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Номер 4.26, страница 23 - 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. 4. Импульс, работа, энергия. Законы сохранения - номер 4.26, страница 23.
App\Models\Task {#1024 // 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" => 1288041 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "23" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/04-26" "field_display_title" => "4.26" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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 {#333 #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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1060 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1060 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1065 #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 {#333} "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 {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065} ] #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 {#1066 #items: array:3 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1067 #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 {#1068 #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 {#1069 #items: array:1 [ 0 => App\Models\Book {#1070 #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 {#1071 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1097 …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 {#1098 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1101 …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 {#1071 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1097 …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 {#1098 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1101 …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 {#1103 #items: array:1 [ 0 => App\Models\Branch {#1104 #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 {#1105 …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 {#1106 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 …2} ] #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 {#1105 …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 {#1106 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 …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" => 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 {#1068} "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 {#1069} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1103} ] #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 {#1141 #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" => 1287871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4. Импульс, работа, энергия. Законы сохранения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "20" "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 {#1143 #items: array:1 [ 0 => App\Models\Book {#1144 #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 {#1145 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1171 …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 {#1172 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1175 …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 {#1145 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1171 …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 {#1172 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1175 …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 {#1177 #items: array:1 [ 0 => App\Models\Branch {#1178 #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 {#1179 …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 {#1180 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1214 …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 {#1179 …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 {#1180 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1214 …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" => 1287871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4. Импульс, работа, энергия. Законы сохранения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1142} "field_page_start" => "20" "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 {#1143} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1177} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:3 [ 0 => App\Models\Element {#1220 #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" => 