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Номер 8.21, страница 46 - 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 8.21, страница 46.
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" => 1288207 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "46" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/08-21" "field_display_title" => "8.21" "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" => 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" => 1478020 "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" => "1288207" "type" => "task" ] "text" => "<p><strong>8.21*</strong>. Гладкий шелковый шнур длиной $\text{L}$ и массой $\text{m}$, лежащий перпендикулярно краю стола, начинает без начальной скорости соскальзывать с края стола. Найдите скорость $\text{v}$ и ускорение $\text{a}$ шнура в тот момент, когда он соскользнул со стола наполовину. Какова сила $\text{T}$ натяжения шнура у края стола в этот момент?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "21-1.jpg" "alt" => null "width" => "1436" "height" => 376 "path" => "/media/fizika_08/gelfgat-u/0-08/21-1.webp?ts=1756049797" ] ] ] #original: array:7 [ "id" => 1478020 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1172} "task" => array:2 [ "refs" => "1288207" "type" => "task" ] "text" => "<p><strong>8.21*</strong>. Гладкий шелковый шнур длиной $\text{L}$ и массой $\text{m}$, лежащий перпендикулярно краю стола, начинает без начальной скорости соскальзывать с края стола. Найдите скорость $\text{v}$ и ускорение $\text{a}$ шнура в тот момент, когда он соскользнул со стола наполовину. Какова сила $\text{T}$ натяжения шнура у края стола в этот момент?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "21-1.jpg" "alt" => null "width" => "1436" "height" => 376 "path" => "/media/fizika_08/gelfgat-u/0-08/21-1.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" => 1478547 "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" => "1288207" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "21-1.jpg" "alt" => null "width" => "1437" "height" => 458 "path" => "/media/fizika_08/gelfgat-u/1-08/21-1.webp?ts=1756049862" ] ] ] #original: array:6 [ "id" => 1478547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1164} "task" => array:2 [ "refs" => "1288207" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "21-1.jpg" "alt" => null "width" => "1437" "height" => 458 "path" => "/media/fizika_08/gelfgat-u/1-08/21-1.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" => 1737900 "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" => "1288207" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Длина шнура: $\text{L}$</p><p>Масса шнура: $\text{m}$</p><p>Начальная скорость: $v_0 = 0$</p><p>Стол гладкий (трения нет).</p><p><strong>Найти:</strong></p><p>Скорость $\text{v}$, ускорение $\text{a}$ и силу натяжения $\text{T}$ в момент, когда шнур соскользнул наполовину.</p><p><strong>Решение:</strong></p><p>Обозначим линейную плотность шнура как $\rho = \frac{m}{L}$. В момент, когда со стола соскользнула половина шнура, на столе лежит часть длиной $\frac{L}{2}$ и массой $\frac{m}{2}$, а свисает часть также длиной $\frac{L}{2}$ и массой $\frac{m}{2}$.</p><p><strong>1.</strong> Нахождение ускорения $\text{a}$</p><p>Рассмотрим систему как единое целое. Движущей силой является сила тяжести, действующая на свисающую часть шнура. Эта сила приложена к центру масс свисающей части.</p><p>Масса свисающей части: $m_{свис} = \frac{m}{2}$.</p><p>Сила тяжести, действующая на нее: $F_{тяж} = m_{свис}g = \frac{m}{2}g$.</p><p>Эта сила сообщает ускорение $\text{a}$ всему шнуру массой $\text{m}$. Согласно второму закону Ньютона:</p><p>$F_{тяж} = m \cdot a$</p><p>$\frac{m}{2}g = m \cdot a$</p><p>Отсюда находим ускорение:</p><p>$a = \frac{g}{2}$</p><p><strong>Ответ:</strong> Ускорение шнура в данный момент $a = \frac{g}{2}$.</p><p><strong>2.