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Номер 2.23, страница 13 - гдз по физике 9 класс задачник Артеменков, Ломаченков
Авторы: Артеменков Д. А., Ломаченков И. А., Панебратцев Ю. А.
Тип: Задачник
Серия: сферы
Издательство: Просвещение
Год издания: 2011 - 2026
Цвет обложки: синий
ISBN: 978-5-09-038216-8
Популярные ГДЗ в 9 классе
II. Механические колебания и волны - номер 2.23, страница 13.
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" => 1424137 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-23" "field_display_title" => "2.23" "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" => 1424065 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. Механические колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "10" "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 {#1081 #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" => 2549 "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 [ "field_accusative_case" => "физику" "field_creative_case" => "физикой" "field_dative_case" => "физике" "field_genitive_case" => "физики" "field_nominative_case" => "физика" "field_prepositional_case" => "физике" ] "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 [ "field_accusative_case" => "физику" "field_creative_case" => "физикой" "field_dative_case" => "физике" "field_genitive_case" => "физики" "field_nominative_case" => "физика" "field_prepositional_case" => "физике" ] "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:1 [ 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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "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 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => 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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:3 [ 0 => App\Models\Term {#1056 #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" => 3745 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Артеменков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Денис" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3745 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Артеменков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Денис" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => 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:12 [ "id" => 5014 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ломаченков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Иван" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 5014 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ломаченков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Иван" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1058 #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" => 5412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Панебратцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Анатольевич" "field_surname_rp" => "Панебратцева" ] #original: array:12 [ "id" => 5412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Панебратцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Анатольевич" "field_surname_rp" => "Панебратцева" ] #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 {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => 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:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "zadachnik" ] #original: array:10 [ "id" => 6518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Задачник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "задачник" "field_creative_case" => "задачником" "field_dative_case" => "задачнику" "field_genitive_case" => "задачника" "field_nominative_case" => "задачник" "field_prepositional_case" => "задачнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:8 [ "id" => 4795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "сферы" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "sfery" ] #original: array:8 [ "id" => 4795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "сферы" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "sfery" ] #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_umk" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "2" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1072 #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" => 37 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "переработанное и дополненное" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "pererabotannoe i dopolnennoe" ] #original: array:6 [ "id" => 37 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "переработанное и дополненное" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "pererabotannoe i dopolnennoe" ] #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 {#1073 #items: array:1 [ 0 => App\Models\Term {#1058} ] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2011" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие для общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "278" "field_priority" => null "field_default_folder" => "/fizika_09/artemenkov-z/" "field_isbn" => "978-5-09-038216-8" "field_cover" => array:1 [ 0 => "/media/fizika_09/artemenkov-z/covers/cover1.webp?ts=1758483621" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/artemenkov-z/covers/cover1.webp?ts=1758483621" "alt" => "" "width" => "1888" "height" => "2427" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/artemenkov-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\Term {#1078 #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 [ "field_accusative_case" => null "field_creative_case" => null 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Определите период колебаний груза. Постройте график колебаний груза на пружине, если амплитуда колебаний 3 см. При построении графика принять, что в момент времени $t=0, x=0$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "23-1.jpg" "alt" => null "width" => "1725" "height" => 324 "path" => "/media/fizika_09/artemenkov-z/0-2/23-1.webp?ts=1758726373" ] ] ] #original: array:7 [ "id" => 1673755 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1094} "task" => array:2 [ "refs" => "1424137" "type" => "task" ] "text" => "<p><strong>2.23</strong> На пружине жёсткостью 25 Н/м подвешен груз массой 160 г. Определите период колебаний груза. Постройте график колебаний груза на пружине, если амплитуда колебаний 3 см. При построении графика принять, что в момент времени $t=0, x=0$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "23-1.jpg" "alt" => null "width" => "1725" "height" => 324 "path" => "/media/fizika_09/artemenkov-z/0-2/23-1.webp?ts=1758726373" ] ] ] #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 {#1092 #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" => 1695297 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Edition {#1093 #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" => 6166 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Term {#1095 #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_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1099 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6166 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1097} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1098} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1099} "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" => "1424137" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Жёсткость пружины, $k = 25 \text{ Н/м}$</p><p>Масса груза, $m = 160 \text{ г}$</p><p>Амплитуда колебаний, $A = 3 \text{ см}$</p><p>Начальные условия: при $t=0$, $x=0$</p><p><em>Перевод в систему СИ:</em></p><p>Масса: $m = 160 \text{ г} = 0.16 \text{ кг}$</p><p>Амплитуда: $A = 3 \text{ см} = 0.03 \text{ м}$</p><p><strong>Найти:</strong></p><p><strong>1.</strong> Период колебаний груза, $T$</p><p><strong>2.</strong> Построить график колебаний $x(t)$</p><p><strong>Решение:</strong></p><p><strong>1.</strong> Определение периода колебаний</p><p>Колебания груза на пружине являются гармоническими. Период $T$ таких колебаний для пружинного маятника определяется по формуле Томсона:</p><p>$T = 2\pi\sqrt{\frac{m}{k}}$</p><p>Подставим известные значения в систему СИ:</p><p>$T = 2\pi\sqrt{\frac{0.16 \text{ кг}}{25 \text{ Н/м}}} = 2\pi\sqrt{0.0064 \text{ с}^2} = 2\pi \cdot 0.08 \text{ с} = 0.16\pi \text{ с}$</p><p>Вычислим приближенное значение периода:</p><p>$T \approx 0.16 \cdot 3.14159 \approx 0.503 \text{ с}$</p><p>Округлим до десятых: $T \approx 0.5 \text{ с}$.</p><p><strong>2.</strong> Построение графика колебаний</p><p>Уравнение гармонических колебаний в общем виде:</p><p>$x(t) = A \sin(\omega t + \phi_0)$</p><p>где $x(t)$ — смещение от положения равновесия в момент времени $t$, $A$ — амплитуда, $\omega$ — циклическая частота, $\phi_0$ — начальная фаза.</p><p>Амплитуда дана по условию: $A = 3 \text{ см}$.</p><p>Циклическая частота $\omega$ связана с периодом $T$ соотношением $\omega = \frac{2\pi}{T}$.</p><p>$\omega = \frac{2\pi}{0.16\pi \text{ с}} = \frac{2}{0.16} \text{ рад/с} = 12.5 \text{ рад/с}$</p><p>Начальную фазу $\phi_0$ найдём из начальных условий: в момент времени $t=0$ смещение $x=0$.</p><p>$x(0) = A \sin(\omega \cdot 0 + \phi_0) = A \sin(\phi_0)$</p><p>$0 = A \sin(\phi_0) \implies \sin(\phi_0) = 0$</p><p>Простейшее решение этого уравнения — $\phi_0 = 0$. Это означает, что в начальный момент времени тело проходит положение равновесия, двигаясь в сторону положительных значений координаты $x$.</p><p>Таким образом, уравнение движения груза имеет вид (будем измерять смещение $x$ в сантиметрах):</p><p>$x(t) = 3 \sin(12.5 t)$</p><p>Графиком этой функции является синусоида. Ось ординат — смещение $x$ (в см), ось абсцисс — время $t$ (в с). График проходит через начало координат, имеет амплитуду 3 см и период $T \approx 0.5$ с.