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Номер 8.20, страница 45 - гдз по физике 8-11 класс учебник, задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Учебник, задачник
Издательство: Илекса
Год издания: 2005 - 2026
Цвет обложки: синий мужчина в красном, летит на черном шаре
ISBN: 978-5-89237-332-6
Популярные ГДЗ в 8 классе
Задачи. Механика. 8. Комбинированные задачи по механике - номер 8.20, страница 45.
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" => 1288206 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "45" "field_page_end" => "46" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/08-20" "field_display_title" => "8.20" "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" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "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 {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #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" => 5401 "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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …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" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "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" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "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" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "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" => 5401 "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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …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" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "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" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "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" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "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" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "5" "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 {#1038} "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 {#1187 #items: array:3 [ 0 => App\Models\Branch {#1197 #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" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1198 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "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 {#1234 #items: array:1 [ 0 => App\Models\Book {#1199 #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" => 5401 "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 {#1201 #items: array:1 [ 0 => App\Models\Term {#1200 #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 {#1202 #items: array:4 [ 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:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #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 {#1204 #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 => "*" ] } 2 => App\Models\Term {#1205 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #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 {#1206 #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" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #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 {#1207 #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:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #original: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #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 {#1209 #items: array:3 [ 0 => App\Models\Term {#1210 #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" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "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 {#1211 #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" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #original: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #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 {#1212 #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" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "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 {#1213 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ 0 => App\Models\Term {#1215 #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" => 6673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник, задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik, zadachnik" ] #original: array:10 [ "id" => 6673 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник, задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik, zadachnik" ] #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 {#1216 #items: array:1 [ 0 => App\Models\Term {#1217 #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 {#1218 #items: array:1 [ 0 => App\Models\Term {#1219 #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 {#1220 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1221 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1222 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1223 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Term {#1225 #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" => 35 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "исправленное" "field_cases" => array:6 [ …6] "field_translit" => "ispravlennoe" ] #original: array:6 [ "id" => 35 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "исправленное" "field_cases" => array:6 [ …6] "field_translit" => "ispravlennoe" ] #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 {#1226 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "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" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ 0 => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" "alt" => "" "width" => "1813" "height" => "2441" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1227 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1228 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1229 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:1 [ 0 => App\Models\Term {#1231 #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" => 379 "subject" => 556 "class_subject" => 492 ] ] #original: array:50 [ "id" => 5401 "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 {#1201} "field_class" => Illuminate\Database\Eloquent\Collection {#1202} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1207} "field_author" => Illuminate\Database\Eloquent\Collection {#1209} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1213} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1214} "field_country" => Illuminate\Database\Eloquent\Collection {#1216} "field_city" => Illuminate\Database\Eloquent\Collection {#1218} "field_series" => Illuminate\Database\Eloquent\Collection {#1220} "field_umk" => Illuminate\Database\Eloquent\Collection {#1221} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1222} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1223} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1224} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1226} "field_under_the_edition_degree" => null "field_cover_description" => "мужчина в красном, летит на черном шаре" "field_publication_year" => "2005" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "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" => "1001" "field_priority" => null "field_default_folder" => "/fizika_08/gelfgat-u/" "field_isbn" => "978-5-89237-332-6" "field_cover" => array:1 [ 0 => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_08/gelfgat-u/covers/cover1.webp?ts=1755898771" "alt" => "" "width" => "1813" "height" => "2441" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1227} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1228} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1229} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1230} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 379 "subject" => 556 "class_subject" => 492 ] ] #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 {#1232 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1198} "field_page_start" => "5" "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 {#1234} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1232} ] #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 {#1233 #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" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1235 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "5" "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 {#1236 #items: array:1 [ 0 => App\Models\Book {#1199} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1235} "field_page_start" => "5" "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 {#1236} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1237} ] #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 {#1238 #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" => 1287875 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "8. Комбинированные задачи по механике" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1239 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "43" "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 {#1240 #items: array:1 [ 0 => App\Models\Book {#1199} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1317 #items: array:1 [ 0 => App\Models\Branch {#1241 #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" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_page_start" => "5" "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 {#1242 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1277 …2} ] #original: array:24 [ "id" => 1287867 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_page_start" => "5" "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 {#1242 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1277 …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" => 1287875 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "8. Комбинированные задачи по механике" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1239} "field_page_start" => "43" "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 {#1240} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1317} ] #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 {#1148 #items: array:3 [ 0 => App\Models\Element {#1180 #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" => 1478018 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Edition {#1179 #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" => 5410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1178 …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 {#1176 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1178 …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 {#1176 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1174 …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" => "1288206" "type" => "task" ] "text" => "<p><strong>8.20**.</strong> Веревка массой $\text{m}$ и длиной $\text{L}$ переброшена через маленький блок и уравновешена. От легкого толчка блок начал вращаться. Какова скорость $\text{v}$ веревки в тот момент, когдаmс одной стороны блока свешивается большая часть веревки длиной $\text{x}$? С какой силой $\text{F}$ веревка давит на блок в этот момент?</p>" "img" => array:2 [ 0 => array:5 [ "name" => "20-1.jpg" "alt" => null "width" => "1431" "height" => 210 "path" => "/media/fizika_08/gelfgat-u/0-08/20-1.webp?ts=1756049797" ] 1 => array:5 [ "name" => "20-2.jpg" "alt" => null "width" => "1425" "height" => 229 "path" => "/media/fizika_08/gelfgat-u/0-08/20-2.webp?ts=1756049797" ] ] ] #original: array:7 [ "id" => 1478018 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1172} "task" => array:2 [ "refs" => "1288206" "type" => "task" ] "text" => "<p><strong>8.20**.</strong> Веревка массой $\text{m}$ и длиной $\text{L}$ переброшена через маленький блок и уравновешена. От легкого толчка блок начал вращаться. Какова скорость $\text{v}$ веревки в тот момент, когдаmс одной стороны блока свешивается большая часть веревки длиной $\text{x}$? С какой силой $\text{F}$ веревка давит на блок в этот момент?</p>" "img" => array:2 [ 0 => array:5 [ "name" => "20-1.jpg" "alt" => null "width" => "1431" "height" => 210 "path" => "/media/fizika_08/gelfgat-u/0-08/20-1.webp?ts=1756049797" ] 1 => array:5 [ "name" => "20-2.jpg" "alt" => null "width" => "1425" "height" => 229 "path" => "/media/fizika_08/gelfgat-u/0-08/20-2.webp?ts=1756049797" ] ] ] #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 {#1173 #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" => 1478545 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Edition {#1171 #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" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 …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" => "1288206" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "20-1.jpg" "alt" => null "width" => "1432" "height" => 1183 "path" => "/media/fizika_08/gelfgat-u/1-08/20-1.webp?ts=1756049862" ] ] ] #original: array:6 [ "id" => 1478545 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1164} "task" => array:2 [ "refs" => "1288206" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "20-1.jpg" "alt" => null "width" => "1432" "height" => 1183 "path" => "/media/fizika_08/gelfgat-u/1-08/20-1.webp?ts=1756049862" ] ] ] #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\Element {#1162 #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" => 1737899 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Edition {#1165 #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" => 6336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1160 …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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_content_source" => null ] #original: array:21 [ "id" => 6336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1160 …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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159 …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" => "1288206" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса веревки: $\text{m}$<br>Длина веревки: $\text{L}$<br>Длина большей части веревки: $\text{x}$<br>Ускорение свободного падения: $\text{g}$</p><p>Все данные представлены в общем виде и не требуют перевода в систему СИ.