1477816 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ 0 => App\Models\Edition {#1222 #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 {#1223 …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 {#1225 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1226 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1227 …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 {#1223 …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 {#1225 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1226 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1227 …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" => "1288041" "type" => "task" ] "text" => "<p><strong>4.26*.</strong> Обезьяна качалась на длинной тонкой лиане, причем максимальный угол отклонения лианы от вертикали был равен $\alpha$. Когда обезьяна находилась в нижней точке траектории, лиана случайно зацепилась серединой за сук. Каким будет теперь максимальный угол $\beta$ отклонения лианы от вертикали, если $\alpha < 60^\circ$? Что изменится, если $60^\circ < \alpha < 90^\circ$?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "26-1.jpg" "alt" => null "width" => "1435" "height" => 367 "path" => "/media/fizika_08/gelfgat-u/0-04/26-1.webp?ts=1756049698" ] ] ] #original: array:7 [ "id" => 1477816 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1221} "task" => array:2 [ "refs" => "1288041" "type" => "task" ] "text" => "<p><strong>4.26*.</strong> Обезьяна качалась на длинной тонкой лиане, причем максимальный угол отклонения лианы от вертикали был равен $\alpha$. Когда обезьяна находилась в нижней точке траектории, лиана случайно зацепилась серединой за сук. Каким будет теперь максимальный угол $\beta$ отклонения лианы от вертикали, если $\alpha < 60^\circ$? Что изменится, если $60^\circ < \alpha < 90^\circ$?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "26-1.jpg" "alt" => null "width" => "1435" "height" => 367 "path" => "/media/fizika_08/gelfgat-u/0-04/26-1.webp?ts=1756049698" ] ] ] #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 {#1228 #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" => 1478210 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Edition {#1230 #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 {#1231 …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 {#1233 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1235 …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 {#1231 …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 {#1233 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1235 …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" => "1288041" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "26-1.jpg" "alt" => null "width" => "996" "height" => 388 "path" => "/media/fizika_08/gelfgat-u/1-04/26-1.webp?ts=1756049759" ] 1 => array:5 [ "name" => "26-2.jpg" "alt" => null "width" => "441" "height" => 616 "path" => "/media/fizika_08/gelfgat-u/1-04/26-2.webp?ts=1756049759" ] ] ] #original: array:6 [ "id" => 1478210 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1229} "task" => array:2 [ "refs" => "1288041" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "26-1.jpg" "alt" => null "width" => "996" "height" => 388 "path" => "/media/fizika_08/gelfgat-u/1-04/26-1.webp?ts=1756049759" ] 1 => array:5 [ "name" => "26-2.jpg" "alt" => null "width" => "441" "height" => 616 "path" => "/media/fizika_08/gelfgat-u/1-04/26-2.webp?ts=1756049759" ] ] ] #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 {#1236 #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" => 1737734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Edition {#1238 #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 {#1239 …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 {#1241 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1243 …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 {#1239 …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 {#1241 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1243 …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" => "1288041" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Начальный максимальный угол отклонения лианы: $\alpha$</p><p>Конечное плечо вращения (длина лианы) вдвое меньше начального.</p><p><strong>Найти:</strong></p><p><strong>1.</strong> Новый максимальный угол отклонения $\beta$ при условии $\alpha \le 60^\circ$.