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Номер 7.2, страница 37 - 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. 7. Механические колебания и волны - номер 7.2, страница 37.
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" => 1288150 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/07-2" "field_display_title" => "7.2" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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 {#1047 #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 {#1083 #items: array:1 [ 0 => App\Models\Book {#1048 #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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #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 {#1051 #items: array:4 [ 0 => App\Models\Term {#1052 #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 {#1053 #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 {#1054 #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 {#1055 #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 {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 #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 {#1058 #items: array:3 [ 0 => App\Models\Term {#1059 #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 {#1060 #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 {#1061 #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 {#1062 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #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 {#1067 #items: array:1 [ 0 => App\Models\Term {#1068 #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 {#1069 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1074 #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 {#1075 #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 {#1076 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Term {#1080 #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 {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1056} "field_author" => Illuminate\Database\Eloquent\Collection {#1058} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1062} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_country" => Illuminate\Database\Eloquent\Collection {#1065} "field_city" => Illuminate\Database\Eloquent\Collection {#1067} "field_series" => Illuminate\Database\Eloquent\Collection {#1069} "field_umk" => Illuminate\Database\Eloquent\Collection {#1070} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_publication_number" => "6" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1073} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1075} "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 {#1076} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1077} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1078} "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1079} "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 {#1081 #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 {#1047} "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 {#1083} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1081} ] #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 {#1114 #items: array:3 [ 0 => App\Models\Branch {#1124 #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 {#1125 #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 {#1126 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1127 #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 {#1125} "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 {#1126} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1127} ] #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 {#1128 #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 {#1129 #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 {#1130 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #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 {#1129} "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 {#1130} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1131} ] #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 {#1132 #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" => 1287874 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "7. Механические колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "37" "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 {#1134 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Branch {#1142 #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 {#1144 …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 {#1145 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 …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 {#1144 …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 {#1145 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 …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" => 1287874 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "7. Механические колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1133} "field_page_start" => "37" "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 {#1134} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1143} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:3 [ 0 => App\Models\Element {#1104 #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" => 1477925 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Edition {#1105 #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 {#1102 …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 {#1101 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 …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 {#1102 …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 {#1101 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 …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" => "1288150" "type" => "task" ] "text" => "<p><strong>7.2.</strong> Маятник совершает гармонические колебания с амплитудой 6 см. Какую часть периода маятник находится не далее 3 см от положения равновесия?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1420" "height" => 224 "path" => "/media/fizika_08/gelfgat-u/0-07/2-1.webp?ts=1756049770" ] ] ] #original: array:7 [ "id" => 1477925 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "1288150" "type" => "task" ] "text" => "<p><strong>7.2.</strong> Маятник совершает гармонические колебания с амплитудой 6 см. Какую часть периода маятник находится не далее 3 см от положения равновесия?