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Номер 3, страница 100 - гдз по физике 9 класс учебник Изергин
Авторы: Изергин Э. Т.
Тип: Учебник
Серия: инновационная школа
Издательство: Русское слово
Год издания: 2019 - 2026
Цвет обложки: синий
ISBN: 978-5-533-01069-6
Популярные ГДЗ в 9 классе
Глава III. Механические колебания и волны. Параграф 20. Вынужденные колебания. Резонанс - номер 3, страница 100.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1439772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1439560 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => "III" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "67" "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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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" => 4169 "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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #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 {#1036 #items: array:1 [ 0 => App\Models\Term {#1037 #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 {#1038 #items: array:1 [ 0 => App\Models\Term {#1039 #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" => 5176 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Русское слово" "field_cases" => null "field_translit" => "russkoe slovo" ] #original: array:6 [ "id" => 5176 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Русское слово" "field_cases" => null "field_translit" => "russkoe slovo" ] #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 {#1040 #items: array:1 [ 0 => App\Models\Term {#1041 #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" => 4491 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Изергин" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Тимофеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4491 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Изергин" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Тимофеевич" "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 {#1042 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Term {#1044 #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" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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 {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 #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 {#1047 #items: array:1 [ 0 => App\Models\Term {#1048 #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 {#1049 #items: array:1 [ 0 => App\Models\Term {#1050 #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" => 4715 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "инновационная школа" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "innovacionnaya shkola" ] #original: array:8 [ "id" => 4715 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "инновационная школа" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "innovacionnaya shkola" ] #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 {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => 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:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "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_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => 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_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 9 класса общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "388" "field_priority" => null "field_default_folder" => "/fizika_09/izergin-u/" "field_isbn" => "978-5-533-01069-6" "field_cover" => array:1 [ 0 => "/media/fizika_09/izergin-u/covers/cover1.webp?ts=1759162386" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/izergin-u/covers/cover1.webp?ts=1759162386" "alt" => "" "width" => "1881" "height" => "2502" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/izergin-uchebnik" "field_cover_color" => 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: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 "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "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" => "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" => 380 "subject" => 556 "class_subject" => 493 ] ] #original: array:50 [ "id" => 4169 "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 {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038} "field_author" => Illuminate\Database\Eloquent\Collection {#1040} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1042} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1043} "field_country" => Illuminate\Database\Eloquent\Collection {#1045} "field_city" => Illuminate\Database\Eloquent\Collection {#1047} "field_series" => Illuminate\Database\Eloquent\Collection {#1049} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 9 класса общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "388" "field_priority" => null "field_default_folder" => "/fizika_09/izergin-u/" "field_isbn" => "978-5-533-01069-6" "field_cover" => array:1 [ 0 => "/media/fizika_09/izergin-u/covers/cover1.webp?ts=1759162386" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/izergin-u/covers/cover1.webp?ts=1759162386" "alt" => "" "width" => "1881" "height" => "2502" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1057} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059} "field_url" => "/9-klass/fizika/izergin-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #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 {#1062 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1439560 