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Номер 4, страница 55, часть 2 - 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: 2
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
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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" => 1435142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_page_end" => null "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad/32-4" "field_display_title" => "4" "field_outside_task" => null "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" => 1434829 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "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" => "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 {#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" => 1434829 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "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 {#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 {#1113 #items: array:3 [ 0 => App\Models\Branch {#1123 #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" => 1434829 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ 0 => App\Models\Term {#1124 #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" => "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 {#1160 #items: array:1 [ 0 => App\Models\Book {#1126 #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 {#1128 #items: array:1 [ 0 => App\Models\Term {#1125 #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 {#1129 #items: array:1 [ 0 => App\Models\Term {#1130 #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 {#1131 #items: array:1 [ 0 => App\Models\Term {#1132 #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 {#1133 #items: array:4 [ 0 => App\Models\Term {#1134 #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 {#1135 #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 {#1136 #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 {#1137 #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 {#1138 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\Term {#1140 #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 {#1141 #items: array:1 [ 0 => App\Models\Term {#1142 #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 {#1143 #items: array:1 [ 0 => App\Models\Term {#1144 #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 {#1145 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1147 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1149 #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 {#1150 #items: array:1 [ 0 => App\Models\Term {#1151 #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 {#1152 #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 {#1153 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1154 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1155 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Term {#1157 #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 {#1128} "field_class" => Illuminate\Database\Eloquent\Collection {#1129} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1131} "field_author" => Illuminate\Database\Eloquent\Collection {#1133} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1138} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1139} "field_country" => Illuminate\Database\Eloquent\Collection {#1141} "field_city" => Illuminate\Database\Eloquent\Collection {#1143} "field_series" => Illuminate\Database\Eloquent\Collection {#1145} "field_umk" => Illuminate\Database\Eloquent\Collection {#1146} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1147} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1148} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1150} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1152} "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 {#1153} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1154} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1155} "field_url" => "/9-klass/fizika/grachev-rabochaya-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1156} "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 {#1158 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1434829 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1127} "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 {#1160} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1158} ] #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 {#1159 #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" => 1434841 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Term {#1161 #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" => "48" "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 {#1163 #items: array:1 [ 0 => App\Models\Book {#1126} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1434841 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1164} "field_page_start" => "48" "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 {#1163} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1162} ] #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 {#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" => 1434843 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Преобразование энергии при механических колебаниях" "field_branch_order" => "32" "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" => 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" => "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 {#1168 #items: array:1 [ 0 => App\Models\Book {#1126} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ 0 => App\Models\Branch {#1176 #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" => 1434841 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_page_start" => "48" "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 {#1179 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1180 …2} ] #original: array:24 [ "id" => 1434841 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_page_start" => "48" "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 {#1179 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1180 …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" => 1434843 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Преобразование энергии при механических колебаниях" "field_branch_order" => "32" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1169} "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 {#1168} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1177} ] #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 {#1074 #items: array:2 [ 0 => App\Models\Element {#1105 #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" => 1698530 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1103 #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 {#1101 …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 {#1102 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1099 …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 {#1101 …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 {#1102 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1099 …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" => "1435142" "type" => "task" ] "text" => "<p><strong>4.</strong> Пружинный маятник состоит из лёгкой пружины жёсткостью $\text{k}$ и грузика массой $\text{m}$. В начальный момент времени грузик сместили из положения равновесия на расстояние $\text{A}$, а затем отпустили без начальной скорости. Маятник совершает свободные гармонические колебания. Заполните таблицу, ответив на вопросы.</p><p><strong>Чему равен модуль максимальной скорости грузика?</strong></p><p>$V_{max} = A \sqrt{\frac{k}{m}}$</p><p><strong>Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях жёсткость пружины увеличить в 4 раза?</strong></p><p><strong>Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях массу грузика увеличить в 4 раза?</strong></p><p><strong>Как нужно изменить начальное смещение грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях увеличился в 2 раза?</strong></p><p><strong>Как нужно изменить массу грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 3 раза?</strong></p><p><strong>Как нужно изменить жёсткость пружины, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 2 раза?</strong></p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1949" "height" => 1370 "path" => "/media/fizika_09/grachev-rt/0-32/4-1.webp?ts=1759402778" ] ] ] #original: array:7 [ "id" => 1698530 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "1435142" "type" => "task" ] "text" => "<p><strong>4.</strong> Пружинный маятник состоит из лёгкой пружины жёсткостью $\text{k}$ и грузика массой $\text{m}$. В начальный момент времени грузик сместили из положения равновесия на расстояние $\text{A}$, а затем отпустили без начальной скорости. Маятник совершает свободные гармонические колебания. Заполните таблицу, ответив на вопросы.</p><p><strong>Чему равен модуль максимальной скорости грузика?</strong></p><p>$V_{max} = A \sqrt{\frac{k}{m}}$</p><p><strong>Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях жёсткость пружины увеличить в 4 раза?</strong></p><p><strong>Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях массу грузика увеличить в 4 раза?