</p><p><strong>2.</strong> Что изменится и каким будет угол $\beta$ при условии $60^\circ < \alpha < 90^\circ$.</p><p><strong>Решение:</strong></p><p>Для решения задачи воспользуемся законом сохранения механической энергии. Обезьяну будем считать материальной точкой, а лиану — нерастяжимой и невесомой нитью.</p><p><strong>1.</strong> <em>Движение до зацепа.</em></p><p>Пусть начальная длина лианы равна $\text{L}$, а масса обезьяны — $\text{m}$. Выберем нулевой уровень потенциальной энергии в нижней точке траектории. Когда обезьяна находится на максимальной высоте, отклонившись на угол $\alpha$, ее скорость равна нулю. Высота подъема относительно нижней точки равна $h_1 = L - L\cos\alpha = L(1-\cos\alpha)$. Полная механическая энергия в этот момент равна потенциальной энергии:</p><p>$E_1 = mgh_1 = mgL(1-\cos\alpha)$</p><p>В нижней точке траектории высота $h=0$, а скорость максимальна ($v_{max}$). Полная энергия равна кинетической энергии:</p><p>$E_2 = \frac{1}{2}mv_{max}^2$</p><p>Согласно закону сохранения энергии, $E_1 = E_2$, откуда:</p><p>$mgL(1-\cos\alpha) = \frac{1}{2}mv_{max}^2$</p><p>Скорость в нижней точке: $v_{max}^2 = 2gL(1-\cos\alpha)$.</p><p><strong>2.</strong> <em>Движение после зацепа.</em></p><p>В нижней точке лиана зацепляется за сук серединой. Новый центр вращения смещается, и эффективная длина лианы становится $L' = L/2$. Скорость обезьяны в момент зацепа не меняется и равна $v_{max}$. Эта скорость является начальной для второго этапа движения.</p><p>Обезьяна начинает подниматься по дуге окружности радиусом $L'$. Максимальный угол отклонения $\beta$ будет достигнут на высоте $h_2$, где скорость снова станет равной нулю. Высота $h_2$ относительно нижней точки равна $h_2 = L' - L'\cos\beta = \frac{L}{2}(1-\cos\beta)$.</p><p>Полная энергия в нижней точке (сразу после зацепа) равна $E_3 = \frac{1}{2}mv_{max}^2 = mgL(1-\cos\alpha)$.</p><p>Полная энергия в верхней точке новой траектории: $E_4 = mgh_2 = mg\frac{L}{2}(1-\cos\beta)$.</p><p>По закону сохранения энергии $E_3 = E_4$:</p><p>$mgL(1-\cos\alpha) = mg\frac{L}{2}(1-\cos\beta)$</p><p>$2(1-\cos\alpha) = 1-\cos\beta$</p><p>Отсюда находим связь между углами $\alpha$ и $\beta$:</p><p>$\cos\beta = 2\cos\alpha - 1$</p><p>Это соотношение справедливо только в том случае, если лиана на всем пути до верхней точки остается натянутой. Сила натяжения лианы $\text{T}$ должна быть неотрицательной ($T \ge 0$).</p><p><strong>3.</strong> <em>Анализ силы натяжения.</em></p><p>Запишем второй закон Ньютона для обезьяны в проекции на радиальное направление в произвольной точке второго этапа движения (угол $\theta$):</p><p>$T - mg\cos\theta = \frac{mv^2}{L'}$</p><p>где $\text{v}$ — скорость на угле $\theta$. Найдем $v^2$ из закона сохранения энергии для второго этапа:</p><p>$\frac{1}{2}mv_{max}^2 = \frac{1}{2}mv^2 + mgL'(1-\cos\theta)$</p><p>$mgL(1-\cos\alpha) = \frac{1}{2}mv^2 + mg\frac{L}{2}(1-\cos\theta)$</p><p>$v^2 = 2gL(1-\cos\alpha) - gL(1-\cos\theta) = gL(1 - 2\cos\alpha + \cos\theta)$</p><p>Подставим $v^2$ и $L'=L/2$ в уравнение для натяжения:</p><p>$T = mg\cos\theta + \frac{mgL(1 - 2\cos\alpha + \cos\theta)}{L/2} = mg\cos\theta + 2mg(1 - 2\cos\alpha + \cos\theta)$</p><p>$T = mg(3\cos\theta - 4\cos\alpha + 2)$</p><p>Лиана останется натянутой, если $T \ge 0$ на всей траектории. Минимальное значение натяжение имеет в верхней точке, где $\cos\theta$ минимален. Если обезьяна достигает точки, где $v=0$ (угол $\beta$), то натяжение в этой точке будет $T(\beta) = mg(3\cos\beta - 4\cos\alpha + 2)$. Подставив $\cos\beta = 2\cos\alpha - 1$, получим:</p><p>$T(\beta) = mg(3(2\cos\alpha - 1) - 4\cos\alpha + 2) = mg(6\cos\alpha - 3 - 4\cos\alpha + 2) = mg(2\cos\alpha - 1)$</p><p>Условие $T(\beta) \ge 0$ выполняется, если $2\cos\alpha - 1 \ge 0$, то есть $\cos\alpha \ge 1/2$. Для углов в диапазоне $0 < \alpha < 90^\circ$ это соответствует условию $\alpha \le 60^\circ$.</p><p><strong>Каким будет теперь максимальный угол $\beta$ отклонения лианы от вертикали, если $\alpha \le 60^\circ$?</strong></p><p>Как показано выше, при $\alpha \le 60^\circ$ натяжение лианы в верхней точке траектории неотрицательно. Это означает, что обезьяна достигает высоты, где ее скорость обращается в ноль, и лиана при этом не провисает. Следовательно, полученная из закона сохранения энергии формула справедлива.