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => "III" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "67" "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1062} ] #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 {#1063 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1064 #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" => 1439567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Вынужденные колебания. Резонанс" "field_branch_order" => "20" "field_url" => null "field_branch_type" => 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:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "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_page_start" => "98" "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 {#1067 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Branch {#1069 #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" => 1439560 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => "III" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Term {#1033} ] #escapeWhenCastingToString: false } "field_page_start" => "67" "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 {#1071 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1439560 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => "III" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_page_start" => "67" "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 {#1071} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1072} ] #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" => 1439567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Вынужденные колебания. Резонанс" "field_branch_order" => "20" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1065} "field_page_start" => "98" "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 {#1067} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1073 #items: array:2 [ 0 => App\Models\Element {#1074 #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" => 1707317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\Edition {#1076 #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" => 6139 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => 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 [ …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, image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "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 {#1079 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6139 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1077} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "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 {#1079} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081} "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" => "1439772" "type" => "task" ] "text" => "<p><strong>3.</strong> Изобразите графики зависимости $x(t)$ для свободных незатухающих и затухающих колебаний. Что можно сказать о периоде и частоте затухающих колебаний?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1420" "height" => 251 "path" => "/media/fizika_09/izergin-u/0-20/3-1.webp?ts=1759529230" ] ] ] #original: array:7 [ "id" => 1707317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1075} "task" => array:2 [ "refs" => "1439772" "type" => "task" ] "text" => "<p><strong>3.</strong> Изобразите графики зависимости $x(t)$ для свободных незатухающих и затухающих колебаний. Что можно сказать о периоде и частоте затухающих колебаний?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1420" "height" => 251 "path" => "/media/fizika_09/izergin-u/0-20/3-1.webp?ts=1759529230" ] ] ] #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 {#1082 #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" => 1813335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1083 #items: array:1 [ 0 => App\Models\Edition {#1084 #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" => 6646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Term {#1086 #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" => "0" "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 {#1087 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => "0" "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 {#1087} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1089} "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" => "1439772" "type" => "task" ] "text" => "<p><strong>Решение</strong></p><p><strong>Графики зависимости x(t) для свободных незатухающих и затухающих колебаний</strong></p><p>Свободные колебания — это колебания, которые происходят в системе, выведенной из положения равновесия и предоставленной самой себе. Они могут быть незатухающими (в идеализированных системах без потерь энергии) и затухающими (в реальных системах с трением или сопротивлением).</p><p><em>Свободные незатухающие колебания</em><br>Это гармонические колебания с постоянной амплитудой и периодом. Уравнение, описывающее смещение $\text{x}$ от положения равновесия в зависимости от времени $\text{t}$, имеет вид:<br>$x(t) = A \cos(\omega_0 t + \phi_0)$<br>Здесь $\text{A}$ — постоянная амплитуда, $\omega_0$ — собственная циклическая частота, $\phi_0$ — начальная фаза. График таких колебаний — это синусоида (или косинусоида).</p><svg width="400" height="200" xmlns="http://www.w3.org/2000/svg" style="background-color:white; border:1px solid black; margin:10px 0;"><title>График свободных незатухающих колебаний</title><!-- Оси --><line x1="20" y1="100" x2="380" y2="100" stroke="black" stroke-width="1.5" marker-end="url(#arrow)"></line><line x1="30" y1="180" x2="30" y2="20" stroke="black" stroke-width="1.5" marker-end="url(#arrow)"></line><defs><marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"><path d="M 0 0 L 10 5 L 0 10 z" fill="black"></path></marker></defs><!