</p><p><strong>Найти:</strong></p><p><strong>1.</strong> Скорость веревки $\text{v}$.<br>2. Силу давления веревки на блок $\text{F}$.</p><p><strong>Решение:</strong></p><p><strong>1.</strong> Какова скорость v веревки в тот момент, когда с одной стороны блока свешивается большая часть веревки длиной x?</p><p>Для нахождения скорости веревки воспользуемся законом сохранения механической энергии. Система состоит из веревки и Земли. Так как блок маленький и трение отсутствует, полная механическая энергия системы сохраняется.</p><p>Выберем за нулевой уровень потенциальной энергии горизонтальную линию, проходящую через блок.</p><p><em>Начальное состояние:</em> Веревка находится в равновесии, значит, с каждой стороны блока свисает часть веревки длиной $L/2$. Скорость веревки равна нулю ($v_0 = 0$).<br>Масса каждой половины веревки равна $m/2$. Центр масс каждой половины находится на расстоянии $(L/2)/2 = L/4$ ниже блока. Начальная потенциальная энергия системы: $E_{p0} = -\frac{m}{2}g\frac{L}{4} - \frac{m}{2}g\frac{L}{4} = -\frac{mgL}{4}$ Начальная кинетическая энергия: $E_{k0} = 0$. Полная начальная энергия: $E_0 = E_{k0} + E_{p0} = -\frac{mgL}{4}$.</p><p><em>Конечное состояние:</em> С одной стороны свисает часть веревки длиной $\text{x}$, а с другой — длиной $L-x$. Вся веревка движется со скоростью $\text{v}$.<br>Линейная плотность веревки $\rho = m/L$. Масса длинной части: $m_1 = \rho x = \frac{m}{L}x$. Ее центр масс находится на расстоянии $x/2$ ниже блока. Масса короткой части: $m_2 = \rho(L-x) = \frac{m}{L}(L-x)$. Ее центр масс находится на расстоянии $(L-x)/2$ ниже блока. Конечная потенциальная энергия системы: $E_p = -m_1 g \frac{x}{2} - m_2 g \frac{L-x}{2} = -\frac{m}{L}x \cdot g \frac{x}{2} - \frac{m}{L}(L-x) \cdot g \frac{L-x}{2} = -\frac{mg}{2L}(x^2 + (L-x)^2)$. Конечная кинетическая энергия всей веревки массой $\text{m}$: $E_k = \frac{1}{2}mv^2$. Полная конечная энергия: $E = E_k + E_p = \frac{1}{2}mv^2 - \frac{mg}{2L}(x^2 + (L-x)^2)$.</p><p>По закону сохранения энергии $E_0 = E$: $-\frac{mgL}{4} = \frac{1}{2}mv^2 - \frac{mg}{2L}(x^2 + (L-x)^2)$</p><p>Выразим $v^2$: $\frac{1}{2}mv^2 = \frac{mg}{2L}(x^2 + (L-x)^2) - \frac{mgL}{4}$ $v^2 = \frac{g}{L}(x^2 + L^2 - 2Lx + x^2) - \frac{gL}{2}$ $v^2 = \frac{g}{L}(2x^2 - 2Lx + L^2) - \frac{gL}{2}$ $v^2 = \frac{2gx^2}{L} - 2gx + gL - \frac{gL}{2}$ $v^2 = \frac{2gx^2}{L} - 2gx + \frac{gL}{2}$ $v^2 = \frac{g}{L}(2x^2 - 2Lx + \frac{L^2}{2}) = \frac{g}{2L}(4x^2 - 4Lx + L^2)$ $v^2 = \frac{g}{2L}(2x - L)^2$</p><p>Извлекаем квадратный корень. Так как $\text{x}$ — большая часть, то $x > L/2$, следовательно, $2x-L > 0$. $v = (2x-L)\sqrt{\frac{g}{2L}}$</p><p><strong>Ответ:</strong> Скорость веревки равна $v = (2x-L)\sqrt{\frac{g}{2L}}$.</p><p><strong>2.</strong> С какой силой F веревка давит на блок в этот момент?</p><p>Сила давления $\text{F}$, с которой веревка давит на блок, по третьему закону Ньютона равна по модулю и противоположна по направлению силе $F_{опоры}$, с которой блок действует на веревку. Найдем силу $F_{опоры}$, используя второй закон Ньютона для центра масс всей веревки.</p><p>$m\vec{a}_{cm} = \sum \vec{F}_{ext}$</p><p>На систему (веревку) действуют две внешние силы: сила тяжести $m\vec{g}$, направленная вниз, и сила реакции опоры со стороны блока $\vec{F}_{опоры}$, направленная вверх. В проекции на вертикальную ось, направленную вниз: $m a_{cm} = mg - F_{опоры}$ Отсюда, искомая сила $F = F_{опоры} = mg - m a_{cm}$.</p><p>Найдем ускорение центра масс веревки $a_{cm}$. Сначала определим положение центра масс $y_{cm}$ (отсчитывая от блока вниз): $y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m} = \frac{(\frac{m}{L}x)\frac{x}{2} + (\frac{m}{L}(L-x))\frac{L-x}{2}}{m} = \frac{x^2 + (L-x)^2}{2L}$</p><p>Скорость центра масс $v_{cm}$ равна производной от координаты по времени: $v_{cm} = \frac{dy_{cm}}{dt} = \frac{d}{dt}\left(\frac{x^2 + (L-x)^2}{2L}\right) = \frac{1}{2L}(2x\frac{dx}{dt} + 2(L-x)(-\frac{dx}{dt})) = \frac{v}{L}(2x-L)$, где $\frac{dx}{dt} = v$ — скорость движения веревки.