</p><p>$\cos\beta = 2\cos\alpha - 1$</p><p>Например, если $\alpha = 60^\circ$, то $\cos\alpha = 1/2$, и $\cos\beta = 2(1/2) - 1 = 0$, что соответствует $\beta = 90^\circ$.</p><p><strong>Ответ:</strong> Если $\alpha \le 60^\circ$, максимальный угол отклонения будет $\beta = \arccos(2\cos\alpha - 1)$.</p><p><strong>Что изменится, если $60^\circ < \alpha < 90^\circ$?</strong></p><p>В этом случае $\cos\alpha < 1/2$, и величина $2\cos\alpha - 1$ становится отрицательной. Это означает, что натяжение $T(\beta)$ в теоретической верхней точке (где $v=0$) было бы отрицательным, что физически невозможно для лианы.</p><p>Изменится то, что обезьяна не достигнет точки, где ее скорость равна нулю. Лиана провиснет раньше, в тот момент, когда сила натяжения станет равной нулю ($T=0$). После этого обезьяна будет двигаться по параболической траектории как тело, брошенное под углом.</p><p>Максимальный угол отклонения лианы $\beta$ будет соответствовать углу, при котором натяжение обращается в ноль. Найдем этот угол из условия $T=0$:</p><p>$T = mg(3\cos\beta - 4\cos\alpha + 2) = 0$</p><p>$3\cos\beta = 4\cos\alpha - 2$</p><p>$\cos\beta = \frac{4\cos\alpha - 2}{3}$</p><p>Именно этот угол и будет максимальным углом отклонения лианы.</p><p><strong>Ответ:</strong> Если $60^\circ < \alpha < 90^\circ$, то изменится характер движения: обезьяна не достигнет точки полной остановки, так как лиана провиснет раньше. Максимальный угол отклонения лианы будет определяться моментом, когда ее натяжение становится равным нулю, и будет равен $\beta = \arccos\left(\frac{4\cos\alpha - 2}{3}\right)$.</p>" ] #original: array:6 [ "id" => 1737734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1237} "task" => array:2 [ "refs" => "1288041" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Начальный максимальный угол отклонения лианы: $\alpha$</p><p>Конечное плечо вращения (длина лианы) вдвое меньше начального.</p><p><strong>Найти:</strong></p><p><strong>1.</strong> Новый максимальный угол отклонения $\beta$ при условии $\alpha \le 60^\circ$.</p><p><strong>2.</strong> Что изменится и каким будет угол $\beta$ при условии $60^\circ < \alpha < 90^\circ$.</p><p><strong>Решение:</strong></p><p>Для решения задачи воспользуемся законом сохранения механической энергии. Обезьяну будем считать материальной точкой, а лиану — нерастяжимой и невесомой нитью.</p><p><strong>1.</strong> <em>Движение до зацепа.</em></p><p>Пусть начальная длина лианы равна $\text{L}$, а масса обезьяны — $\text{m}$. Выберем нулевой уровень потенциальной энергии в нижней точке траектории. Когда обезьяна находится на максимальной высоте, отклонившись на угол $\alpha$, ее скорость равна нулю. Высота подъема относительно нижней точки равна $h_1 = L - L\cos\alpha = L(1-\cos\alpha)$. Полная механическая энергия в этот момент равна потенциальной энергии:</p><p>$E_1 = mgh_1 = mgL(1-\cos\alpha)$</p><p>В нижней точке траектории высота $h=0$, а скорость максимальна ($v_{max}$). Полная энергия равна кинетической энергии:</p><p>$E_2 = \frac{1}{2}mv_{max}^2$</p><p>Согласно закону сохранения энергии, $E_1 = E_2$, откуда:</p><p>$mgL(1-\cos\alpha) = \frac{1}{2}mv_{max}^2$</p><p>Скорость в нижней точке: $v_{max}^2 = 2gL(1-\cos\alpha)$.</p><p><strong>2.</strong> <em>Движение после зацепа.</em></p><p>В нижней точке лиана зацепляется за сук серединой. Новый центр вращения смещается, и эффективная длина лианы становится $L' = L/2$. Скорость обезьяны в момент зацепа не меняется и равна $v_{max}$. Эта скорость является начальной для второго этапа движения.</p><p>Обезьяна начинает подниматься по дуге окружности радиусом $L'$. Максимальный угол отклонения $\beta$ будет достигнут на высоте $h_2$, где скорость снова станет равной нулю. Высота $h_2$ относительно нижней точки равна $h_2 = L' - L'\cos\beta = \frac{L}{2}(1-\cos\beta)$.</p><p>Полная энергия в нижней точке (сразу после зацепа) равна $E_3 = \frac{1}{2}mv_{max}^2 = mgL(1-\cos\alpha)$.</p><p>Полная энергия в верхней точке новой траектории: $E_4 = mgh_2 = mg\frac{L}{2}(1-\cos\beta)$.</p><p>По закону сохранения энергии $E_3 = E_4$:</p><p>$mgL(1-\cos\alpha) = mg\frac{L}{2}(1-\cos\beta)$</p><p>$2(1-\cos\alpha) = 1-\cos\beta$</p><p>Отсюда находим связь между углами $\alpha$ и $\beta$:</p><p>$\cos\beta = 2\cos\alpha - 1$</p><p>Это соотношение справедливо только в том случае, если лиана на всем пути до верхней точки остается натянутой. Сила натяжения лианы $\text{T}$ должна быть неотрицательной ($T \ge 0$).