-- Метки на осях --><text x="385" y="105" font-family="KaTeX_Math" font-style="italic">t</text><text x="10" y="25" font-family="KaTeX_Math" font-style="italic">x(t)</text><text x="35" y="35" font-family="KaTeX_Math" font-style="italic">A</text><line x1="25" y1="30" x2="35" y2="30" stroke="black" stroke-dasharray="2,2"></line><text x="35" y="175" font-family="KaTeX_Math" font-style="italic">-A</text><line x1="25" y1="170" x2="35" y2="170" stroke="black" stroke-dasharray="2,2"></line><!-- График --><path d="M 30 30 C 70 30, 70 170, 110 170 S 150 30, 190 30 S 230 170, 270 170 S 310 30, 350 30" stroke="#007bff" stroke-width="2" fill="none"></path><!-- Период --><line x1="110" y1="110" x2="270" y2="110" stroke="black" stroke-width="1"></line><line x1="110" y1="105" x2="110" y2="115" stroke="black" stroke-width="1"></line><line x1="270" y1="105" x2="270" y2="115" stroke="black" stroke-width="1"></line><text x="185" y="125" font-family="KaTeX_Math" font-style="italic">T₀</text></svg><p><em>Свободные затухающие колебания</em><br>В реальных условиях энергия колебательной системы постепенно расходуется на преодоление сил сопротивления, поэтому амплитуда колебаний со временем уменьшается. Уравнение затухающих колебаний:<br>$x(t) = A_0 e^{-\beta t} \cos(\omega t + \phi_0)$<br>Здесь $A_0$ — начальная амплитуда, $\beta$ — коэффициент затухания, $\omega$ — частота затухающих колебаний. Амплитуда $A(t) = A_0 e^{-\beta t}$ убывает по экспоненциальному закону.</p><svg width="400" height="200" xmlns="http://www.w3.org/2000/svg" style="background-color:white; border:1px solid black; margin:10px 0;"><title>График свободных затухающих колебаний</title><!-- Оси --><line x1="20" y1="100" x2="380" y2="100" stroke="black" stroke-width="1.5" marker-end="url(#arrow)"></line><line x1="30" y1="180" x2="30" y2="20" stroke="black" stroke-width="1.5" marker-end="url(#arrow)"></line><!-- Метки на осях --><text x="385" y="105" font-family="KaTeX_Math" font-style="italic">t</text><text x="10" y="25" font-family="KaTeX_Math" font-style="italic">x(t)</text><text x="35" y="35" font-family="KaTeX_Math" font-style="italic">A₀</text><!-- Огибающие --><path d="M 30 30 C 130 60, 230 80, 380 95" stroke="red" stroke-dasharray="4" stroke-width="1.5" fill="none"></path><path d="M 30 170 C 130 140, 230 120, 380 105" stroke="red" stroke-dasharray="4" stroke-width="1.5" fill="none"></path><!-- График --><path d="M 30 30 C 55 30, 65 158, 95 145 C 125 132, 125 45, 155 58 C 185 71, 185 132, 215 122 C 245 112, 245 78, 275 88 C 305 98, 305 112, 335 106 C 365 100, 370 95, 380 98" stroke="#007bff" stroke-width="2" fill="none"></path><!-- Период --><line x1="155" y1="110" x2="335" y2="110" stroke="black" stroke-width="1"></line><line x1="155" y1="105" x2="155" y2="115" stroke="black" stroke-width="1"></line><line x1="335" y1="105" x2="335" y2="115" stroke="black" stroke-width="1"></line><text x="240" y="125" font-family="KaTeX_Math" font-style="italic">T</text></svg><p><strong>Что можно сказать о периоде и частоте затухающих колебаний?</strong></p><p>Наличие сил сопротивления (затухания) не только уменьшает амплитуду, но и изменяет частоту и период колебаний. Циклическая частота затухающих колебаний $\omega$ связана с собственной частотой $\omega_0$ (частотой колебаний той же системы без затухания) и коэффициентом затухания $\beta$ следующим образом:<br>$\omega = \sqrt{\omega_0^2 - \beta^2}$<br>Из этой формулы следует, что частота затухающих колебаний $\omega$ всегда <strong>меньше</strong> собственной частоты $\omega_0$.<br>$\omega < \omega_0$<br>Период колебаний $\text{T}$ обратно пропорционален частоте ($T = 2\pi / \omega$). Следовательно, для периода затухающих колебаний $\text{T}$ и периода собственных незатухающих колебаний $T_0 = 2\pi/\omega_0$ справедливо:<br>$T = \frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\omega_0^2 - \beta^2}}$<br>Так как знаменатель в выражении для $\text{T}$ меньше, чем $\omega_0$, то период затухающих колебаний $\text{T}$ всегда <strong>больше</strong> периода собственных колебаний $T_0$.<br>$T > T_0$<br>Таким образом, затухание в системе приводит к замедлению колебаний: их частота уменьшается, а период увеличивается.</p><p><strong>Ответ:</strong> График свободных незатухающих колебаний представляет собой синусоиду с постоянной амплитудой. График свободных затухающих колебаний — это синусоида, амплитуда которой убывает со временем по экспоненциальному закону (см. графики выше). Период затухающих колебаний больше, а их частота — меньше, чем соответствующие период и частота свободных незатухающих колебаний для той же колебательной системы.</p>" ] #original: array:6 [ "id" => 1813335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1083} "task" => array:2 [ "refs" => "1439772" "type" => "task" ] "text" => "<p><strong>Решение</strong></p><p><strong>Графики зависимости x(t) для свободных незатухающих и затухающих колебаний</strong></p><p>Свободные колебания — это колебания, которые происходят в системе, выведенной из положения равновесия и предоставленной самой себе. Они могут быть незатухающими (в идеализированных системах без потерь энергии) и затухающими (в реальных системах с трением или сопротивлением).</p><p><em>Свободные незатухающие колебания</em><br>Это гармонические колебания с постоянной амплитудой и периодом. Уравнение, описывающее смещение $\text{x}$ от положения равновесия в зависимости от времени $\text{t}$, имеет вид:<br>$x(t) = A \cos(\omega_0 t + \phi_0)$<br>Здесь $\text{A}$ — постоянная амплитуда, $\omega_0$ — собственная циклическая частота, $\phi_0$ — начальная фаза. График таких колебаний — это синусоида (или косинусоида).