</p><p>Ключевые точки для одного периода:</p><ul><li>$t = 0$: $x = 0$ (положение равновесия)</li><li>$t = T/4 \approx 0.125 \text{ с}$: $x = 3 \text{ см}$ (максимальное смещение)</li><li>$t = T/2 \approx 0.25 \text{ с}$: $x = 0$ (положение равновесия)</li><li>$t = 3T/4 \approx 0.375 \text{ с}$: $x = -3 \text{ см}$ (минимальное смещение)</li><li>$t = T \approx 0.5 \text{ с}$: $x = 0$ (завершение первого полного колебания)</li></ul><p>График зависимости смещения от времени $x(t)$:</p><svg width="480" height="240" viewBox="0 0 480 240" xmlns="http://www.w3.org/2000/svg" style="background-color:white; border:1px solid black; margin-top:10px;"><line x1="40" y1="120" x2="460" y2="120" stroke="black" stroke-width="2"></line><line x1="40" y1="20" x2="40" y2="220" stroke="black" stroke-width="2"></line><polygon points="460,120 450,115 450,125" fill="black"></polygon><polygon points="40,20 35,30 45,30" fill="black"></polygon><text x="465" y="125" font-family="sans-serif" font-size="14">t, c</text><text x="10" y="25" font-family="sans-serif" font-size="14">x, см</text><text x="35" y="135" font-family="sans-serif" font-size="12">0</text><line x1="140" y1="115" x2="140" y2="125" stroke="black"></line><text x="130" y="140" font-family="sans-serif" font-size="12">0,25</text><line x1="240" y1="115" x2="240" y2="125" stroke="black"></line><text x="235" y="140" font-family="sans-serif" font-size="12">0,5</text><line x1="340" y1="115" x2="340" y2="125" stroke="black"></line><text x="330" y="140" font-family="sans-serif" font-size="12">0,75</text><line x1="440" y1="115" x2="440" y2="125" stroke="black"></line><text x="435" y="140" font-family="sans-serif" font-size="12">1,0</text><line x1="35" y1="40" x2="45" y2="40" stroke="black"></line><text x="20" y="45" font-family="sans-serif" font-size="12">3</text><line x1="35" y1="200" x2="45" y2="200" stroke="black"></line><text x="15" y="205" font-family="sans-serif" font-size="12">-3</text><polyline points="40.0,120.0 44.0,96.3 48.0,73.6 52.0,52.8 56.0,34.9 60.0,20.8 64.0,11.5 68.0,7.5 72.0,9.1 76.0,15.9 80.0,27.3 84.0,42.4 88.0,59.9 92.0,78.7 96.0,97.6 100.0,115.4 104.0,131.2 108.0,144.1 112.0,153.6 116.0,159.5 120.0,161.7 124.0,160.5 128.0,156.1 132.0,149.1 136.0,139.9 140.0,129.2 144.0,117.6 148.0,105.7 152.0,94.2 156.0,83.9 160.0,75.2 164.0,68.7 168.0,64.8 172.0,63.7 176.0,65.6 180.0,70.3 184.0,77.7 188.0,87.4 192.0,98.9 196.0,111.7 200.0,124.9 204.0,137.9 208.0,149.9 212.0,160.2 216.0,168.1 220.0,173.2 224.0,175.2 228.0,174.1 232.0,170.1 236.0,163.6 240.0,155.1 244.0,145.1 248.0,134.1 252.0,122.7 256.0,111.6 260.0,101.3 264.0,92.5 268.0,85.6 272.0,81.0 276.0,78.9 280.0,80.0 284.0,84.0 288.0,90.8 292.0,100.0 296.0,111.1 300.0,123.5 304.0,136.5 308.0,149.4 312.0,161.4 316.0,171.8 320.0,179.9 324.0,185.2 328.0,187.3 332.0,186.2 336.0,181.9 340.0,175.1 344.0,166.1 348.0,155.4 352.0,143.7 356.0,131.5 360.0,119.5 364.0,108.4 368.0,98.6 372.0,90.7 376.0,85.1 380.0,81.9 384.0,81.4 388.0,83.5 392.0,88.1 396.0,95.0 400.0,103.8 404.0,114.1 408.0,125.4 412.0,137.0 416.0,148.1 420.0,158.2 424.0,166.5 428.0,172.6 432.0,176.1 436.0,176.9 440.0,175.0" stroke="blue" stroke-width="2" fill="none"></polyline></svg><p><strong>Ответ:</strong></p><p>Период колебаний груза составляет $T = 0.16\pi \text{ с} \approx 0.5 \text{ с}$.</p><p>Уравнение колебаний, описывающее движение груза: $x(t) = 3 \sin(12.5 t)$, где смещение $x$ выражено в сантиметрах, а время $t$ — в секундах. График колебаний представлен в решении.</p>" ] #original: array:6 [ "id" => 1695297 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102} "task" => array:2 [ "refs" => "1424137" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Жёсткость пружины, $k = 25 \text{ Н/м}$</p><p>Масса груза, $m = 160 \text{ г}$</p><p>Амплитуда колебаний, $A = 3 \text{ см}$</p><p>Начальные условия: при $t=0$, $x=0$</p><p><em>Перевод в систему СИ:</em></p><p>Масса: $m = 160 \text{ г} = 0.16 \text{ кг}$</p><p>Амплитуда: $A = 3 \text{ см} = 0.03 \text{ м}$</p><p><strong>Найти:</strong></p><p><strong>1.</strong> Период колебаний груза, $T$</p><p><strong>2.</strong> Построить график колебаний $x(t)$</p><p><strong>Решение:</strong></p><p><strong>1.</strong> Определение периода колебаний</p><p>Колебания груза на пружине являются гармоническими. Период $T$ таких колебаний для пружинного маятника определяется по формуле Томсона:</p><p>$T = 2\pi\sqrt{\frac{m}{k}}$</p><p>Подставим известные значения в систему СИ:</p><p>$T = 2\pi\sqrt{\frac{0.16 \text{ кг}}{25 \text{ Н/м}}} = 2\pi\sqrt{0.0064 \text{ с}^2} = 2\pi \cdot 0.08 \text{ с} = 0.16\pi \text{ с}$</p><p>Вычислим приближенное значение периода:</p><p>$T \approx 0.16 \cdot 3.14159 \approx 0.503 \text{ с}$</p><p>Округлим до десятых: $T \approx 0.5 \text{ с}$.