</p><p>Ускорение центра масс $a_{cm}$ равно производной от скорости по времени: $a_{cm} = \frac{dv_{cm}}{dt} = \frac{d}{dt}\left(\frac{v(2x-L)}{L}\right) = \frac{1}{L}\left(\frac{dv}{dt}(2x-L) + v(2\frac{dx}{dt})\right) = \frac{1}{L}(a(2x-L) + 2v^2)$ где $a = \frac{dv}{dt}$ — ускорение веревки. Ускорение найдем из второго закона Ньютона для веревки как целого. Движущая сила — разность весов свисающих частей: $ma = m_1 g - m_2 g = \frac{m}{L}xg - \frac{m}{L}(L-x)g = \frac{mg}{L}(2x-L)$ $a = \frac{g}{L}(2x-L)$</p><p>Подставим выражения для $\text{a}$ и $v^2$ в формулу для $a_{cm}$: $a_{cm} = \frac{1}{L}\left(\frac{g}{L}(2x-L)(2x-L) + 2\frac{g}{2L}(2x-L)^2\right)$ $a_{cm} = \frac{1}{L}\left(\frac{g}{L}(2x-L)^2 + \frac{g}{L}(2x-L)^2\right) = \frac{2g}{L^2}(2x-L)^2$</p><p>Теперь найдем силу давления $\text{F}$: $F = mg - m a_{cm} = mg - m\frac{2g}{L^2}(2x-L)^2 = mg\left(1 - \frac{2(2x-L)^2}{L^2}\right)$</p><p><strong>Ответ:</strong> Сила, с которой веревка давит на блок, равна $F = mg\left(1 - \frac{2(2x-L)^2}{L^2}\right)$.</p>" ] #original: array:6 [ "id" => 1737899 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156} "task" => array:2 [ "refs" => "1288206" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Масса веревки: $\text{m}$<br>Длина веревки: $\text{L}$<br>Длина большей части веревки: $\text{x}$<br>Ускорение свободного падения: $\text{g}$</p><p>Все данные представлены в общем виде и не требуют перевода в систему СИ.</p><p><strong>Найти:</strong></p><p><strong>1.</strong> Скорость веревки $\text{v}$.<br>2. Силу давления веревки на блок $\text{F}$.</p><p><strong>Решение:</strong></p><p><strong>1.</strong> Какова скорость v веревки в тот момент, когда с одной стороны блока свешивается большая часть веревки длиной x?</p><p>Для нахождения скорости веревки воспользуемся законом сохранения механической энергии. Система состоит из веревки и Земли. Так как блок маленький и трение отсутствует, полная механическая энергия системы сохраняется.</p><p>Выберем за нулевой уровень потенциальной энергии горизонтальную линию, проходящую через блок.</p><p><em>Начальное состояние:</em> Веревка находится в равновесии, значит, с каждой стороны блока свисает часть веревки длиной $L/2$. Скорость веревки равна нулю ($v_0 = 0$).<br>Масса каждой половины веревки равна $m/2$. Центр масс каждой половины находится на расстоянии $(L/2)/2 = L/4$ ниже блока. Начальная потенциальная энергия системы: $E_{p0} = -\frac{m}{2}g\frac{L}{4} - \frac{m}{2}g\frac{L}{4} = -\frac{mgL}{4}$ Начальная кинетическая энергия: $E_{k0} = 0$. Полная начальная энергия: $E_0 = E_{k0} + E_{p0} = -\frac{mgL}{4}$.</p><p><em>Конечное состояние:</em> С одной стороны свисает часть веревки длиной $\text{x}$, а с другой — длиной $L-x$. Вся веревка движется со скоростью $\text{v}$.<br>Линейная плотность веревки $\rho = m/L$. Масса длинной части: $m_1 = \rho x = \frac{m}{L}x$. Ее центр масс находится на расстоянии $x/2$ ниже блока. Масса короткой части: $m_2 = \rho(L-x) = \frac{m}{L}(L-x)$. Ее центр масс находится на расстоянии $(L-x)/2$ ниже блока. Конечная потенциальная энергия системы: $E_p = -m_1 g \frac{x}{2} - m_2 g \frac{L-x}{2} = -\frac{m}{L}x \cdot g \frac{x}{2} - \frac{m}{L}(L-x) \cdot g \frac{L-x}{2} = -\frac{mg}{2L}(x^2 + (L-x)^2)$. Конечная кинетическая энергия всей веревки массой $\text{m}$: $E_k = \frac{1}{2}mv^2$. Полная конечная энергия: $E = E_k + E_p = \frac{1}{2}mv^2 - \frac{mg}{2L}(x^2 + (L-x)^2)$.</p><p>По закону сохранения энергии $E_0 = E$: $-\frac{mgL}{4} = \frac{1}{2}mv^2 - \frac{mg}{2L}(x^2 + (L-x)^2)$</p><p>Выразим $v^2$: $\frac{1}{2}mv^2 = \frac{mg}{2L}(x^2 + (L-x)^2) - \frac{mgL}{4}$ $v^2 = \frac{g}{L}(x^2 + L^2 - 2Lx + x^2) - \frac{gL}{2}$ $v^2 = \frac{g}{L}(2x^2 - 2Lx + L^2) - \frac{gL}{2}$ $v^2 = \frac{2gx^2}{L} - 2gx + gL - \frac{gL}{2}$ $v^2 = \frac{2gx^2}{L} - 2gx + \frac{gL}{2}$ $v^2 = \frac{g}{L}(2x^2 - 2Lx + \frac{L^2}{2}) = \frac{g}{2L}(4x^2 - 4Lx + L^2)$ $v^2 = \frac{g}{2L}(2x - L)^2$</p><p>Извлекаем квадратный корень. Так как $\text{x}$ — большая часть, то $x > L/2$, следовательно, $2x-L > 0$. $v = (2x-L)\sqrt{\frac{g}{2L}}$</p><p><strong>Ответ:</strong> Скорость веревки равна $v = (2x-L)\sqrt{\frac{g}{2L}}$.</p><p><strong>2.</strong> С какой силой F веревка давит на блок в этот момент?</p><p>Сила давления $\text{F}$, с которой веревка давит на блок, по третьему закону Ньютона равна по модулю и противоположна по направлению силе $F_{опоры}$, с которой блок действует на веревку. Найдем силу $F_{опоры}$, используя второй закон Ньютона для центра масс всей веревки.</p><p>$m\vec{a}_{cm} = \sum \vec{F}_{ext}$</p><p>На систему (веревку) действуют две внешние силы: сила тяжести $m\vec{g}$, направленная вниз, и сила реакции опоры со стороны блока $\vec{F}_{опоры}$, направленная вверх. В проекции на вертикальную ось, направленную вниз: $m a_{cm} = mg - F_{опоры}$ Отсюда, искомая сила $F = F_{опоры} = mg - m a_{cm}$.