</p><svg width="400" height="200" xmlns="http://www.w3.org/2000/svg" style="background-color:white; border:1px solid black; margin:10px 0;"><title>График свободных незатухающих колебаний</title><!-- Оси --><line x1="20" y1="100" x2="380" y2="100" stroke="black" stroke-width="1.5" marker-end="url(#arrow)"></line><line x1="30" y1="180" x2="30" y2="20" stroke="black" stroke-width="1.5" marker-end="url(#arrow)"></line><defs><marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"><path d="M 0 0 L 10 5 L 0 10 z" fill="black"></path></marker></defs><!-- Метки на осях --><text x="385" y="105" font-family="KaTeX_Math" font-style="italic">t</text><text x="10" y="25" font-family="KaTeX_Math" font-style="italic">x(t)</text><text x="35" y="35" font-family="KaTeX_Math" font-style="italic">A</text><line x1="25" y1="30" x2="35" y2="30" stroke="black" stroke-dasharray="2,2"></line><text x="35" y="175" font-family="KaTeX_Math" font-style="italic">-A</text><line x1="25" y1="170" x2="35" y2="170" stroke="black" stroke-dasharray="2,2"></line><!-- График --><path d="M 30 30 C 70 30, 70 170, 110 170 S 150 30, 190 30 S 230 170, 270 170 S 310 30, 350 30" stroke="#007bff" stroke-width="2" fill="none"></path><!-- Период --><line x1="110" y1="110" x2="270" y2="110" stroke="black" stroke-width="1"></line><line x1="110" y1="105" x2="110" y2="115" stroke="black" stroke-width="1"></line><line x1="270" y1="105" x2="270" y2="115" stroke="black" stroke-width="1"></line><text x="185" y="125" font-family="KaTeX_Math" font-style="italic">T₀</text></svg><p><em>Свободные затухающие колебания</em><br>В реальных условиях энергия колебательной системы постепенно расходуется на преодоление сил сопротивления, поэтому амплитуда колебаний со временем уменьшается. Уравнение затухающих колебаний:<br>$x(t) = A_0 e^{-\beta t} \cos(\omega t + \phi_0)$<br>Здесь $A_0$ — начальная амплитуда, $\beta$ — коэффициент затухания, $\omega$ — частота затухающих колебаний. Амплитуда $A(t) = A_0 e^{-\beta t}$ убывает по экспоненциальному закону.</p><svg width="400" height="200" xmlns="http://www.w3.org/2000/svg" style="background-color:white; border:1px solid black; margin:10px 0;"><title>График свободных затухающих колебаний</title><!-- Оси --><line x1="20" y1="100" x2="380" y2="100" stroke="black" stroke-width="1.5" marker-end="url(#arrow)"></line><line x1="30" y1="180" x2="30" y2="20" stroke="black" stroke-width="1.5" marker-end="url(#arrow)"></line><!-- Метки на осях --><text x="385" y="105" font-family="KaTeX_Math" font-style="italic">t</text><text x="10" y="25" font-family="KaTeX_Math" font-style="italic">x(t)</text><text x="35" y="35" font-family="KaTeX_Math" font-style="italic">A₀</text><!-- Огибающие --><path d="M 30 30 C 130 60, 230 80, 380 95" stroke="red" stroke-dasharray="4" stroke-width="1.5" fill="none"></path><path d="M 30 170 C 130 140, 230 120, 380 105" stroke="red" stroke-dasharray="4" stroke-width="1.5" fill="none"></path><!-- График --><path d="M 30 30 C 55 30, 65 158, 95 145 C 125 132, 125 45, 155 58 C 185 71, 185 132, 215 122 C 245 112, 245 78, 275 88 C 305 98, 305 112, 335 106 C 365 100, 370 95, 380 98" stroke="#007bff" stroke-width="2" fill="none"></path><!-- Период --><line x1="155" y1="110" x2="335" y2="110" stroke="black" stroke-width="1"></line><line x1="155" y1="105" x2="155" y2="115" stroke="black" stroke-width="1"></line><line x1="335" y1="105" x2="335" y2="115" stroke="black" stroke-width="1"></line><text x="240" y="125" font-family="KaTeX_Math" font-style="italic">T</text></svg><p><strong>Что можно сказать о периоде и частоте затухающих колебаний?</strong></p><p>Наличие сил сопротивления (затухания) не только уменьшает амплитуду, но и изменяет частоту и период колебаний. Циклическая частота затухающих колебаний $\omega$ связана с собственной частотой $\omega_0$ (частотой колебаний той же системы без затухания) и коэффициентом затухания $\beta$ следующим образом:<br>$\omega = \sqrt{\omega_0^2 - \beta^2}$<br>Из этой формулы следует, что частота затухающих колебаний $\omega$ всегда <strong>меньше</strong> собственной частоты $\omega_0$.<br>$\omega < \omega_0$<br>Период колебаний $\text{T}$ обратно пропорционален частоте ($T = 2\pi / \omega$). Следовательно, для периода затухающих колебаний $\text{T}$ и периода собственных незатухающих колебаний $T_0 = 2\pi/\omega_0$ справедливо:<br>$T = \frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\omega_0^2 - \beta^2}}$<br>Так как знаменатель в выражении для $\text{T}$ меньше, чем $\omega_0$, то период затухающих колебаний $\text{T}$ всегда <strong>больше</strong> периода собственных колебаний $T_0$.<br>$T > T_0$<br>Таким образом, затухание в системе приводит к замедлению колебаний: их частота уменьшается, а период увеличивается.</p><p><strong>Ответ:</strong> График свободных незатухающих колебаний представляет собой синусоиду с постоянной амплитудой. График свободных затухающих колебаний — это синусоида, амплитуда которой убывает со временем по экспоненциальному закону (см. графики выше). Период затухающих колебаний больше, а их частота — меньше, чем соответствующие период и частота свободных незатухающих колебаний для той же колебательной системы.