</strong> Нахождение силы натяжения $\text{T}$</p><p>Сила натяжения $\text{T}$ у края стола — это сила, с которой свисающая часть тянет часть, лежащую на столе. Рассмотрим часть шнура, которая находится на столе. Ее масса $m_{стол} = \frac{m}{2}$. Единственная горизонтальная сила, действующая на эту часть, — это сила натяжения $\text{T}$. По второму закону Ньютона для этой части шнура:</p><p>$T = m_{стол} \cdot a$</p><p>Подставим найденное ранее ускорение $a = \frac{g}{2}$:</p><p>$T = \frac{m}{2} \cdot \frac{g}{2} = \frac{mg}{4}$</p><p><em>Проверка:</em> Можно рассмотреть свисающую часть. На нее действуют две силы: сила тяжести $\frac{mg}{2}$ (вниз) и сила натяжения $\text{T}$ (вверх). Результирующая сила сообщает этой части ускорение $\text{a}$ (вниз):</p><p>$\frac{mg}{2} - T = m_{свис} \cdot a = \frac{m}{2} \cdot a$</p><p>$T = \frac{mg}{2} - \frac{m}{2}a = \frac{m}{2}(g - a)$</p><p>Подставляя $a = \frac{g}{2}$, получаем:</p><p>$T = \frac{m}{2}(g - \frac{g}{2}) = \frac{m}{2} \cdot \frac{g}{2} = \frac{mg}{4}$. Результаты совпали.</p><p><strong>Ответ:</strong> Сила натяжения шнура у края стола $T = \frac{mg}{4}$.</p><p><strong>3.</strong> Нахождение скорости $\text{v}$</p><p>Поскольку стол гладкий, трение отсутствует, и можно применить закон сохранения механической энергии. За нулевой уровень потенциальной энергии примем поверхность стола.</p><p>Начальное состояние: шнур покоится ($v_0 = 0$), его центр масс находится на столе. Начальная механическая энергия $E_1 = 0$.</p><p>Конечное состояние: соскользнула половина шнура. Весь шнур движется со скоростью $\text{v}$.</p><p>Кинетическая энергия всей системы: $E_k = \frac{1}{2}mv^2$.</p><p>Потенциальная энергия системы определяется положением свисающей части. Масса свисающей части $\frac{m}{2}$. Ее центр масс находится на расстоянии $\text{h}$ ниже уровня стола. Поскольку длина свисающей части $\frac{L}{2}$, ее центр масс находится посредине, то есть на глубине $h = \frac{L/2}{2} = \frac{L}{4}$.</p><p>Потенциальная энергия свисающей части: $E_p = -m_{свис}gh = -\frac{m}{2}g\frac{L}{4} = -\frac{mgL}{8}$.</p><p>Полная механическая энергия в конечном состоянии: $E_2 = E_k + E_p = \frac{1}{2}mv^2 - \frac{mgL}{8}$.</p><p>По закону сохранения энергии $E_1 = E_2$:</p><p>$0 = \frac{1}{2}mv^2 - \frac{mgL}{8}$</p><p>$\frac{1}{2}mv^2 = \frac{mgL}{8}$</p><p>$v^2 = \frac{gL}{4}$</p><p>$v = \sqrt{\frac{gL}{4}} = \frac{1}{2}\sqrt{gL}$</p><p><strong>Ответ:</strong> Скорость шнура в данный момент $v = \frac{1}{2}\sqrt{gL}$.</p>" ] #original: array:6 [ "id" => 1737900 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156} "task" => array:2 [ "refs" => "1288207" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Длина шнура: $\text{L}$</p><p>Масса шнура: $\text{m}$</p><p>Начальная скорость: $v_0 = 0$</p><p>Стол гладкий (трения нет).</p><p><strong>Найти:</strong></p><p>Скорость $\text{v}$, ускорение $\text{a}$ и силу натяжения $\text{T}$ в момент, когда шнур соскользнул наполовину.</p><p><strong>Решение:</strong></p><p>Обозначим линейную плотность шнура как $\rho = \frac{m}{L}$. В момент, когда со стола соскользнула половина шнура, на столе лежит часть длиной $\frac{L}{2}$ и массой $\frac{m}{2}$, а свисает часть также длиной $\frac{L}{2}$ и массой $\frac{m}{2}$.</p><p><strong>1.</strong> Нахождение ускорения $\text{a}$</p><p>Рассмотрим систему как единое целое. Движущей силой является сила тяжести, действующая на свисающую часть шнура. Эта сила приложена к центру масс свисающей части.</p><p>Масса свисающей части: $m_{свис} = \frac{m}{2}$.</p><p>Сила тяжести, действующая на нее: $F_{тяж} = m_{свис}g = \frac{m}{2}g$.</p><p>Эта сила сообщает ускорение $\text{a}$ всему шнуру массой $\text{m}$. Согласно второму закону Ньютона:</p><p>$F_{тяж} = m \cdot a$</p><p>$\frac{m}{2}g = m \cdot a$</p><p>Отсюда находим ускорение:</p><p>$a = \frac{g}{2}$</p><p><strong>Ответ:</strong> Ускорение шнура в данный момент $a = \frac{g}{2}$.</p><p><strong>2.