</p><p>Найдем ускорение центра масс веревки $a_{cm}$. Сначала определим положение центра масс $y_{cm}$ (отсчитывая от блока вниз): $y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m} = \frac{(\frac{m}{L}x)\frac{x}{2} + (\frac{m}{L}(L-x))\frac{L-x}{2}}{m} = \frac{x^2 + (L-x)^2}{2L}$</p><p>Скорость центра масс $v_{cm}$ равна производной от координаты по времени: $v_{cm} = \frac{dy_{cm}}{dt} = \frac{d}{dt}\left(\frac{x^2 + (L-x)^2}{2L}\right) = \frac{1}{2L}(2x\frac{dx}{dt} + 2(L-x)(-\frac{dx}{dt})) = \frac{v}{L}(2x-L)$, где $\frac{dx}{dt} = v$ — скорость движения веревки.</p><p>Ускорение центра масс $a_{cm}$ равно производной от скорости по времени: $a_{cm} = \frac{dv_{cm}}{dt} = \frac{d}{dt}\left(\frac{v(2x-L)}{L}\right) = \frac{1}{L}\left(\frac{dv}{dt}(2x-L) + v(2\frac{dx}{dt})\right) = \frac{1}{L}(a(2x-L) + 2v^2)$ где $a = \frac{dv}{dt}$ — ускорение веревки. Ускорение найдем из второго закона Ньютона для веревки как целого. Движущая сила — разность весов свисающих частей: $ma = m_1 g - m_2 g = \frac{m}{L}xg - \frac{m}{L}(L-x)g = \frac{mg}{L}(2x-L)$ $a = \frac{g}{L}(2x-L)$</p><p>Подставим выражения для $\text{a}$ и $v^2$ в формулу для $a_{cm}$: $a_{cm} = \frac{1}{L}\left(\frac{g}{L}(2x-L)(2x-L) + 2\frac{g}{2L}(2x-L)^2\right)$ $a_{cm} = \frac{1}{L}\left(\frac{g}{L}(2x-L)^2 + \frac{g}{L}(2x-L)^2\right) = \frac{2g}{L^2}(2x-L)^2$</p><p>Теперь найдем силу давления $\text{F}$: $F = mg - m a_{cm} = mg - m\frac{2g}{L^2}(2x-L)^2 = mg\left(1 - \frac{2(2x-L)^2}{L^2}\right)$</p><p><strong>Ответ:</strong> Сила, с которой веревка давит на блок, равна $F = mg\left(1 - \frac{2(2x-L)^2}{L^2}\right)$.</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 } 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#observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1464 #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" => 1288204 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "45" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/08-18" "field_display_title" => "8.18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1461 …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" => 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#attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1492 #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 [ …24] #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" => 1340396 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "45" "field_url" => 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№8.20 (с. 45)
Условие. №8.20 (с. 45)
скриншот условия
8.20**. Веревка массой $\text{m}$ и длиной $\text{L}$ переброшена через маленький блок и уравновешена. От легкого толчка блок начал вращаться. Какова скорость $\text{v}$ веревки в тот момент, когдаmс одной стороны блока свешивается большая часть веревки длиной $\text{x}$? С какой силой $\text{F}$ веревка давит на блок в этот момент?
Решение. №8.20 (с. 45)
Решение 2. №8.20 (с. 45)
Дано:
Масса веревки: $\text{m}$
Длина веревки: $\text{L}$
Длина большей части веревки: $\text{x}$
Ускорение свободного падения: $\text{g}$
Все данные представлены в общем виде и не требуют перевода в систему СИ.
Найти:
1. Скорость веревки $\text{v}$.
2. Силу давления веревки на блок $\text{F}$.
Решение:
1. Какова скорость v веревки в тот момент, когда с одной стороны блока свешивается большая часть веревки длиной x?
Для нахождения скорости веревки воспользуемся законом сохранения механической энергии. Система состоит из веревки и Земли. Так как блок маленький и трение отсутствует, полная механическая энергия системы сохраняется.
Выберем за нулевой уровень потенциальной энергии горизонтальную линию, проходящую через блок.
Начальное состояние: Веревка находится в равновесии, значит, с каждой стороны блока свисает часть веревки длиной $L/2$. Скорость веревки равна нулю ($v_0 = 0$).
Масса каждой половины веревки равна $m/2$. Центр масс каждой половины находится на расстоянии $(L/2)/2 = L/4$ ниже блока. Начальная потенциальная энергия системы: $E_{p0} = -\frac{m}{2}g\frac{L}{4} - \frac{m}{2}g\frac{L}{4} = -\frac{mgL}{4}$ Начальная кинетическая энергия: $E_{k0} = 0$. Полная начальная энергия: $E_0 = E_{k0} + E_{p0} = -\frac{mgL}{4}$.
Конечное состояние: С одной стороны свисает часть веревки длиной $\text{x}$, а с другой — длиной $L-x$. Вся веревка движется со скоростью $\text{v}$.
Линейная плотность веревки $\rho = m/L$. Масса длинной части: $m_1 = \rho x = \frac{m}{L}x$. Ее центр масс находится на расстоянии $x/2$ ниже блока. Масса короткой части: $m_2 = \rho(L-x) = \frac{m}{L}(L-x)$. Ее центр масс находится на расстоянии $(L-x)/2$ ниже блока. Конечная потенциальная энергия системы: $E_p = -m_1 g \frac{x}{2} - m_2 g \frac{L-x}{2} = -\frac{m}{L}x \cdot g \frac{x}{2} - \frac{m}{L}(L-x) \cdot g \frac{L-x}{2} = -\frac{mg}{2L}(x^2 + (L-x)^2)$. Конечная кинетическая энергия всей веревки массой $\text{m}$: $E_k = \frac{1}{2}mv^2$. Полная конечная энергия: $E = E_k + E_p = \frac{1}{2}mv^2 - \frac{mg}{2L}(x^2 + (L-x)^2)$.