</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" => "1439773" "type" => "task" ] "previous" => array:2 [ "refs" => "1439771" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1090 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:1 [ 0 => App\Models\BookPage {#1095 #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" => 1439418 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_url" => "/9-klass/fizika/izergin-uchebnik/page-100" "field_display_title" => "100" "field_folder" => "1" "field_image_name" => "100" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Book {#1099 #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" => 4169 "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 {#1100 #items: array:1 [ 0 => App\Models\Term {#1101 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Term {#1103 #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_publisher" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Term {#1105 #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_author" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Term {#1107 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] …17 } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1108 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Term {#1110 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Term {#1112 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Term {#1114 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Term {#1116 …30} ] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1118 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 …30} ] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 9 класса общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "388" "field_priority" => null "field_default_folder" => "/fizika_09/izergin-u/" "field_isbn" => "978-5-533-01069-6" "field_cover" => array:1 [ 0 => "/media/fizika_09/izergin-u/covers/cover1.webp?ts=1759162386" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/izergin-u/covers/cover1.webp?ts=1759162386" "alt" => "" "width" => "1881" "height" => "2502" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1124 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/fizika/izergin-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 …30} ] #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" => 380 "subject" => 556 "class_subject" => 493 ] ] #original: array:50 [ "id" => 4169 "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 {#1100} "field_class" => Illuminate\Database\Eloquent\Collection {#1102} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1104} "field_author" => Illuminate\Database\Eloquent\Collection {#1106} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1108} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1109} "field_country" => Illuminate\Database\Eloquent\Collection {#1111} "field_city" => Illuminate\Database\Eloquent\Collection {#1113} "field_series" => Illuminate\Database\Eloquent\Collection {#1115} "field_umk" => Illuminate\Database\Eloquent\Collection {#1117} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1118} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1119} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1122} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 9 класса общеобразовательных организаций" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "388" "field_priority" => null "field_default_folder" => "/fizika_09/izergin-u/" "field_isbn" => "978-5-533-01069-6" "field_cover" => array:1 [ 0 => "/media/fizika_09/izergin-u/covers/cover1.webp?ts=1759162386" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_09/izergin-u/covers/cover1.webp?ts=1759162386" "alt" => "" "width" => "1881" "height" => "2502" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1123} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1124} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1125} "field_url" => "/9-klass/fizika/izergin-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1126} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 556 "class_subject" => 493 ] ] #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 "edition_groups" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ 0 => App\Models\Element {#1130 #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" => 1707656 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Edition {#1132 …30} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1439418" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "100-1.jpg" "alt" => null "width" => "1901" "height" => 2557 "path" => "/media/fizika_09/izergin-u/0-1/100-1.webp?ts=1759529683" ] ] ] #original: array:6 [ "id" => 1707656 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131} "book_page" => array:2 [ "refs" => "1439418" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "100-1.jpg" "alt" => null "width" => "1901" "height" => 2557 "path" => "/media/fizika_09/izergin-u/0-1/100-1.webp?ts=1759529683" ] ] ] #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" => "1439419" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1439417" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1138 #items: array:4 [ 0 => App\Models\Task {#1139 #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" => 1439770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #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 {#1142 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Branch {#1144 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:2 [ 0 => App\Models\Branch {#1150 …30} 1 => App\Models\Branch {#1154 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:2 [ 0 => App\Models\Element {#1160 …30} 1 => App\Models\Element {#1162 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1439771" "type" => "task" ] "previous" => array:2 [ "refs" => "1439769" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Book {#1099} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1439418" "type" => "book_page" ] ] #original: array:24 [ "id" => 1439770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1140} "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 {#1142} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1143} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1149} "content" => Illuminate\Database\Eloquent\Collection {#1159} "next" => array:2 [ "refs" => "1439771" "type" => "task" ] "previous" => array:2 [ "refs" => "1439769" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1170} "page" => array:2 [ "refs" => "1439418" "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 {#1171 #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" => 1439771 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #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 {#1173 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:2 [ 0 => App\Models\Branch {#1150 …30} 1 => App\Models\Branch {#1154 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:2 [ 0 => App\Models\Element {#1177 …30} 1 => App\Models\Element {#1179 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1439772" "type" => "task" ] "previous" => array:2 [ "refs" => "1439770" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Book {#1099} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1439418" "type" => "book_page" ] ] #original: array:24 [ "id" => 1439771 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1172} "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 {#1173} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1174} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1175} "content" => Illuminate\Database\Eloquent\Collection {#1176} "next" => array:2 [ "refs" => "1439772" "type" => "task" ] "previous" => array:2 [ "refs" => "1439770" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1181} "page" => array:2 [ "refs" => "1439418" "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 {#1182 #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" => 1439772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #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 {#1184 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:2 [ 0 => App\Models\Branch {#1150 …30} 1 => App\Models\Branch {#1154 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:2 [ 0 => App\Models\Element {#1188 …30} 1 => App\Models\Element {#1190 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1439773" "type" => "task" ] "previous" => array:2 [ "refs" => "1439771" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Book {#1099} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1439418" "type" => "book_page" ] ] #original: array:24 [ "id" => 1439772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1183} "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 {#1184} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1185} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1186} "content" => Illuminate\Database\Eloquent\Collection {#1187} "next" => array:2 [ "refs" => "1439773" "type" => "task" ] "previous" => array:2 [ "refs" => "1439771" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1192} "page" => array:2 [ "refs" => "1439418" "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 {#1193 #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" => 1439773 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #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 {#1195 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:2 [ 0 => App\Models\Branch {#1150 …30} 1 => App\Models\Branch {#1154 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:2 [ 0 => App\Models\Element {#1199 …30} 1 => App\Models\Element {#1201 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1439774" "type" => "task" ] "previous" => array:2 [ "refs" => "1439772" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Book {#1099} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1439418" "type" => "book_page" ] ] #original: array:24 [ "id" => 1439773 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1194} "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 {#1195} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1196} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1197} "content" => Illuminate\Database\Eloquent\Collection {#1198} "next" => array:2 [ "refs" => "1439774" "type" => "task" ] "previous" => array:2 [ "refs" => "1439772" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1203} "page" => array:2 [ "refs" => "1439418" "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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1439418 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_url" => "/9-klass/fizika/izergin-uchebnik/page-100" "field_display_title" => "100" "field_folder" => "1" "field_image_name" => "100" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1098} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1097} "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 {#1128} "content" => Illuminate\Database\Eloquent\Collection {#1129} "next" => array:2 [ "refs" => "1439419" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1439417" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1138} ] #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" => 1439772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "100" "field_page_end" => null "field_url" => "/9-klass/fizika/izergin-uchebnik/20-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1063} "content" => Illuminate\Database\Eloquent\Collection {#1073} "next" => array:2 [ "refs" => "1439773" "type" => "task" ] "previous" => array:2 [ "refs" => "1439771" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1090} "page" => array:2 [ "refs" => "1439418" "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 (с. 100)
Условие. №3 (с. 100)
скриншот условия
3. Изобразите графики зависимости $x(t)$ для свободных незатухающих и затухающих колебаний. Что можно сказать о периоде и частоте затухающих колебаний?
