="Times New Roman" font-size="12">1</text> <!-- v1 (2 m/s -> 20px) tangent clockwise --> <line x1="0" y1="0" x2="-10" y2="-17.3" stroke="blue" stroke-width="2" marker-end="url(#arrow-blue)"></line> <text x="-25" y="-15" fill="blue" font-size="12">$\vec{v}_1$</text> <!-- a1 (2 m/s^2 -> 20px) towards center --> <line x1="0" y1="0" x2="-34.6" y2="20" stroke="red" stroke-width="2" marker-end="url(#arrow-red)"></line> <text x="-25" y="10" fill="red" font-size="12">$\vec{a}_1$</text> <!-- F1 (140 N -> 28px) towards center --> <line x1="2" y1="-1.15" x2="-25.8" y2="15.0" stroke="green" stroke-width="2" marker-end="url(#arrow-green)"></line> <text x="-2" y="18" fill="green" font-size="12">$\vec{F}_1$</text> </g> <!-- Point 2: Son (R=3m -> 60px at 90 deg) --> <g transform="translate(0, 60)"> <circle cx="0" cy="0" r="3" fill="black"></circle> <text x="5" y="5" font-family="Times New Roman" font-size="12">2</text> <!-- v2 (3 m/s -> 30px) left --> <line x1="0" y1="0" x2="-30" y2="0" stroke="blue" stroke-width="2" marker-end="url(#arrow-blue)"></line> <text x="-20" y="-8" fill="blue" font-size="12">$\vec{v}_2$</text> <!-- a2 (3 m/s^2 -> 30px) towards center --> <line x1="0" y1="0" x2="0" y2="-60" stroke="red" stroke-width="2" marker-end="url(#arrow-red)"></line> <text x="5" y="-30" fill="red" font-size="12">$\vec{a}_2$</text> <!-- F2 (120 N -> 24px) towards center --> <line x1="3" y1="0" x2="3" y2="-24" stroke="green" stroke-width="2" marker-end="url(#arrow-green)"></line> <text x="8" y="-10" fill="green" font-size="12">$\vec{F}_2$</text> </g> <!-- Point 3: Daughter (R=4m -> 80px at 180 deg) --> <g transform="translate(-80, 0)"> <circle cx="0" cy="0" r="3" fill="black"></circle> <text x="5" y="5" font-family="Times New Roman" font-size="12">3</text> <!-- v3 (4 m/s -> 40px) down --> <line x1="0" y1="0" x2="0" y2="40" stroke="blue" stroke-width="2" marker-end="url(#arrow-blue)"></line> <text x="8" y="25" fill="blue" font-size="12">$\vec{v}_3$</text> <!-- a3 (4 m/s^2 -> 40px) towards center --> <line x1="0" y1="0" x2="80" y2="0" stroke="red" stroke-width="2" marker-end="url(#arrow-red)"></line> <text x="35" y="-5" fill="red" font-size="12">$\vec{a}_3$</text> <!-- F3 (120 N -> 24px) towards center --> <line x1="0" y1="3" x2="24" y2="3" stroke="green" stroke-width="2" marker-end="url(#arrow-green)"></line> <text x="5" y="18" fill="green" font-size="12">$\vec{F}_3$</text> </g></svg><p><strong>Ответ:</strong> Все требуемые величины были вычислены, таблица заполнена, а векторы скоростей, ускорений и сил изображены на схеме выше.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1435002" "type" => "task" ] "previous" => array:2 [ "refs" => "1435000" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Book {#1168} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\BookPage {#1116 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1435714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/page-3-79" "field_display_title" => "79" "field_folder" => "3" "field_image_name" => "79" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Branch {#1130 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1434847 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_page_start" => "3" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1126 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1094 …2} ] #original: array:24 [ "id" => 1434847 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_page_start" => "3" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1126 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1094 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1095 #items: array:1 [ 0 => App\Models\Book {#1168} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1083 #items: array:1 [ 0 => App\Models\Element {#1082 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1699101 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1080 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1699101 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1080 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1435715" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1435713" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1544 #items: array:4 [ 0 => App\Models\Task {#1086 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1435425 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => null "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/exs1-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1087 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1089 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1278 …2} "content" => Illuminate\Database\Eloquent\Collection {#1295 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1312 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1435425 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => null "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/exs1-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1087 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1089 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1278 …2} "content" => Illuminate\Database\Eloquent\Collection {#1295 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1312 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1313 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1435426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => null "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/exs1-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1314 