</strong></p><p><strong>Как нужно изменить начальное смещение грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях увеличился в 2 раза?</strong></p><p><strong>Как нужно изменить массу грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 3 раза?</strong></p><p><strong>Как нужно изменить жёсткость пружины, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 2 раза?</strong></p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1949" "height" => 1370 "path" => "/media/fizika_09/grachev-rt/0-32/4-1.webp?ts=1759402778" ] ] ] #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 {#1098 #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" => 1734516 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1096 #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 {#1095 …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 {#1092 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091 …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 {#1095 …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 {#1092 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091 …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" => "1435142" "type" => "task" ] "text" => "<p>Для решения задачи воспользуемся законом сохранения полной механической энергии для пружинного маятника. В общем виде он записывается так:</p> <p>$E = E_k + E_p = \frac{mv^2}{2} + \frac{kx^2}{2} = const$</p> <p>где $m$ – масса грузика, $v$ – его скорость, $k$ – жёсткость пружины, $x$ – смещение грузика от положения равновесия.</p> <p>В начальный момент времени грузик смещен на расстояние $A$ (амплитуда колебаний) и его скорость равна нулю. Вся энергия маятника является потенциальной:</p> <p>$E = \frac{kA^2}{2}$</p> <p>Максимальная скорость $v_{max}$ достигается при прохождении положения равновесия ($x=0$), когда вся энергия маятника является кинетической:</p> <p>$E = \frac{mv_{max}^2}{2}$</p> <p>Приравнивая эти два выражения для полной энергии, получаем:</p> <p>$\frac{mv_{max}^2}{2} = \frac{kA^2}{2}$</p> <p>Отсюда можно выразить модуль максимальной скорости:</p> <p>$v_{max} = A\sqrt{\frac{k}{m}}$</p> <p>Эта формула будет использована для ответа на все вопросы.</p> <hr> <p><strong>Чему равен модуль максимальной скорости грузика?</strong></p> <p><strong>Решение:</strong></p> <p>Как показано выше, исходя из закона сохранения энергии, модуль максимальной скорости грузика определяется параметрами системы: амплитудой колебаний $A$, жёсткостью пружины $k$ и массой грузика $m$. Формула для максимальной скорости имеет вид:</p> <p>$v_{max} = A\sqrt{\frac{k}{m}}$</p> <p><strong>Ответ:</strong> Модуль максимальной скорости грузика равен $A\sqrt{\frac{k}{m}}$.</p> <hr> <p><strong>Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях жёсткость пружины увеличить в 4 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Пусть начальная максимальная скорость $v_{max,1} = A\sqrt{\frac{k_1}{m}}$. По условию, жёсткость пружины увеличивают в 4 раза, то есть $k_2 = 4k_1$. Новая максимальная скорость $v_{max,2}$ будет равна:</p> <p>$v_{max,2} = A\sqrt{\frac{k_2}{m}} = A\sqrt{\frac{4k_1}{m}} = 2 \cdot \left(A\sqrt{\frac{k_1}{m}}\right) = 2v_{max,1}$</p> <p>Таким образом, максимальная скорость увеличится в 2 раза.</p> <p><strong>Ответ:</strong> Модуль максимальной скорости увеличится в 2 раза.</p> <hr> <p><strong>Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях массу грузика увеличить в 4 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Пусть начальная максимальная скорость $v_{max,1} = A\sqrt{\frac{k}{m_1}}$. По условию, массу грузика увеличивают в 4 раза, то есть $m_2 = 4m_1$. Новая максимальная скорость $v_{max,2}$ будет равна:</p> <p>$v_{max,2} = A\sqrt{\frac{k}{m_2}} = A\sqrt{\frac{k}{4m_1}} = \frac{1}{\sqrt{4}} \cdot \left(A\sqrt{\frac{k}{m_1}}\right) = \frac{1}{2}v_{max,1}$</p> <p>Таким образом, максимальная скорость уменьшится в 2 раза.</p> <p><strong>Ответ:</strong> Модуль максимальной скорости уменьшится в 2 раза.</p> <hr> <p><strong>Как нужно изменить начальное смещение грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях увеличился в 2 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Из формулы $v_{max} = A\sqrt{\frac{k}{m}}$ видно, что максимальная скорость прямо пропорциональна начальному смещению (амплитуде) $A$, так как $\sqrt{\frac{k}{m}}$ — константа при прочих равных условиях. Чтобы максимальная скорость увеличилась в 2 раза, необходимо увеличить амплитуду в 2 раза.</p> <p>Пусть $v_{max,2} = 2v_{max,1}$.</p> <p>$A_2\sqrt{\frac{k}{m}} = 2 \cdot A_1\sqrt{\frac{k}{m}}$</p> <p>$A_2 = 2A_1$</p> <p><strong>Ответ:</strong> Начальное смещение грузика нужно увеличить в 2 раза.