Решение. №3 (с. 100)
Решение
Графики зависимости x(t) для свободных незатухающих и затухающих колебаний
Свободные колебания — это колебания, которые происходят в системе, выведенной из положения равновесия и предоставленной самой себе. Они могут быть незатухающими (в идеализированных системах без потерь энергии) и затухающими (в реальных системах с трением или сопротивлением).
Свободные незатухающие колебания
Это гармонические колебания с постоянной амплитудой и периодом. Уравнение, описывающее смещение $\text{x}$ от положения равновесия в зависимости от времени $\text{t}$, имеет вид:
$x(t) = A \cos(\omega_0 t + \phi_0)$
Здесь $\text{A}$ — постоянная амплитуда, $\omega_0$ — собственная циклическая частота, $\phi_0$ — начальная фаза. График таких колебаний — это синусоида (или косинусоида).
Свободные затухающие колебания
В реальных условиях энергия колебательной системы постепенно расходуется на преодоление сил сопротивления, поэтому амплитуда колебаний со временем уменьшается. Уравнение затухающих колебаний:
$x(t) = A_0 e^{-\beta t} \cos(\omega t + \phi_0)$
Здесь $A_0$ — начальная амплитуда, $\beta$ — коэффициент затухания, $\omega$ — частота затухающих колебаний. Амплитуда $A(t) = A_0 e^{-\beta t}$ убывает по экспоненциальному закону.
Что можно сказать о периоде и частоте затухающих колебаний?
Наличие сил сопротивления (затухания) не только уменьшает амплитуду, но и изменяет частоту и период колебаний. Циклическая частота затухающих колебаний $\omega$ связана с собственной частотой $\omega_0$ (частотой колебаний той же системы без затухания) и коэффициентом затухания $\beta$ следующим образом:
$\omega = \sqrt{\omega_0^2 - \beta^2}$
Из этой формулы следует, что частота затухающих колебаний $\omega$ всегда меньше собственной частоты $\omega_0$.
$\omega < \omega_0$
Период колебаний $\text{T}$ обратно пропорционален частоте ($T = 2\pi / \omega$). Следовательно, для периода затухающих колебаний $\text{T}$ и периода собственных незатухающих колебаний $T_0 = 2\pi/\omega_0$ справедливо:
$T = \frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{\omega_0^2 - \beta^2}}$
Так как знаменатель в выражении для $\text{T}$ меньше, чем $\omega_0$, то период затухающих колебаний $\text{T}$ всегда больше периода собственных колебаний $T_0$.
$T > T_0$
Таким образом, затухание в системе приводит к замедлению колебаний: их частота уменьшается, а период увеличивается.
Ответ: График свободных незатухающих колебаний представляет собой синусоиду с постоянной амплитудой. График свободных затухающих колебаний — это синусоида, амплитуда которой убывает со временем по экспоненциальному закону (см. графики выше). Период затухающих колебаний больше, а их частота — меньше, чем соответствующие период и частота свободных незатухающих колебаний для той же колебательной системы.
Другие задания:
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