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1316 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1317 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1355 …2} "content" => Illuminate\Database\Eloquent\Collection {#1372 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1389 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1435426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => null "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/exs1-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1314 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1316 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1317 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1355 …2} "content" => Illuminate\Database\Eloquent\Collection {#1372 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1389 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1390 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1435427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => null "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/exs1-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1391 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1393 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1394 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1432 …2} "content" => Illuminate\Database\Eloquent\Collection {#1449 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1466 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1435427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => null "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/exs1-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1391 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1393 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1394 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1432 …2} "content" => Illuminate\Database\Eloquent\Collection {#1449 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1466 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1467 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1435428 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => "80" …19 ] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1435714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/page-3-79" "field_display_title" => "79" "field_folder" => "3" "field_image_name" => "79" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1095} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1092} "content" => Illuminate\Database\Eloquent\Collection {#1083} "next" => array:2 [ "refs" => "1435715" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1435713" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1544} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1435001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "79" "field_page_end" => "80" "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/16-4" "field_display_title" => "4" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1076} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1155} "content" => Illuminate\Database\Eloquent\Collection {#1112} "next" => array:2 [ "refs" => "1435002" "type" => "task" ] "previous" => array:2 [ "refs" => "1435000" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1146} "page" => array:2 [ "refs" => "1435714" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№4 (с. 79)
Условие. №4 (с. 79)
скриншот условия
4. На равномерно вращающейся вокруг вертикальной оси O карусели стоят отец (1), сын (2) и дочь (3). Масса отца $m_1 = 70$ кг, масса сына $m_2 = 40$ кг, масса дочери $m_3 = 30$ кг. На рис. 54 схематически показан вид на карусель сверху. Дочь (3) находится на расстоянии $R_3 = 4$ м от оси вращения. Скорость сына $\vec{v}_2$ направлена так, как показано на рисунке, а её модуль равен 3 м/с. Серыми стрелками показаны единичный вектор ускорения и вектор силы, модуль которого равен 100 Н.
Используя линейку, определите удаления отца ($R_1$) и сына ($R_2$) от оси вращения. Вычислите модули векторов скоростей отца ($\vec{v}_1$) и дочери ($\vec{v}_3$), модули центростремительных ускорений ($a_{цс}$) отца, сына и дочери, а также модули сил трения ($F_{тр}$), действующих на них. Заполните таблицу и изобразите (разным цветом), соблюдая масштабы, векторы скоростей, ускорений и сил трения, обозначив их.
R, м$|\vec{v}|$, м/с$|\vec{a}_{цс}|$, м/с$^2$$|\vec{F}_{тр}|$, Н
Отец: (пусто) (пусто) (пусто) (пусто)
Сын: (пусто) 3 (пусто) (пусто)
Дочь: 4 (пусто) (пусто) (пусто)
$|\vec{a}|=1$ м/с$^2$
$|\vec{F}|=100$ Н
Рис. 54
Решение. №4 (с. 79)
Дано:
Масса отца (1): $m_1 = 70$ кг
Масса сына (2): $m_2 = 40$ кг
Масса дочери (3): $m_3 = 30$ кг
Расстояние дочери от оси вращения: $R_3 = 4$ м
Модуль скорости сына: $v_2 = 3$ м/с
Все данные уже представлены в системе СИ.
Найти:
$R_1, R_2$ – расстояния отца и сына от оси вращения;
$v_1, v_3$ – модули скоростей отца и дочери;
$a_{цс1}, a_{цс2}, a_{цс3}$ – модули центростремительных ускорений отца, сына и дочери;
$F_{тр1}, F_{тр2}, F_{тр3}$ – модули сил трения, действующих на отца, сына и дочь.
Решение:
1. Определение расстояний $R_1$ и $R_2$
Для определения расстояний отца ($R_1$) и сына ($R_2$) воспользуемся методом масштабирования, приняв за основу известное расстояние дочери ($R_3 = 4$ м). Измерив на рисунке расстояния от центра O до точек 1, 2 и 3, можно установить, что расстояние до сына составляет примерно 3/4 от расстояния до дочери, а расстояние до отца - примерно 1/2 от расстояния до дочери.