</strong> Нахождение силы натяжения $\text{T}$</p><p>Сила натяжения $\text{T}$ у края стола — это сила, с которой свисающая часть тянет часть, лежащую на столе. Рассмотрим часть шнура, которая находится на столе. Ее масса $m_{стол} = \frac{m}{2}$. Единственная горизонтальная сила, действующая на эту часть, — это сила натяжения $\text{T}$. По второму закону Ньютона для этой части шнура:</p><p>$T = m_{стол} \cdot a$</p><p>Подставим найденное ранее ускорение $a = \frac{g}{2}$:</p><p>$T = \frac{m}{2} \cdot \frac{g}{2} = \frac{mg}{4}$</p><p><em>Проверка:</em> Можно рассмотреть свисающую часть. На нее действуют две силы: сила тяжести $\frac{mg}{2}$ (вниз) и сила натяжения $\text{T}$ (вверх). Результирующая сила сообщает этой части ускорение $\text{a}$ (вниз):</p><p>$\frac{mg}{2} - T = m_{свис} \cdot a = \frac{m}{2} \cdot a$</p><p>$T = \frac{mg}{2} - \frac{m}{2}a = \frac{m}{2}(g - a)$</p><p>Подставляя $a = \frac{g}{2}$, получаем:</p><p>$T = \frac{m}{2}(g - \frac{g}{2}) = \frac{m}{2} \cdot \frac{g}{2} = \frac{mg}{4}$. Результаты совпали.</p><p><strong>Ответ:</strong> Сила натяжения шнура у края стола $T = \frac{mg}{4}$.</p><p><strong>3.</strong> Нахождение скорости $\text{v}$</p><p>Поскольку стол гладкий, трение отсутствует, и можно применить закон сохранения механической энергии. За нулевой уровень потенциальной энергии примем поверхность стола.</p><p>Начальное состояние: шнур покоится ($v_0 = 0$), его центр масс находится на столе. Начальная механическая энергия $E_1 = 0$.</p><p>Конечное состояние: соскользнула половина шнура. Весь шнур движется со скоростью $\text{v}$.</p><p>Кинетическая энергия всей системы: $E_k = \frac{1}{2}mv^2$.</p><p>Потенциальная энергия системы определяется положением свисающей части. Масса свисающей части $\frac{m}{2}$. Ее центр масс находится на расстоянии $\text{h}$ ниже уровня стола. Поскольку длина свисающей части $\frac{L}{2}$, ее центр масс находится посредине, то есть на глубине $h = \frac{L/2}{2} = \frac{L}{4}$.</p><p>Потенциальная энергия свисающей части: $E_p = -m_{свис}gh = -\frac{m}{2}g\frac{L}{4} = -\frac{mgL}{8}$.</p><p>Полная механическая энергия в конечном состоянии: $E_2 = E_k + E_p = \frac{1}{2}mv^2 - \frac{mgL}{8}$.</p><p>По закону сохранения энергии $E_1 = E_2$:</p><p>$0 = \frac{1}{2}mv^2 - \frac{mgL}{8}$</p><p>$\frac{1}{2}mv^2 = \frac{mgL}{8}$</p><p>$v^2 = \frac{gL}{4}$</p><p>$v = \sqrt{\frac{gL}{4}} = \frac{1}{2}\sqrt{gL}$</p><p><strong>Ответ:</strong> Скорость шнура в данный момент $v = \frac{1}{2}\sqrt{gL}$.</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 [ 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array:6 [ "id" => 1518518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1142 …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" => "1340398" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340396" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1427 #items: array:6 [ 0 => App\Models\Task {#1439 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: 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#observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1466 #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" => 1288209 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "46" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/08-23" "field_display_title" => "8.23" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1465 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1472 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1474 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1470 …2} "content" => Illuminate\Database\Eloquent\Collection {#1469 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1487 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288209 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "46" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/08-23" "field_display_title" => "8.23" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1465 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection 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№8.21 (с. 46)
Условие. №8.21 (с. 46)
скриншот условия
8.21*. Гладкий шелковый шнур длиной $\text{L}$ и массой $\text{m}$, лежащий перпендикулярно краю стола, начинает без начальной скорости соскальзывать с края стола. Найдите скорость $\text{v}$ и ускорение $\text{a}$ шнура в тот момент, когда он соскользнул со стола наполовину. Какова сила $\text{T}$ натяжения шнура у края стола в этот момент?
Решение. №8.21 (с. 46)
Решение 2. №8.21 (с. 46)
Дано:
Длина шнура: $\text{L}$
Масса шнура: $\text{m}$
Начальная скорость: $v_0 = 0$
Стол гладкий (трения нет).