</p><p><strong>3.</strong> <em>Анализ силы натяжения.</em></p><p>Запишем второй закон Ньютона для обезьяны в проекции на радиальное направление в произвольной точке второго этапа движения (угол $\theta$):</p><p>$T - mg\cos\theta = \frac{mv^2}{L'}$</p><p>где $\text{v}$ — скорость на угле $\theta$. Найдем $v^2$ из закона сохранения энергии для второго этапа:</p><p>$\frac{1}{2}mv_{max}^2 = \frac{1}{2}mv^2 + mgL'(1-\cos\theta)$</p><p>$mgL(1-\cos\alpha) = \frac{1}{2}mv^2 + mg\frac{L}{2}(1-\cos\theta)$</p><p>$v^2 = 2gL(1-\cos\alpha) - gL(1-\cos\theta) = gL(1 - 2\cos\alpha + \cos\theta)$</p><p>Подставим $v^2$ и $L'=L/2$ в уравнение для натяжения:</p><p>$T = mg\cos\theta + \frac{mgL(1 - 2\cos\alpha + \cos\theta)}{L/2} = mg\cos\theta + 2mg(1 - 2\cos\alpha + \cos\theta)$</p><p>$T = mg(3\cos\theta - 4\cos\alpha + 2)$</p><p>Лиана останется натянутой, если $T \ge 0$ на всей траектории. Минимальное значение натяжение имеет в верхней точке, где $\cos\theta$ минимален. Если обезьяна достигает точки, где $v=0$ (угол $\beta$), то натяжение в этой точке будет $T(\beta) = mg(3\cos\beta - 4\cos\alpha + 2)$. Подставив $\cos\beta = 2\cos\alpha - 1$, получим:</p><p>$T(\beta) = mg(3(2\cos\alpha - 1) - 4\cos\alpha + 2) = mg(6\cos\alpha - 3 - 4\cos\alpha + 2) = mg(2\cos\alpha - 1)$</p><p>Условие $T(\beta) \ge 0$ выполняется, если $2\cos\alpha - 1 \ge 0$, то есть $\cos\alpha \ge 1/2$. Для углов в диапазоне $0 < \alpha < 90^\circ$ это соответствует условию $\alpha \le 60^\circ$.</p><p><strong>Каким будет теперь максимальный угол $\beta$ отклонения лианы от вертикали, если $\alpha \le 60^\circ$?</strong></p><p>Как показано выше, при $\alpha \le 60^\circ$ натяжение лианы в верхней точке траектории неотрицательно. Это означает, что обезьяна достигает высоты, где ее скорость обращается в ноль, и лиана при этом не провисает. Следовательно, полученная из закона сохранения энергии формула справедлива.</p><p>$\cos\beta = 2\cos\alpha - 1$</p><p>Например, если $\alpha = 60^\circ$, то $\cos\alpha = 1/2$, и $\cos\beta = 2(1/2) - 1 = 0$, что соответствует $\beta = 90^\circ$.</p><p><strong>Ответ:</strong> Если $\alpha \le 60^\circ$, максимальный угол отклонения будет $\beta = \arccos(2\cos\alpha - 1)$.</p><p><strong>Что изменится, если $60^\circ < \alpha < 90^\circ$?</strong></p><p>В этом случае $\cos\alpha < 1/2$, и величина $2\cos\alpha - 1$ становится отрицательной. Это означает, что натяжение $T(\beta)$ в теоретической верхней точке (где $v=0$) было бы отрицательным, что физически невозможно для лианы.</p><p>Изменится то, что обезьяна не достигнет точки, где ее скорость равна нулю. Лиана провиснет раньше, в тот момент, когда сила натяжения станет равной нулю ($T=0$). После этого обезьяна будет двигаться по параболической траектории как тело, брошенное под углом.</p><p>Максимальный угол отклонения лианы $\beta$ будет соответствовать углу, при котором натяжение обращается в ноль. Найдем этот угол из условия $T=0$:</p><p>$T = mg(3\cos\beta - 4\cos\alpha + 2) = 0$</p><p>$3\cos\beta = 4\cos\alpha - 2$</p><p>$\cos\beta = \frac{4\cos\alpha - 2}{3}$</p><p>Именно этот угол и будет максимальным углом отклонения лианы.</p><p><strong>Ответ:</strong> Если $60^\circ < \alpha < 90^\circ$, то изменится характер движения: обезьяна не достигнет точки полной остановки, так как лиана провиснет раньше. Максимальный угол отклонения лианы будет определяться моментом, когда ее натяжение становится равным нулю, и будет равен $\beta = \arccos\left(\frac{4\cos\alpha - 2}{3}\right)$.</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 } "next" => array:2 [ "refs" => "1288042" "type" => "task" ] "previous" => array:2 [ "refs" => "1288040" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:1 [ 0 => App\Models\Book {#1245 #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 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"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 {#1248 #items: array:4 [ 0 => App\Models\Term {#1249 #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 {#1250 #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 {#1251 #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 {#1252 #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 {#1253 #items: array:1 [ 0 => App\Models\Term {#1254 #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 {#1255 #items: array:3 [ 0 => App\Models\Term {#1256 #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 {#1257 #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" …1 ] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1258 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1259 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:1 [ 0 => App\Models\Term {#1261 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1262 #items: array:1 [ 0 => App\Models\Term {#1263 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1264 #items: array:1 [ 0 => App\Models\Term {#1265 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1266 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1267 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1268 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1269 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:1 [ 0 => App\Models\Term {#1271 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1272 #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 {#1273 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1274 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1275 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1276 #items: array:1 [ 0 => App\Models\Term {#1277 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1246} "field_class" => Illuminate\Database\Eloquent\Collection {#1248} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1253} "field_author" => Illuminate\Database\Eloquent\Collection {#1255} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1259} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1260} "field_country" => Illuminate\Database\Eloquent\Collection {#1262} "field_city" => Illuminate\Database\Eloquent\Collection {#1264} "field_series" => Illuminate\Database\Eloquent\Collection {#1266} "field_umk" => Illuminate\Database\Eloquent\Collection {#1267} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1268} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1269} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1270} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1272} "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 {#1273} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1274} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1275} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1276} "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 } "page" => Illuminate\Database\Eloquent\Collection {#1638 #items: array:1 [ 0 => App\Models\BookPage {#1281 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1340374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "23" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-23" "field_display_title" => "23" "field_folder" => "1" "field_image_name" => "23" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1285 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1283 #items: array:1 [ 0 => App\Models\Book {#1286 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1319 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1320 #items: array:1 [ 0 => App\Models\Element {#1321 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1340375" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340373" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1329 #items: array:8 [ 0 => App\Models\Task {#1330 #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] } 1 => App\Models\Task {#1545 #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] } 2 => App\Models\Task {#1558 #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] } 3 => App\Models\Task {#1571 #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] } 4 => App\Models\Task {#1584 #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] } 5 => App\Models\Task {#1597 #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 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№4.26 (с. 23)
Условие. №4.26 (с. 23)
скриншот условия
4.26*. Обезьяна качалась на длинной тонкой лиане, причем максимальный угол отклонения лианы от вертикали был равен $\alpha$. Когда обезьяна находилась в нижней точке траектории, лиана случайно зацепилась серединой за сук. Каким будет теперь максимальный угол $\beta$ отклонения лианы от вертикали, если $\alpha < 60^\circ$? Что изменится, если $60^\circ < \alpha < 90^\circ$?