</p> <hr> <p><strong>Как нужно изменить массу грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 3 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Требуется, чтобы $v_{max,2} = \frac{1}{3}v_{max,1}$. Запишем это условие, используя формулу для максимальной скорости:</p> <p>$A\sqrt{\frac{k}{m_2}} = \frac{1}{3} \cdot A\sqrt{\frac{k}{m_1}}$</p> <p>Сокращая одинаковые множители, получаем:</p> <p>$\sqrt{\frac{1}{m_2}} = \frac{1}{3}\sqrt{\frac{1}{m_1}} \Rightarrow \frac{1}{\sqrt{m_2}} = \frac{1}{3\sqrt{m_1}}$</p> <p>$\sqrt{m_2} = 3\sqrt{m_1}$</p> <p>Возводя обе части в квадрат, находим соотношение масс:</p> <p>$m_2 = 9m_1$</p> <p><strong>Ответ:</strong> Массу грузика нужно увеличить в 9 раз.</p> <hr> <p><strong>Как нужно изменить жёсткость пружины, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 2 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Требуется, чтобы $v_{max,2} = \frac{1}{2}v_{max,1}$. Запишем это условие, используя формулу для максимальной скорости:</p> <p>$A\sqrt{\frac{k_2}{m}} = \frac{1}{2} \cdot A\sqrt{\frac{k_1}{m}}$</p> <p>Сокращая одинаковые множители, получаем:</p> <p>$\sqrt{k_2} = \frac{1}{2}\sqrt{k_1}$</p> <p>Возводя обе части в квадрат, находим соотношение жёсткостей:</p> <p>$k_2 = \frac{1}{4}k_1$</p> <p><strong>Ответ:</strong> Жёсткость пружины нужно уменьшить в 4 раза.</p>" ] #original: array:6 [ "id" => 1734516 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "1435142" "type" => "task" ] "text" => "<p>Для решения задачи воспользуемся законом сохранения полной механической энергии для пружинного маятника. В общем виде он записывается так:</p> <p>$E = E_k + E_p = \frac{mv^2}{2} + \frac{kx^2}{2} = const$</p> <p>где $m$ – масса грузика, $v$ – его скорость, $k$ – жёсткость пружины, $x$ – смещение грузика от положения равновесия.</p> <p>В начальный момент времени грузик смещен на расстояние $A$ (амплитуда колебаний) и его скорость равна нулю. Вся энергия маятника является потенциальной:</p> <p>$E = \frac{kA^2}{2}$</p> <p>Максимальная скорость $v_{max}$ достигается при прохождении положения равновесия ($x=0$), когда вся энергия маятника является кинетической:</p> <p>$E = \frac{mv_{max}^2}{2}$</p> <p>Приравнивая эти два выражения для полной энергии, получаем:</p> <p>$\frac{mv_{max}^2}{2} = \frac{kA^2}{2}$</p> <p>Отсюда можно выразить модуль максимальной скорости:</p> <p>$v_{max} = A\sqrt{\frac{k}{m}}$</p> <p>Эта формула будет использована для ответа на все вопросы.</p> <hr> <p><strong>Чему равен модуль максимальной скорости грузика?</strong></p> <p><strong>Решение:</strong></p> <p>Как показано выше, исходя из закона сохранения энергии, модуль максимальной скорости грузика определяется параметрами системы: амплитудой колебаний $A$, жёсткостью пружины $k$ и массой грузика $m$. Формула для максимальной скорости имеет вид:</p> <p>$v_{max} = A\sqrt{\frac{k}{m}}$</p> <p><strong>Ответ:</strong> Модуль максимальной скорости грузика равен $A\sqrt{\frac{k}{m}}$.</p> <hr> <p><strong>Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях жёсткость пружины увеличить в 4 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Пусть начальная максимальная скорость $v_{max,1} = A\sqrt{\frac{k_1}{m}}$. По условию, жёсткость пружины увеличивают в 4 раза, то есть $k_2 = 4k_1$. Новая максимальная скорость $v_{max,2}$ будет равна:</p> <p>$v_{max,2} = A\sqrt{\frac{k_2}{m}} = A\sqrt{\frac{4k_1}{m}} = 2 \cdot \left(A\sqrt{\frac{k_1}{m}}\right) = 2v_{max,1}$</p> <p>Таким образом, максимальная скорость увеличится в 2 раза.</p> <p><strong>Ответ:</strong> Модуль максимальной скорости увеличится в 2 раза.</p> <hr> <p><strong>Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях массу грузика увеличить в 4 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Пусть начальная максимальная скорость $v_{max,1} = A\sqrt{\frac{k}{m_1}}$. По условию, массу грузика увеличивают в 4 раза, то есть $m_2 = 4m_1$. Новая максимальная скорость $v_{max,2}$ будет равна:</p> <p>$v_{max,2} = A\sqrt{\frac{k}{m_2}} = A\sqrt{\frac{k}{4m_1}} = \frac{1}{\sqrt{4}} \cdot \left(A\sqrt{\frac{k}{m_1}}\right) = \frac{1}{2}v_{max,1}$</p> <p>Таким образом, максимальная скорость уменьшится в 2 раза.</p> <p><strong>Ответ:</strong> Модуль максимальной скорости уменьшится в 2 раза.</p> <hr> <p><strong>Как нужно изменить начальное смещение грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях увеличился в 2 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Из формулы $v_{max} = A\sqrt{\frac{k}{m}}$ видно, что максимальная скорость прямо пропорциональна начальному смещению (амплитуде) $A$, так как $\sqrt{\frac{k}{m}}$ — константа при прочих равных условиях. Чтобы максимальная скорость увеличилась в 2 раза, необходимо увеличить амплитуду в 2 раза.