Найти:
Скорость $\text{v}$, ускорение $\text{a}$ и силу натяжения $\text{T}$ в момент, когда шнур соскользнул наполовину.
Решение:
Обозначим линейную плотность шнура как $\rho = \frac{m}{L}$. В момент, когда со стола соскользнула половина шнура, на столе лежит часть длиной $\frac{L}{2}$ и массой $\frac{m}{2}$, а свисает часть также длиной $\frac{L}{2}$ и массой $\frac{m}{2}$.
1. Нахождение ускорения $\text{a}$
Рассмотрим систему как единое целое. Движущей силой является сила тяжести, действующая на свисающую часть шнура. Эта сила приложена к центру масс свисающей части.
Масса свисающей части: $m_{свис} = \frac{m}{2}$.
Сила тяжести, действующая на нее: $F_{тяж} = m_{свис}g = \frac{m}{2}g$.
Эта сила сообщает ускорение $\text{a}$ всему шнуру массой $\text{m}$. Согласно второму закону Ньютона:
$F_{тяж} = m \cdot a$
$\frac{m}{2}g = m \cdot a$
Отсюда находим ускорение:
$a = \frac{g}{2}$
Ответ: Ускорение шнура в данный момент $a = \frac{g}{2}$.
2. Нахождение силы натяжения $\text{T}$
Сила натяжения $\text{T}$ у края стола — это сила, с которой свисающая часть тянет часть, лежащую на столе. Рассмотрим часть шнура, которая находится на столе. Ее масса $m_{стол} = \frac{m}{2}$. Единственная горизонтальная сила, действующая на эту часть, — это сила натяжения $\text{T}$. По второму закону Ньютона для этой части шнура:
$T = m_{стол} \cdot a$
Подставим найденное ранее ускорение $a = \frac{g}{2}$:
$T = \frac{m}{2} \cdot \frac{g}{2} = \frac{mg}{4}$
Проверка: Можно рассмотреть свисающую часть. На нее действуют две силы: сила тяжести $\frac{mg}{2}$ (вниз) и сила натяжения $\text{T}$ (вверх). Результирующая сила сообщает этой части ускорение $\text{a}$ (вниз):
$\frac{mg}{2} - T = m_{свис} \cdot a = \frac{m}{2} \cdot a$
$T = \frac{mg}{2} - \frac{m}{2}a = \frac{m}{2}(g - a)$
Подставляя $a = \frac{g}{2}$, получаем:
$T = \frac{m}{2}(g - \frac{g}{2}) = \frac{m}{2} \cdot \frac{g}{2} = \frac{mg}{4}$. Результаты совпали.
Ответ: Сила натяжения шнура у края стола $T = \frac{mg}{4}$.
3. Нахождение скорости $\text{v}$
Поскольку стол гладкий, трение отсутствует, и можно применить закон сохранения механической энергии. За нулевой уровень потенциальной энергии примем поверхность стола.
Начальное состояние: шнур покоится ($v_0 = 0$), его центр масс находится на столе. Начальная механическая энергия $E_1 = 0$.
Конечное состояние: соскользнула половина шнура. Весь шнур движется со скоростью $\text{v}$.
Кинетическая энергия всей системы: $E_k = \frac{1}{2}mv^2$.
Потенциальная энергия системы определяется положением свисающей части. Масса свисающей части $\frac{m}{2}$. Ее центр масс находится на расстоянии $\text{h}$ ниже уровня стола. Поскольку длина свисающей части $\frac{L}{2}$, ее центр масс находится посредине, то есть на глубине $h = \frac{L/2}{2} = \frac{L}{4}$.
Потенциальная энергия свисающей части: $E_p = -m_{свис}gh = -\frac{m}{2}g\frac{L}{4} = -\frac{mgL}{8}$.
Полная механическая энергия в конечном состоянии: $E_2 = E_k + E_p = \frac{1}{2}mv^2 - \frac{mgL}{8}$.
По закону сохранения энергии $E_1 = E_2$:
$0 = \frac{1}{2}mv^2 - \frac{mgL}{8}$
$\frac{1}{2}mv^2 = \frac{mgL}{8}$
$v^2 = \frac{gL}{4}$
$v = \sqrt{\frac{gL}{4}} = \frac{1}{2}\sqrt{gL}$
Ответ: Скорость шнура в данный момент $v = \frac{1}{2}\sqrt{gL}$.
Другие задания:
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