</p><p><strong>2.</strong> Построение графика колебаний</p><p>Уравнение гармонических колебаний в общем виде:</p><p>$x(t) = A \sin(\omega t + \phi_0)$</p><p>где $x(t)$ — смещение от положения равновесия в момент времени $t$, $A$ — амплитуда, $\omega$ — циклическая частота, $\phi_0$ — начальная фаза.</p><p>Амплитуда дана по условию: $A = 3 \text{ см}$.</p><p>Циклическая частота $\omega$ связана с периодом $T$ соотношением $\omega = \frac{2\pi}{T}$.</p><p>$\omega = \frac{2\pi}{0.16\pi \text{ с}} = \frac{2}{0.16} \text{ рад/с} = 12.5 \text{ рад/с}$</p><p>Начальную фазу $\phi_0$ найдём из начальных условий: в момент времени $t=0$ смещение $x=0$.</p><p>$x(0) = A \sin(\omega \cdot 0 + \phi_0) = A \sin(\phi_0)$</p><p>$0 = A \sin(\phi_0) \implies \sin(\phi_0) = 0$</p><p>Простейшее решение этого уравнения — $\phi_0 = 0$. Это означает, что в начальный момент времени тело проходит положение равновесия, двигаясь в сторону положительных значений координаты $x$.</p><p>Таким образом, уравнение движения груза имеет вид (будем измерять смещение $x$ в сантиметрах):</p><p>$x(t) = 3 \sin(12.5 t)$</p><p>Графиком этой функции является синусоида. Ось ординат — смещение $x$ (в см), ось абсцисс — время $t$ (в с). График проходит через начало координат, имеет амплитуду 3 см и период $T \approx 0.5$ с.</p><p>Ключевые точки для одного периода:</p><ul><li>$t = 0$: $x = 0$ (положение равновесия)</li><li>$t = T/4 \approx 0.125 \text{ с}$: $x = 3 \text{ см}$ (максимальное смещение)</li><li>$t = T/2 \approx 0.25 \text{ с}$: $x = 0$ (положение равновесия)</li><li>$t = 3T/4 \approx 0.375 \text{ с}$: $x = -3 \text{ см}$ (минимальное смещение)</li><li>$t = T \approx 0.5 \text{ с}$: $x = 0$ (завершение первого полного колебания)</li></ul><p>График зависимости смещения от времени $x(t)$:</p><svg width="480" height="240" viewBox="0 0 480 240" xmlns="http://www.w3.org/2000/svg" style="background-color:white; border:1px solid black; margin-top:10px;"><line x1="40" y1="120" x2="460" y2="120" stroke="black" stroke-width="2"></line><line x1="40" y1="20" x2="40" y2="220" stroke="black" stroke-width="2"></line><polygon points="460,120 450,115 450,125" fill="black"></polygon><polygon points="40,20 35,30 45,30" fill="black"></polygon><text x="465" y="125" font-family="sans-serif" font-size="14">t, c</text><text x="10" y="25" font-family="sans-serif" font-size="14">x, см</text><text x="35" y="135" font-family="sans-serif" font-size="12">0</text><line x1="140" y1="115" x2="140" y2="125" stroke="black"></line><text x="130" y="140" font-family="sans-serif" font-size="12">0,25</text><line x1="240" y1="115" x2="240" y2="125" stroke="black"></line><text x="235" y="140" font-family="sans-serif" font-size="12">0,5</text><line x1="340" y1="115" x2="340" y2="125" stroke="black"></line><text x="330" y="140" font-family="sans-serif" font-size="12">0,75</text><line x1="440" y1="115" x2="440" y2="125" stroke="black"></line><text x="435" y="140" font-family="sans-serif" font-size="12">1,0</text><line x1="35" y1="40" x2="45" y2="40" stroke="black"></line><text x="20" y="45" font-family="sans-serif" font-size="12">3</text><line x1="35" y1="200" x2="45" y2="200" stroke="black"></line><text x="15" y="205" font-family="sans-serif" font-size="12">-3</text><polyline points="40.0,120.0 44.0,96.3 48.0,73.6 52.0,52.8 56.0,34.9 60.0,20.8 64.0,11.5 68.0,7.5 72.0,9.1 76.0,15.9 80.0,27.3 84.0,42.4 88.0,59.9 92.0,78.7 96.0,97.6 100.0,115.4 104.0,131.2 108.0,144.1 112.0,153.6 116.0,159.5 120.0,161.7 124.0,160.5 128.0,156.1 132.0,149.1 136.0,139.9 140.0,129.2 144.0,117.6 148.0,105.7 152.0,94.2 156.0,83.9 160.0,75.2 164.0,68.7 168.0,64.8 172.0,63.7 176.0,65.6 180.0,70.3 184.0,77.7 188.0,87.4 192.0,98.9 196.0,111.7 200.0,124.9 204.0,137.9 208.0,149.9 212.0,160.2 216.0,168.1 220.0,173.2 224.0,175.2 228.0,174.1 232.0,170.1 236.0,163.6 240.0,155.1 244.0,145.1 248.0,134.1 252.0,122.7 256.0,111.6 260.0,101.3 264.0,92.5 268.0,85.6 272.0,81.0 276.0,78.9 280.0,80.0 284.0,84.0 288.0,90.8 292.0,100.0 296.0,111.1 300.0,123.5 304.0,136.5 308.0,149.4 312.0,161.4 316.0,171.8 320.0,179.9 324.0,185.2 328.0,187.3 332.0,186.2 336.0,181.9 340.0,175.1 344.0,166.1 348.0,155.4 352.0,143.7 356.0,131.5 360.0,119.5 364.0,108.4 368.0,98.6 372.0,90.7 376.0,85.1 380.0,81.9 384.0,81.4 388.0,83.5 392.0,88.1 396.0,95.0 400.0,103.8 404.0,114.1 408.0,125.4 412.0,137.0 416.0,148.1 420.0,158.2 424.0,166.5 428.0,172.6 432.0,176.1 436.0,176.9 440.0,175.0" stroke="blue" stroke-width="2" fill="none"></polyline></svg><p><strong>Ответ:</strong></p><p>Период колебаний груза составляет $T = 0.16\pi \text{ с} \approx 0.5 \text{ с}$.</p><p>Уравнение колебаний, описывающее движение груза: $x(t) = 3 \sin(12.5 t)$, где смещение $x$ выражено в сантиметрах, а время $t$ — в секундах. График колебаний представлен в решении.