</p> <p>Пусть $v_{max,2} = 2v_{max,1}$.</p> <p>$A_2\sqrt{\frac{k}{m}} = 2 \cdot A_1\sqrt{\frac{k}{m}}$</p> <p>$A_2 = 2A_1$</p> <p><strong>Ответ:</strong> Начальное смещение грузика нужно увеличить в 2 раза.</p> <hr> <p><strong>Как нужно изменить массу грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 3 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Требуется, чтобы $v_{max,2} = \frac{1}{3}v_{max,1}$. Запишем это условие, используя формулу для максимальной скорости:</p> <p>$A\sqrt{\frac{k}{m_2}} = \frac{1}{3} \cdot A\sqrt{\frac{k}{m_1}}$</p> <p>Сокращая одинаковые множители, получаем:</p> <p>$\sqrt{\frac{1}{m_2}} = \frac{1}{3}\sqrt{\frac{1}{m_1}} \Rightarrow \frac{1}{\sqrt{m_2}} = \frac{1}{3\sqrt{m_1}}$</p> <p>$\sqrt{m_2} = 3\sqrt{m_1}$</p> <p>Возводя обе части в квадрат, находим соотношение масс:</p> <p>$m_2 = 9m_1$</p> <p><strong>Ответ:</strong> Массу грузика нужно увеличить в 9 раз.</p> <hr> <p><strong>Как нужно изменить жёсткость пружины, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 2 раза?</strong></p> <p><strong>Решение:</strong></p> <p>Требуется, чтобы $v_{max,2} = \frac{1}{2}v_{max,1}$. Запишем это условие, используя формулу для максимальной скорости:</p> <p>$A\sqrt{\frac{k_2}{m}} = \frac{1}{2} \cdot A\sqrt{\frac{k_1}{m}}$</p> <p>Сокращая одинаковые множители, получаем:</p> <p>$\sqrt{k_2} = \frac{1}{2}\sqrt{k_1}$</p> <p>Возводя обе части в квадрат, находим соотношение жёсткостей:</p> <p>$k_2 = \frac{1}{4}k_1$</p> <p><strong>Ответ:</strong> Жёсткость пружины нужно уменьшить в 4 раза.</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" => "1435143" "type" => "task" ] "previous" => array:2 [ "refs" => "1435141" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Book {#1126} ] #escapeWhenCastingToString: false } "page" => 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array:1 [ …1] ] #original: array:6 [ "id" => 1699077 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1205 …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" => "1435691" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1435689" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:5 [ 0 => App\Models\Task {#1212 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false 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№4 (с. 55)
Условие. №4 (с. 55)
скриншот условия
4. Пружинный маятник состоит из лёгкой пружины жёсткостью $\text{k}$ и грузика массой $\text{m}$. В начальный момент времени грузик сместили из положения равновесия на расстояние $\text{A}$, а затем отпустили без начальной скорости. Маятник совершает свободные гармонические колебания. Заполните таблицу, ответив на вопросы.
Чему равен модуль максимальной скорости грузика?
$V_{max} = A \sqrt{\frac{k}{m}}$
Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях жёсткость пружины увеличить в 4 раза?
Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях массу грузика увеличить в 4 раза?
Как нужно изменить начальное смещение грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях увеличился в 2 раза?
Как нужно изменить массу грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 3 раза?
Как нужно изменить жёсткость пружины, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 2 раза?
Решение. №4 (с. 55)
Для решения задачи воспользуемся законом сохранения полной механической энергии для пружинного маятника. В общем виде он записывается так:
$E = E_k + E_p = \frac{mv^2}{2} + \frac{kx^2}{2} = const$
где $m$ – масса грузика, $v$ – его скорость, $k$ – жёсткость пружины, $x$ – смещение грузика от положения равновесия.
В начальный момент времени грузик смещен на расстояние $A$ (амплитуда колебаний) и его скорость равна нулю. Вся энергия маятника является потенциальной:
$E = \frac{kA^2}{2}$
Максимальная скорость $v_{max}$ достигается при прохождении положения равновесия ($x=0$), когда вся энергия маятника является кинетической:
$E = \frac{mv_{max}^2}{2}$
Приравнивая эти два выражения для полной энергии, получаем:
$\frac{mv_{max}^2}{2} = \frac{kA^2}{2}$
Отсюда можно выразить модуль максимальной скорости:
$v_{max} = A\sqrt{\frac{k}{m}}$
Эта формула будет использована для ответа на все вопросы.
Чему равен модуль максимальной скорости грузика?
Решение:
Как показано выше, исходя из закона сохранения энергии, модуль максимальной скорости грузика определяется параметрами системы: амплитудой колебаний $A$, жёсткостью пружины $k$ и массой грузика $m$. Формула для максимальной скорости имеет вид:
$v_{max} = A\sqrt{\frac{k}{m}}$
Ответ: Модуль максимальной скорости грузика равен $A\sqrt{\frac{k}{m}}$.
Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях жёсткость пружины увеличить в 4 раза?
Решение:
Пусть начальная максимальная скорость $v_{max,1} = A\sqrt{\frac{k_1}{m}}$. По условию, жёсткость пружины увеличивают в 4 раза, то есть $k_2 = 4k_1$. Новая максимальная скорость $v_{max,2}$ будет равна:
$v_{max,2} = A\sqrt{\frac{k_2}{m}} = A\sqrt{\frac{4k_1}{m}} = 2 \cdot \left(A\sqrt{\frac{k_1}{m}}\right) = 2v_{max,1}$
Таким образом, максимальная скорость увеличится в 2 раза.
Ответ: Модуль максимальной скорости увеличится в 2 раза.
Во сколько раз изменится модуль максимальной скорости грузика, если при прочих равных условиях массу грузика увеличить в 4 раза?
Решение:
Пусть начальная максимальная скорость $v_{max,1} = A\sqrt{\frac{k}{m_1}}$. По условию, массу грузика увеличивают в 4 раза, то есть $m_2 = 4m_1$. Новая максимальная скорость $v_{max,2}$ будет равна:
$v_{max,2} = A\sqrt{\frac{k}{m_2}} = A\sqrt{\frac{k}{4m_1}} = \frac{1}{\sqrt{4}} \cdot \left(A\sqrt{\frac{k}{m_1}}\right) = \frac{1}{2}v_{max,1}$
Таким образом, максимальная скорость уменьшится в 2 раза.
Ответ: Модуль максимальной скорости уменьшится в 2 раза.
Как нужно изменить начальное смещение грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях увеличился в 2 раза?
Решение:
Из формулы $v_{max} = A\sqrt{\frac{k}{m}}$ видно, что максимальная скорость прямо пропорциональна начальному смещению (амплитуде) $A$, так как $\sqrt{\frac{k}{m}}$ — константа при прочих равных условиях. Чтобы максимальная скорость увеличилась в 2 раза, необходимо увеличить амплитуду в 2 раза.
Пусть $v_{max,2} = 2v_{max,1}$.
$A_2\sqrt{\frac{k}{m}} = 2 \cdot A_1\sqrt{\frac{k}{m}}$
$A_2 = 2A_1$
Ответ: Начальное смещение грузика нужно увеличить в 2 раза.
Как нужно изменить массу грузика, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 3 раза?
Решение:
Требуется, чтобы $v_{max,2} = \frac{1}{3}v_{max,1}$. Запишем это условие, используя формулу для максимальной скорости:
$A\sqrt{\frac{k}{m_2}} = \frac{1}{3} \cdot A\sqrt{\frac{k}{m_1}}$
Сокращая одинаковые множители, получаем:
$\sqrt{\frac{1}{m_2}} = \frac{1}{3}\sqrt{\frac{1}{m_1}} \Rightarrow \frac{1}{\sqrt{m_2}} = \frac{1}{3\sqrt{m_1}}$
$\sqrt{m_2} = 3\sqrt{m_1}$
Возводя обе части в квадрат, находим соотношение масс:
$m_2 = 9m_1$
Ответ: Массу грузика нужно увеличить в 9 раз.
Как нужно изменить жёсткость пружины, чтобы модуль максимальной скорости грузика при прочих равных условиях уменьшился в 2 раза?
Решение:
Требуется, чтобы $v_{max,2} = \frac{1}{2}v_{max,1}$. Запишем это условие, используя формулу для максимальной скорости:
$A\sqrt{\frac{k_2}{m}} = \frac{1}{2} \cdot A\sqrt{\frac{k_1}{m}}$
Сокращая одинаковые множители, получаем:
$\sqrt{k_2} = \frac{1}{2}\sqrt{k_1}$
Возводя обе части в квадрат, находим соотношение жёсткостей:
$k_2 = \frac{1}{4}k_1$
Ответ: Жёсткость пружины нужно уменьшить в 4 раза.
Другие задания:
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