По закону сохранения энергии $E_0 = E$: $-\frac{mgL}{4} = \frac{1}{2}mv^2 - \frac{mg}{2L}(x^2 + (L-x)^2)$
Выразим $v^2$: $\frac{1}{2}mv^2 = \frac{mg}{2L}(x^2 + (L-x)^2) - \frac{mgL}{4}$ $v^2 = \frac{g}{L}(x^2 + L^2 - 2Lx + x^2) - \frac{gL}{2}$ $v^2 = \frac{g}{L}(2x^2 - 2Lx + L^2) - \frac{gL}{2}$ $v^2 = \frac{2gx^2}{L} - 2gx + gL - \frac{gL}{2}$ $v^2 = \frac{2gx^2}{L} - 2gx + \frac{gL}{2}$ $v^2 = \frac{g}{L}(2x^2 - 2Lx + \frac{L^2}{2}) = \frac{g}{2L}(4x^2 - 4Lx + L^2)$ $v^2 = \frac{g}{2L}(2x - L)^2$
Извлекаем квадратный корень. Так как $\text{x}$ — большая часть, то $x > L/2$, следовательно, $2x-L > 0$. $v = (2x-L)\sqrt{\frac{g}{2L}}$
Ответ: Скорость веревки равна $v = (2x-L)\sqrt{\frac{g}{2L}}$.
2. С какой силой F веревка давит на блок в этот момент?
Сила давления $\text{F}$, с которой веревка давит на блок, по третьему закону Ньютона равна по модулю и противоположна по направлению силе $F_{опоры}$, с которой блок действует на веревку. Найдем силу $F_{опоры}$, используя второй закон Ньютона для центра масс всей веревки.
$m\vec{a}_{cm} = \sum \vec{F}_{ext}$
На систему (веревку) действуют две внешние силы: сила тяжести $m\vec{g}$, направленная вниз, и сила реакции опоры со стороны блока $\vec{F}_{опоры}$, направленная вверх. В проекции на вертикальную ось, направленную вниз: $m a_{cm} = mg - F_{опоры}$ Отсюда, искомая сила $F = F_{опоры} = mg - m a_{cm}$.
Найдем ускорение центра масс веревки $a_{cm}$. Сначала определим положение центра масс $y_{cm}$ (отсчитывая от блока вниз): $y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m} = \frac{(\frac{m}{L}x)\frac{x}{2} + (\frac{m}{L}(L-x))\frac{L-x}{2}}{m} = \frac{x^2 + (L-x)^2}{2L}$
Скорость центра масс $v_{cm}$ равна производной от координаты по времени: $v_{cm} = \frac{dy_{cm}}{dt} = \frac{d}{dt}\left(\frac{x^2 + (L-x)^2}{2L}\right) = \frac{1}{2L}(2x\frac{dx}{dt} + 2(L-x)(-\frac{dx}{dt})) = \frac{v}{L}(2x-L)$, где $\frac{dx}{dt} = v$ — скорость движения веревки.
Ускорение центра масс $a_{cm}$ равно производной от скорости по времени: $a_{cm} = \frac{dv_{cm}}{dt} = \frac{d}{dt}\left(\frac{v(2x-L)}{L}\right) = \frac{1}{L}\left(\frac{dv}{dt}(2x-L) + v(2\frac{dx}{dt})\right) = \frac{1}{L}(a(2x-L) + 2v^2)$ где $a = \frac{dv}{dt}$ — ускорение веревки. Ускорение найдем из второго закона Ньютона для веревки как целого. Движущая сила — разность весов свисающих частей: $ma = m_1 g - m_2 g = \frac{m}{L}xg - \frac{m}{L}(L-x)g = \frac{mg}{L}(2x-L)$ $a = \frac{g}{L}(2x-L)$
Подставим выражения для $\text{a}$ и $v^2$ в формулу для $a_{cm}$: $a_{cm} = \frac{1}{L}\left(\frac{g}{L}(2x-L)(2x-L) + 2\frac{g}{2L}(2x-L)^2\right)$ $a_{cm} = \frac{1}{L}\left(\frac{g}{L}(2x-L)^2 + \frac{g}{L}(2x-L)^2\right) = \frac{2g}{L^2}(2x-L)^2$
Теперь найдем силу давления $\text{F}$: $F = mg - m a_{cm} = mg - m\frac{2g}{L^2}(2x-L)^2 = mg\left(1 - \frac{2(2x-L)^2}{L^2}\right)$
Ответ: Сила, с которой веревка давит на блок, равна $F = mg\left(1 - \frac{2(2x-L)^2}{L^2}\right)$.
Другие задания:
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