</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" => "1424138" "type" => "task" ] "previous" => array:2 [ "refs" => "1424136" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\BookPage {#1080 #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" => 1424363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_url" => "/9-klass/fizika/artemenkov-zadachnik/page-13" "field_display_title" => "13" "field_folder" => "1" "field_image_name" => "13" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1100 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1101 #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 {#1103 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Element {#1112 #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" => 1674173 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1114 #items: array:1 [ 0 => App\Models\Edition {#1086} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1994" "height" => 2505 "path" => "/media/fizika_09/artemenkov-z/0-1/13-1.webp?ts=1758727968" ] ] ] #original: array:6 [ "id" => 1674173 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1114} "book_page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1994" "height" => 2505 "path" => "/media/fizika_09/artemenkov-z/0-1/13-1.webp?ts=1758727968" ] ] ] #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" => "1424364" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1424362" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:6 [ 0 => App\Models\Task {#1181 #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" => 1424137 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-23" "field_display_title" => "2.23" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1183 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1184 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:2 [ 0 => App\Models\Element {#1085} 1 => App\Models\Element {#1092} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1424138" "type" => "task" ] "previous" => array:2 [ "refs" => "1424136" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #original: array:24 [ "id" => 1424137 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-23" "field_display_title" => "2.23" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1182} "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 {#1183} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1184} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1185} "content" => Illuminate\Database\Eloquent\Collection {#1186} "next" => array:2 [ "refs" => "1424138" "type" => "task" ] "previous" => array:2 [ "refs" => "1424136" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1187} "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #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 {#1188 #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" => 1424138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-24" "field_display_title" => "2.24" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1190 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:2 [ 0 => App\Models\Element {#1203 #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 [ …7] #original: array:7 [ …7] #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\Element {#1205 #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" => "1424139" "type" => "task" ] "previous" => array:2 [ "refs" => "1424137" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #original: array:24 [ "id" => 1424138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-24" "field_display_title" => "2.24" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1189} "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 {#1190} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192} "content" => Illuminate\Database\Eloquent\Collection {#1194} "next" => array:2 [ "refs" => "1424139" "type" => "task" ] "previous" => array:2 [ "refs" => "1424137" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1200} "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #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 {#1195 #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" => 1424139 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-25" "field_display_title" => "2.25" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1193 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1202 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:2 [ 0 => App\Models\Element {#1215 #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 [ …7] #original: array:7 [ …7] #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\Element {#1217 #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" => "1424140" "type" => "task" ] "previous" => array:2 [ "refs" => "1424138" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #original: array:24 [ "id" => 