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1420" "height" => 224 "path" => "/media/fizika_08/gelfgat-u/0-07/2-1.webp?ts=1756049770" ] ] ] #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 {#1099 #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" => 1478442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1096 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_content_type" => "free" "field_content_mode" => "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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …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 {#1095 …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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …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" => "1288150" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "391" "height" => 122 "path" => "/media/fizika_08/gelfgat-u/1-07/2-1.webp?ts=1756049832" ] ] ] #original: array:6 [ "id" => 1478442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1288150" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "391" "height" => 122 "path" => "/media/fizika_08/gelfgat-u/1-07/2-1.webp?ts=1756049832" ] ] ] #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 {#1091 #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" => 1737843 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Edition {#1090 #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 {#1041 …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 {#1086 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 …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 {#1041 …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 {#1086 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 …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" => "1288150" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Амплитуда колебаний, $A = 6 \text{ см}$</p><p>Рассматриваемый диапазон смещений, $|x| \le 3 \text{ см}$</p><p><strong>Перевод в СИ:</strong></p><p>$A = 0.06 \text{ м}$</p><p>$|x| \le 0.03 \text{ м}$</p><p><strong>Найти:</strong></p><p>Часть периода $\frac{\Delta t}{T}$, в течение которой маятник находится в заданном диапазоне.</p><p><strong>Решение:</strong></p><p>Гармонические колебания маятника описываются уравнением $x(t) = A \sin(\omega t)$, где $\text{x}$ - смещение от положения равновесия, $\text{A}$ - амплитуда, $\omega = \frac{2\pi}{T}$ - циклическая частота, а $\text{T}$ - период колебаний. Выбор функции синуса соответствует началу движения из положения равновесия при $t=0$.</p><p>Условие, что маятник находится "не далее 3 см от положения равновесия", математически записывается в виде неравенства: $|x| \le 3$ см.</p><p>Подставим уравнение движения в это неравенство:</p><p>$|A \sin(\omega t)| \le 3$</p><p>Используя данное значение амплитуды $A = 6$ см, получим:</p><p>$|6 \sin(\omega t)| \le 3$</p><p>Разделив обе части на 6, приходим к неравенству для синуса фазы колебаний:</p><p>$|\sin(\omega t)| \le \frac{1}{2}$</p><p>Это эквивалентно двойному неравенству: $-\frac{1}{2} \le \sin(\omega t) \le \frac{1}{2}$.</p><p>Чтобы найти общую продолжительность времени, в течение которого выполняется это условие, сначала определим время $t_1$, необходимое маятнику для перемещения из положения равновесия ($x=0$) до границы указанной области ($x=3$ см).</p><p>$x(t_1) = 3 \implies A \sin(\omega t_1) = 3 \implies 6 \sin(\frac{2\pi}{T} t_1) = 3$</p><p>Отсюда получаем: $\sin(\frac{2\pi}{T} t_1) = \frac{1}{2}$.</p><p>Наименьшее положительное значение фазы $\frac{2\pi}{T} t_1$, синус которого равен $\frac{1}{2}$, составляет $\frac{\pi}{6}$ радиан.</p><p>$\frac{2\pi}{T} t_1 = \frac{\pi}{6}$</p><p>Из этого уравнения выражаем время $t_1$:</p><p>$t_1 = \frac{\pi}{6} \cdot \frac{T}{2\pi} = \frac{T}{12}$</p><p>За один полный период колебаний маятник проходит через область $|x| \le 3$ см четыре раза. В силу симметрии гармонического движения, время прохождения каждого из этих участков одинаково и равно $t_1$. Эти четыре участка соответствуют движению: от $x=0$ до $x=3$ см, от $x=3$ см до $x=0$, от $x=0$ до $x=-3$ см и от $x=-3$ см до $x=0$.</p><p>Таким образом, общее время $\Delta t$, которое маятник проводит в указанной области за один период, равно сумме времен этих четырех проходов:</p><p>$\Delta t = 4 \cdot t_1 = 4 \cdot \frac{T}{12} = \frac{T}{3}$</p><p>Искомая часть периода представляет собой отношение $\frac{\Delta t}{T}$:</p><p>$\frac{\Delta t}{T} = \frac{T/3}{T} = \frac{1}{3}$</p><p><strong>Ответ:</strong> маятник находится не далее 3 см от положения равновесия в течение $\frac{1}{3}$ периода.</p>" ] #original: array:6 [ "id" => 1737843 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1043} "task" => array:2 [ "refs" => "1288150" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Амплитуда колебаний, $A = 6 \text{ см}$</p><p>Рассматриваемый диапазон смещений, $|x| \le 3 \text{ см}$</p><p><strong>Перевод в СИ:</strong></p><p>$A = 0.06 \text{ м}$</p><p>$|x| \le 0.03 \text{ м}$</p><p><strong>Найти:</strong></p><p>Часть периода $\frac{\Delta t}{T}$, в течение которой маятник находится в заданном диапазоне.</p><p><strong>Решение:</strong></p><p>Гармонические колебания маятника описываются уравнением $x(t) = A \sin(\omega t)$, где $\text{x}$ - смещение от положения равновесия, $\text{A}$ - амплитуда, $\omega = \frac{2\pi}{T}$ - циклическая частота, а $\text{T}$ - период колебаний. Выбор функции синуса соответствует началу движения из положения равновесия при $t=0$.</p><p>Условие, что маятник находится "не далее 3 см от положения равновесия", математически записывается в виде неравенства: $|x| \le 3$ см.</p><p>Подставим уравнение движения в это неравенство:</p><p>$|A \sin(\omega t)| \le 3$</p><p>Используя данное значение амплитуды $A = 6$ см, получим:</p><p>$|6 \sin(\omega t)| \le 3$</p><p>Разделив обе части на 6, приходим к неравенству для синуса фазы колебаний:</p><p>$|\sin(\omega t)| \le \frac{1}{2}$</p><p>Это эквивалентно двойному неравенству: $-\frac{1}{2} \le \sin(\omega t) \le \frac{1}{2}$.</p><p>Чтобы найти общую продолжительность времени, в течение которого выполняется это условие, сначала определим время $t_1$, необходимое маятнику для перемещения из положения равновесия ($x=0$) до границы указанной области ($x=3$ см).</p><p>$x(t_1) = 3 \implies A \sin(\omega t_1) = 3 \implies 6 \sin(\frac{2\pi}{T} t_1) = 3$</p><p>Отсюда получаем: $\sin(\frac{2\pi}{T} t_1) = \frac{1}{2}$.</p><p>Наименьшее положительное значение фазы $\frac{2\pi}{T} t_1$, синус которого равен $\frac{1}{2}$, составляет $\frac{\pi}{6}$ радиан.</p><p>$\frac{2\pi}{T} t_1 = \frac{\pi}{6}$</p><p>Из этого уравнения выражаем время $t_1$:</p><p>$t_1 = \frac{\pi}{6} \cdot \frac{T}{2\pi} = \frac{T}{12}$</p><p>За один полный период колебаний маятник проходит через область $|x| \le 3$ см четыре раза. В силу симметрии гармонического движения, время прохождения каждого из этих участков одинаково и равно $t_1$. Эти четыре участка соответствуют движению: от $x=0$ до $x=3$ см, от $x=3$ см до $x=0$, от $x=0$ до $x=-3$ см и от $x=-3$ см до $x=0$.</p><p>Таким образом, общее время $\Delta t$, которое маятник проводит в указанной области за один период, равно сумме времен этих четырех проходов:</p><p>$\Delta t = 4 \cdot t_1 = 4 \cdot \frac{T}{12} = \frac{T}{3}$</p><p>Искомая часть периода представляет собой отношение $\frac{\Delta t}{T}$:</p><p>$\frac{\Delta t}{T} = \frac{T/3}{T} = \frac{1}{3}$</p><p><strong>Ответ:</strong> маятник находится не далее 3 см от положения равновесия в течение $\frac{1}{3}$ периода.