$R_2 \approx \frac{3}{4} R_3 = \frac{3}{4} \cdot 4 \text{ м} = 3 \text{ м}$
$R_1 \approx \frac{1}{2} R_3 = \frac{1}{2} \cdot 4 \text{ м} = 2 \text{ м}$
Ответ: $R_1 = 2$ м, $R_2 = 3$ м.
2. Вычисление угловой скорости и линейных скоростей $v_1$ и $v_3$
Карусель вращается равномерно, следовательно, все точки на ней имеют одинаковую угловую скорость $\omega$. Найдем ее, используя данные для сына:
$v_2 = \omega R_2 \implies \omega = \frac{v_2}{R_2} = \frac{3 \text{ м/с}}{3 \text{ м}} = 1 \text{ рад/с}$
Теперь можем найти модули скоростей отца и дочери:
$v_1 = \omega R_1 = 1 \text{ рад/с} \cdot 2 \text{ м} = 2 \text{ м/с}$
$v_3 = \omega R_3 = 1 \text{ рад/с} \cdot 4 \text{ м} = 4 \text{ м/с}$
Ответ: $v_1 = 2$ м/с, $v_3 = 4$ м/с.
3. Вычисление модулей центростремительных ускорений ($a_{цс}$)
Центростремительное ускорение вычисляется по формуле $a_{цс} = \omega^2 R$ или $a_{цс} = v^2/R$.
Для отца: $a_{цс1} = \omega^2 R_1 = (1 \text{ рад/с})^2 \cdot 2 \text{ м} = 2 \text{ м/с}^2$
Для сына: $a_{цс2} = \omega^2 R_2 = (1 \text{ рад/с})^2 \cdot 3 \text{ м} = 3 \text{ м/с}^2$
Для дочери: $a_{цс3} = \omega^2 R_3 = (1 \text{ рад/с})^2 \cdot 4 \text{ м} = 4 \text{ м/с}^2$
Ответ: $a_{цс1} = 2$ м/с², $a_{цс2} = 3$ м/с², $a_{цс3} = 4$ м/с².
4. Вычисление модулей сил трения ($F_{тр}$)
Сила трения покоя сообщает телам центростремительное ускорение, поэтому, согласно второму закону Ньютона, $F_{тр} = m \cdot a_{цс}$.
Для отца: $F_{тр1} = m_1 \cdot a_{цс1} = 70 \text{ кг} \cdot 2 \text{ м/с}^2 = 140 \text{ Н}$
Для сына: $F_{тр2} = m_2 \cdot a_{цс2} = 40 \text{ кг} \cdot 3 \text{ м/с}^2 = 120 \text{ Н}$
Для дочери: $F_{тр3} = m_3 \cdot a_{цс3} = 30 \text{ кг} \cdot 4 \text{ м/с}^2 = 120 \text{ Н}$
Ответ: $F_{тр1} = 140$ Н, $F_{тр2} = 120$ Н, $F_{тр3} = 120$ Н.
5. Заполнение таблицы и изображение векторов
Сведем все вычисленные значения в таблицу.
| $R$, м | $|\vec{v}|$, м/с | $|\vec{a}_{цс}|$, м/с² | $|\vec{F}_{тр}|$, Н | |
|---|---|---|---|---|
| Отец | 2 | 2 | 2 | 140 |
| Сын | 3 | 3 | 3 | 120 |
| Дочь | 4 | 4 | 4 | 120 |
На рисунке ниже изображены векторы скоростей, ускорений и сил трения с соблюдением масштаба и направлений. Направление вращения карусели — по часовой стрелке, что следует из направления вектора скорости сына $\vec{v}_2$.
- Векторы скорости ($\vec{v}$) показаны синим цветом. Они направлены по касательной к траектории движения.
- Векторы центростремительного ускорения ($\vec{a}_{цс}$) показаны красным цветом. Они направлены к центру вращения О.
- Векторы силы трения ($\vec{F}_{тр}$) показаны зеленым цветом. Они также направлены к центру вращения О.
Ответ: Все требуемые величины были вычислены, таблица заполнена, а векторы скоростей, ускорений и сил изображены на схеме выше.
Другие задания:
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