1424139 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-25" "field_display_title" => "2.25" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1193} "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 {#1202} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1198} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1196} "content" => Illuminate\Database\Eloquent\Collection {#1197} "next" => array:2 [ "refs" => "1424140" "type" => "task" ] "previous" => array:2 [ "refs" => "1424138" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1212} "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #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\Task {#1207 #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" => 1424140 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-26" "field_display_title" => "2.26" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1214 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1209 #items: array:2 [ 0 => App\Models\Element {#1227 #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 [ …7] #original: array:7 [ …7] #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\Element {#1229 #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" => "1424141" "type" => "task" ] "previous" => array:2 [ "refs" => "1424139" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #original: array:24 [ "id" => 1424140 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-26" "field_display_title" => "2.26" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199} "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 {#1214} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1210} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1208} "content" => Illuminate\Database\Eloquent\Collection {#1209} "next" => array:2 [ "refs" => "1424141" "type" => "task" ] "previous" => array:2 [ "refs" => "1424139" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224} "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #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 {#1219 #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" => 1424141 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-27" "field_display_title" => "2.27" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1226 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:2 [ 0 => App\Models\Element {#1239 #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 [ …7] #original: array:7 [ …7] #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\Element {#1241 #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" => "1424142" "type" => "task" ] "previous" => array:2 [ "refs" => "1424140" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #original: array:24 [ "id" => 1424141 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-27" "field_display_title" => "2.27" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1211} "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 {#1226} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1222} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1220} "content" => Illuminate\Database\Eloquent\Collection {#1221} "next" => array:2 [ "refs" => "1424142" "type" => "task" ] "previous" => array:2 [ "refs" => "1424140" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1236} "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #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 {#1231 #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" => 1424142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-28" "field_display_title" => "2.28" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1238 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:2 [ 0 => App\Models\Element {#1251 #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 [ …7] #original: array:7 [ …7] #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\Element {#1253 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1424143" "type" => "task" ] "previous" => array:2 [ "refs" => "1424141" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1424363" "type" => "book_page" ] ] #original: array:24 [ "id" => 1424142 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "13" "field_page_end" => null "field_url" => "/9-klass/fizika/artemenkov-zadachnik/2-28" "field_display_title" => "2.28" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1223} "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 {#1238} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1234} "parent_branches" => 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№2.23 (с. 13)
Условие. №2.23 (с. 13)
скриншот условия
2.23 На пружине жёсткостью 25 Н/м подвешен груз массой 160 г. Определите период колебаний груза. Постройте график колебаний груза на пружине, если амплитуда колебаний 3 см. При построении графика принять, что в момент времени $t=0, x=0$.