Решение. №4.26 (с. 23)
Решение 2. №4.26 (с. 23)
Дано:
Начальный максимальный угол отклонения лианы: $\alpha$
Конечное плечо вращения (длина лианы) вдвое меньше начального.
Найти:
1. Новый максимальный угол отклонения $\beta$ при условии $\alpha \le 60^\circ$.
2. Что изменится и каким будет угол $\beta$ при условии $60^\circ < \alpha < 90^\circ$.
Решение:
Для решения задачи воспользуемся законом сохранения механической энергии. Обезьяну будем считать материальной точкой, а лиану — нерастяжимой и невесомой нитью.
1. Движение до зацепа.
Пусть начальная длина лианы равна $\text{L}$, а масса обезьяны — $\text{m}$. Выберем нулевой уровень потенциальной энергии в нижней точке траектории. Когда обезьяна находится на максимальной высоте, отклонившись на угол $\alpha$, ее скорость равна нулю. Высота подъема относительно нижней точки равна $h_1 = L - L\cos\alpha = L(1-\cos\alpha)$. Полная механическая энергия в этот момент равна потенциальной энергии:
$E_1 = mgh_1 = mgL(1-\cos\alpha)$
В нижней точке траектории высота $h=0$, а скорость максимальна ($v_{max}$). Полная энергия равна кинетической энергии:
$E_2 = \frac{1}{2}mv_{max}^2$
Согласно закону сохранения энергии, $E_1 = E_2$, откуда:
$mgL(1-\cos\alpha) = \frac{1}{2}mv_{max}^2$
Скорость в нижней точке: $v_{max}^2 = 2gL(1-\cos\alpha)$.
2. Движение после зацепа.
В нижней точке лиана зацепляется за сук серединой. Новый центр вращения смещается, и эффективная длина лианы становится $L' = L/2$. Скорость обезьяны в момент зацепа не меняется и равна $v_{max}$. Эта скорость является начальной для второго этапа движения.
Обезьяна начинает подниматься по дуге окружности радиусом $L'$. Максимальный угол отклонения $\beta$ будет достигнут на высоте $h_2$, где скорость снова станет равной нулю. Высота $h_2$ относительно нижней точки равна $h_2 = L' - L'\cos\beta = \frac{L}{2}(1-\cos\beta)$.
Полная энергия в нижней точке (сразу после зацепа) равна $E_3 = \frac{1}{2}mv_{max}^2 = mgL(1-\cos\alpha)$.
Полная энергия в верхней точке новой траектории: $E_4 = mgh_2 = mg\frac{L}{2}(1-\cos\beta)$.
По закону сохранения энергии $E_3 = E_4$:
$mgL(1-\cos\alpha) = mg\frac{L}{2}(1-\cos\beta)$
$2(1-\cos\alpha) = 1-\cos\beta$
Отсюда находим связь между углами $\alpha$ и $\beta$:
$\cos\beta = 2\cos\alpha - 1$
Это соотношение справедливо только в том случае, если лиана на всем пути до верхней точки остается натянутой. Сила натяжения лианы $\text{T}$ должна быть неотрицательной ($T \ge 0$).
3. Анализ силы натяжения.