</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" => "1288151" "type" => "task" ] "previous" => array:2 [ "refs" => "1288149" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\BookPage {#1039 #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" => 1340388 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/page-37" "field_display_title" => "37" "field_folder" => "1" "field_image_name" => "37" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Book {#1048} ] #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 {#1040 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Element {#1117 #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" => 1518509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1518509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 …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" => "1340389" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1340387" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:9 [ 0 => App\Models\Task {#1238 #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" => 1288144 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/06-27" "field_display_title" => "6.27" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1239 …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 {#1240 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1241 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1321 …2} "content" => Illuminate\Database\Eloquent\Collection {#1315 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1314 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288144 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/06-27" "field_display_title" => "6.27" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1239 …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 {#1240 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1241 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1321 …2} "content" => Illuminate\Database\Eloquent\Collection {#1315 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1314 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1279 #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" => 1288145 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/06-28" "field_display_title" => "6.28" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …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 {#1313 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1312 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318 …2} "content" => Illuminate\Database\Eloquent\Collection {#1319 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1300 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288145 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/06-28" "field_display_title" => "6.28" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …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 {#1313 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1312 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318 …2} "content" => Illuminate\Database\Eloquent\Collection {#1319 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1300 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1282 #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" => 1288146 "created_at" => 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#observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1302 #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" => 1288147 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/06-30" "field_display_title" => "6.30" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1304 …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 {#1287 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1283 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1289 …2} "content" => Illuminate\Database\Eloquent\Collection {#1285 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1273 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288147 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/06-30" "field_display_title" => "6.30" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1304 …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 {#1287 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1283 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1289 …2} "content" => Illuminate\Database\Eloquent\Collection {#1285 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1273 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1288 #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" => 1288148 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/06-31" "field_display_title" => "6.31" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1291 …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 {#1274 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1272 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1277 …2} "content" => Illuminate\Database\Eloquent\Collection {#1276 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1259 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288148 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/06-31" "field_display_title" => "6.31" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1291 …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 {#1274 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1272 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1277 …2} "content" => Illuminate\Database\Eloquent\Collection {#1276 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1259 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1275 #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" => 1288149 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/8-klass/fizika/ileksa-gelfgat-uchebnik/07-1" "field_display_title" => "7.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1245 …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 {#1261 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1255 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1263 …2} "content" => Illuminate\Database\Eloquent\Collection {#1262 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1244 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1288149 "created_at" => "2026-04-10 13:58:26" "updated_at" => null …21 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1260 #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] } 7 => App\Models\Task {#1250 #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] } 8 => App\Models\Task {#1348 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" 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+usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№7.2 (с. 37)
Условие. №7.2 (с. 37)
скриншот условия
7.2. Маятник совершает гармонические колебания с амплитудой 6 см. Какую часть периода маятник находится не далее 3 см от положения равновесия?