Решение. №2.23 (с. 13)
Дано:
Жёсткость пружины, $k = 25 \text{ Н/м}$
Масса груза, $m = 160 \text{ г}$
Амплитуда колебаний, $A = 3 \text{ см}$
Начальные условия: при $t=0$, $x=0$
Перевод в систему СИ:
Масса: $m = 160 \text{ г} = 0.16 \text{ кг}$
Амплитуда: $A = 3 \text{ см} = 0.03 \text{ м}$
Найти:
1. Период колебаний груза, $T$
2. Построить график колебаний $x(t)$
Решение:
1. Определение периода колебаний
Колебания груза на пружине являются гармоническими. Период $T$ таких колебаний для пружинного маятника определяется по формуле Томсона:
$T = 2\pi\sqrt{\frac{m}{k}}$
Подставим известные значения в систему СИ:
$T = 2\pi\sqrt{\frac{0.16 \text{ кг}}{25 \text{ Н/м}}} = 2\pi\sqrt{0.0064 \text{ с}^2} = 2\pi \cdot 0.08 \text{ с} = 0.16\pi \text{ с}$
Вычислим приближенное значение периода:
$T \approx 0.16 \cdot 3.14159 \approx 0.503 \text{ с}$
Округлим до десятых: $T \approx 0.5 \text{ с}$.
2. Построение графика колебаний
Уравнение гармонических колебаний в общем виде:
$x(t) = A \sin(\omega t + \phi_0)$
где $x(t)$ — смещение от положения равновесия в момент времени $t$, $A$ — амплитуда, $\omega$ — циклическая частота, $\phi_0$ — начальная фаза.
Амплитуда дана по условию: $A = 3 \text{ см}$.
Циклическая частота $\omega$ связана с периодом $T$ соотношением $\omega = \frac{2\pi}{T}$.
$\omega = \frac{2\pi}{0.16\pi \text{ с}} = \frac{2}{0.16} \text{ рад/с} = 12.5 \text{ рад/с}$
Начальную фазу $\phi_0$ найдём из начальных условий: в момент времени $t=0$ смещение $x=0$.
$x(0) = A \sin(\omega \cdot 0 + \phi_0) = A \sin(\phi_0)$
$0 = A \sin(\phi_0) \implies \sin(\phi_0) = 0$
Простейшее решение этого уравнения — $\phi_0 = 0$. Это означает, что в начальный момент времени тело проходит положение равновесия, двигаясь в сторону положительных значений координаты $x$.
Таким образом, уравнение движения груза имеет вид (будем измерять смещение $x$ в сантиметрах):
$x(t) = 3 \sin(12.5 t)$
Графиком этой функции является синусоида. Ось ординат — смещение $x$ (в см), ось абсцисс — время $t$ (в с). График проходит через начало координат, имеет амплитуду 3 см и период $T \approx 0.5$ с.
Ключевые точки для одного периода:
- $t = 0$: $x = 0$ (положение равновесия)
- $t = T/4 \approx 0.125 \text{ с}$: $x = 3 \text{ см}$ (максимальное смещение)
- $t = T/2 \approx 0.25 \text{ с}$: $x = 0$ (положение равновесия)
- $t = 3T/4 \approx 0.375 \text{ с}$: $x = -3 \text{ см}$ (минимальное смещение)
- $t = T \approx 0.5 \text{ с}$: $x = 0$ (завершение первого полного колебания)
График зависимости смещения от времени $x(t)$:
Ответ:
Период колебаний груза составляет $T = 0.16\pi \text{ с} \approx 0.5 \text{ с}$.
Уравнение колебаний, описывающее движение груза: $x(t) = 3 \sin(12.5 t)$, где смещение $x$ выражено в сантиметрах, а время $t$ — в секундах. График колебаний представлен в решении.
Другие задания:
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