Запишем второй закон Ньютона для обезьяны в проекции на радиальное направление в произвольной точке второго этапа движения (угол $\theta$):
$T - mg\cos\theta = \frac{mv^2}{L'}$
где $\text{v}$ — скорость на угле $\theta$. Найдем $v^2$ из закона сохранения энергии для второго этапа:
$\frac{1}{2}mv_{max}^2 = \frac{1}{2}mv^2 + mgL'(1-\cos\theta)$
$mgL(1-\cos\alpha) = \frac{1}{2}mv^2 + mg\frac{L}{2}(1-\cos\theta)$
$v^2 = 2gL(1-\cos\alpha) - gL(1-\cos\theta) = gL(1 - 2\cos\alpha + \cos\theta)$
Подставим $v^2$ и $L'=L/2$ в уравнение для натяжения:
$T = mg\cos\theta + \frac{mgL(1 - 2\cos\alpha + \cos\theta)}{L/2} = mg\cos\theta + 2mg(1 - 2\cos\alpha + \cos\theta)$
$T = mg(3\cos\theta - 4\cos\alpha + 2)$
Лиана останется натянутой, если $T \ge 0$ на всей траектории. Минимальное значение натяжение имеет в верхней точке, где $\cos\theta$ минимален. Если обезьяна достигает точки, где $v=0$ (угол $\beta$), то натяжение в этой точке будет $T(\beta) = mg(3\cos\beta - 4\cos\alpha + 2)$. Подставив $\cos\beta = 2\cos\alpha - 1$, получим:
$T(\beta) = mg(3(2\cos\alpha - 1) - 4\cos\alpha + 2) = mg(6\cos\alpha - 3 - 4\cos\alpha + 2) = mg(2\cos\alpha - 1)$
Условие $T(\beta) \ge 0$ выполняется, если $2\cos\alpha - 1 \ge 0$, то есть $\cos\alpha \ge 1/2$. Для углов в диапазоне $0 < \alpha < 90^\circ$ это соответствует условию $\alpha \le 60^\circ$.
Каким будет теперь максимальный угол $\beta$ отклонения лианы от вертикали, если $\alpha \le 60^\circ$?
Как показано выше, при $\alpha \le 60^\circ$ натяжение лианы в верхней точке траектории неотрицательно. Это означает, что обезьяна достигает высоты, где ее скорость обращается в ноль, и лиана при этом не провисает. Следовательно, полученная из закона сохранения энергии формула справедлива.
$\cos\beta = 2\cos\alpha - 1$
Например, если $\alpha = 60^\circ$, то $\cos\alpha = 1/2$, и $\cos\beta = 2(1/2) - 1 = 0$, что соответствует $\beta = 90^\circ$.
Ответ: Если $\alpha \le 60^\circ$, максимальный угол отклонения будет $\beta = \arccos(2\cos\alpha - 1)$.
Что изменится, если $60^\circ < \alpha < 90^\circ$?
В этом случае $\cos\alpha < 1/2$, и величина $2\cos\alpha - 1$ становится отрицательной. Это означает, что натяжение $T(\beta)$ в теоретической верхней точке (где $v=0$) было бы отрицательным, что физически невозможно для лианы.
Изменится то, что обезьяна не достигнет точки, где ее скорость равна нулю. Лиана провиснет раньше, в тот момент, когда сила натяжения станет равной нулю ($T=0$). После этого обезьяна будет двигаться по параболической траектории как тело, брошенное под углом.
Максимальный угол отклонения лианы $\beta$ будет соответствовать углу, при котором натяжение обращается в ноль. Найдем этот угол из условия $T=0$:
$T = mg(3\cos\beta - 4\cos\alpha + 2) = 0$
$3\cos\beta = 4\cos\alpha - 2$
$\cos\beta = \frac{4\cos\alpha - 2}{3}$
Именно этот угол и будет максимальным углом отклонения лианы.
Ответ: Если $60^\circ < \alpha < 90^\circ$, то изменится характер движения: обезьяна не достигнет точки полной остановки, так как лиана провиснет раньше. Максимальный угол отклонения лианы будет определяться моментом, когда ее натяжение становится равным нулю, и будет равен $\beta = \arccos\left(\frac{4\cos\alpha - 2}{3}\right)$.
Другие задания:
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