Решение. №7.2 (с. 37)
Решение 2. №7.2 (с. 37)
Дано:
Амплитуда колебаний, $A = 6 \text{ см}$
Рассматриваемый диапазон смещений, $|x| \le 3 \text{ см}$
Перевод в СИ:
$A = 0.06 \text{ м}$
$|x| \le 0.03 \text{ м}$
Найти:
Часть периода $\frac{\Delta t}{T}$, в течение которой маятник находится в заданном диапазоне.
Решение:
Гармонические колебания маятника описываются уравнением $x(t) = A \sin(\omega t)$, где $\text{x}$ - смещение от положения равновесия, $\text{A}$ - амплитуда, $\omega = \frac{2\pi}{T}$ - циклическая частота, а $\text{T}$ - период колебаний. Выбор функции синуса соответствует началу движения из положения равновесия при $t=0$.
Условие, что маятник находится "не далее 3 см от положения равновесия", математически записывается в виде неравенства: $|x| \le 3$ см.
Подставим уравнение движения в это неравенство:
$|A \sin(\omega t)| \le 3$
Используя данное значение амплитуды $A = 6$ см, получим:
$|6 \sin(\omega t)| \le 3$
Разделив обе части на 6, приходим к неравенству для синуса фазы колебаний:
$|\sin(\omega t)| \le \frac{1}{2}$
Это эквивалентно двойному неравенству: $-\frac{1}{2} \le \sin(\omega t) \le \frac{1}{2}$.
Чтобы найти общую продолжительность времени, в течение которого выполняется это условие, сначала определим время $t_1$, необходимое маятнику для перемещения из положения равновесия ($x=0$) до границы указанной области ($x=3$ см).
$x(t_1) = 3 \implies A \sin(\omega t_1) = 3 \implies 6 \sin(\frac{2\pi}{T} t_1) = 3$
Отсюда получаем: $\sin(\frac{2\pi}{T} t_1) = \frac{1}{2}$.
Наименьшее положительное значение фазы $\frac{2\pi}{T} t_1$, синус которого равен $\frac{1}{2}$, составляет $\frac{\pi}{6}$ радиан.
$\frac{2\pi}{T} t_1 = \frac{\pi}{6}$
Из этого уравнения выражаем время $t_1$:
$t_1 = \frac{\pi}{6} \cdot \frac{T}{2\pi} = \frac{T}{12}$
За один полный период колебаний маятник проходит через область $|x| \le 3$ см четыре раза. В силу симметрии гармонического движения, время прохождения каждого из этих участков одинаково и равно $t_1$. Эти четыре участка соответствуют движению: от $x=0$ до $x=3$ см, от $x=3$ см до $x=0$, от $x=0$ до $x=-3$ см и от $x=-3$ см до $x=0$.
Таким образом, общее время $\Delta t$, которое маятник проводит в указанной области за один период, равно сумме времен этих четырех проходов:
$\Delta t = 4 \cdot t_1 = 4 \cdot \frac{T}{12} = \frac{T}{3}$
Искомая часть периода представляет собой отношение $\frac{\Delta t}{T}$:
$\frac{\Delta t}{T} = \frac{T/3}{T} = \frac{1}{3}$
Ответ: маятник находится не далее 3 см от положения равновесия в течение $\frac{1}{3}$ периода.
Другие задания:
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