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Номер 8.2, страница 57 - гдз по физике 10-11 класс задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Задачник
Издательство: Илекса
Год издания: 2008 - 2026
Уровень обучения: профильный
Цвет обложки: красный лупа, парень едет на велосипеде
ISBN: 978-5-89237-252-7
Популярные ГДЗ в 10 классе
Механика. Динамика. 8. Применение законов динамики. Упражнения - номер 8.2, страница 57.
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" => 1319469 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => "58" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/08-2" "field_display_title" => "8.2" "field_outside_task" => "0" "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" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "6" "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" => 5400 "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:2 [ 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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #original: array:6 [ "id" => 5158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Илекса" "field_cases" => null "field_translit" => "ileksa" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:3 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Гельфгат" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Илья" "field_patronymic" => "Маркович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#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" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #original: array:12 [ "id" => 4104 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Генденштейн" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Лев" "field_patronymic" => "Элевич" "field_surname_rp" => "Генденштейна" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4583 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кирик" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Леонид" "field_patronymic" => "Анатольевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 #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 {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 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 {#1067 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1070 #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" => 40 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "профильный" "field_cases" => array:6 [ "field_accusative_case" => "профильный" "field_creative_case" => "профильным" "field_dative_case" => "профильному" "field_genitive_case" => "профильного" "field_nominative_case" => "профильный" "field_prepositional_case" => "профильном" ] "field_translit" => "profilnyy" ] #original: array:6 [ "id" => 40 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "профильный" "field_cases" => array:6 [ "field_accusative_case" => "профильный" "field_creative_case" => "профильным" "field_dative_case" => "профильному" "field_genitive_case" => "профильного" "field_nominative_case" => "профильный" "field_prepositional_case" => "профильном" ] "field_translit" => "profilnyy" ] #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_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "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" => "/10-klass/fizika/gelfgat-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" => 46 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "красный" "field_cases" => array:6 [ "field_accusative_case" => "красный" "field_creative_case" => "красным" "field_dative_case" => "красному" "field_genitive_case" => "красного" "field_nominative_case" => "красный" "field_prepositional_case" => "красном" ] "field_translit" => "krasnyy" ] #original: array:6 [ "id" => 46 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "красный" "field_cases" => array:6 [ "field_accusative_case" => "красный" "field_creative_case" => "красным" "field_dative_case" => "красному" "field_genitive_case" => "красного" "field_nominative_case" => "красный" "field_prepositional_case" => "красном" ] "field_translit" => "krasnyy" ] #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" => 381 "subject" => 556 "class_subject" => 494 ] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1060} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1061} "field_country" => Illuminate\Database\Eloquent\Collection {#1063} "field_city" => Illuminate\Database\Eloquent\Collection {#1065} "field_series" => Illuminate\Database\Eloquent\Collection {#1067} "field_umk" => Illuminate\Database\Eloquent\Collection {#1068} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1073} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ 0 => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/gelfgat-z/covers/cover1.webp?ts=1756049816" "alt" => "" "width" => "1566" "height" => "2412" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1074} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1075} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1076} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1077} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #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 {#1079 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "6" "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} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1079} ] #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 {#1111 #items: array:4 [ 0 => App\Models\Branch {#1121 #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" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "6" "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 {#1123 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1122} "field_page_start" => "6" "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 {#1123} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1124} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1125 #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" => 1319280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "40" "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 {#1127 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_page_start" => "40" "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 {#1127} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1128} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1129 #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" => 1319290 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "8. Применение законов динамики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "57" "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 {#1131 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Branch {#1139 #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" => 1319280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "40" "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 {#1142 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Динамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1141} "field_page_start" => "40" "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 {#1142} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1143} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319290 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "8. Применение законов динамики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "57" "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 {#1131} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1140} ] #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\Branch {#1133 #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" => 1319291 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "57" "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 {#1136 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Branch {#1148 #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" => 1319290 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "8. Применение законов динамики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1150 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "57" "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 {#1151 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Branch {#1139} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319290 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "8. Применение законов динамики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1150} "field_page_start" => "57" "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 {#1151} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1152} ] #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" => 1319291 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1132} "field_page_start" => "57" "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 {#1136} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1149} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: 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#relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1098} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096} "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" => "1319469" "type" => "task" ] "text" => "<p><strong>8.2.</strong> К концам шнура, перекинутого через блок, подвешены грузы с массами $m_1 = 100 \text{ г}$ и $m_2 = 150 \text{ г}$. Найдите ускорение грузов, силу натяжения шнура $\text{T}$ и показание $\text{F}$ динамометра, на котором висит блок.</p><p>☑ 2 м/с$^2$; 1,2 Н; 2,4 Н.</p><p><strong>Решение.</strong> Благодаря нерастяжимости шнура перемещения обоих грузов одинаковы по величине; значит, одинаковы по величине и их ускорения: $a_1 = a_2 = a$. Пренебрегая массами шнура и блока, а также трением в блоке, можно считать силу $\text{T}$ натяжения шнура всюду одинаковой. На рисунке а показаны действующие на грузы силы. Запишем уравнения второго закона Ньютона для каждого из грузов в проекции на ось $\text{y}$:</p><p>$\begin{cases} m_1 a = -m_1 g + T \\ -m_2 a = -m_2 g + T \end{cases}$</p><p>Отсюда $a = \frac{m_2 - m_1}{m_2 + m_1} g = 2,0 \text{ (м/с}^2\text{)}$, $T = \frac{2m_1 m_2 g}{m_1 + m_2} = 1,2 \text{ (Н)}$.</p><p>Для нахождения силы $\text{F}$ упругости пружины динамометра надо рассмотреть силы, действующие на блок (см. рис. б).</p><p>Из условия равновесия блока получаем, что $F = 2T = 2,4 \text{ (Н)}$. На примере данной задачи покажем два эффективных метода проверки полученного результата.</p><p><strong>Первый из них — проверка на симметрию.</strong> Очевидно, если поменять грузы местами, натяжение шнура и величина ускорения грузов не должны измениться. Полученные для $\text{a}$ и $\text{T}$ выражения этому требованию удовлетворяют (при замене $m_1 \leftrightarrow m_2$, значение $\text{T}$ не изменяется, а ускорение меняет только знак).</p><p><strong>Второй метод — проверка на частные и предельные случаи.</strong> Например, при $m_1 = m_2 = m$ ускорение должно быть равно нулю, а $T = mg$. При $m_1 \rightarrow 0$ должно быть $a \rightarrow g$, $T \rightarrow 0$ (груз 2 свободно падает и поэтому находится в состоянии невесомости). Полученные формулы удовлетворяют и этим требованиям. Обратим внимание на то, что вес обоих грузов одинаков, как бы сильно ни отличались их массы: ведь оба груза действуют на шнур с одинаковыми силами. Дело в том, что вес меньшего груза увеличивается (ускорение этого груза направлено вверх), а вес большего груза уменьшается (ускорение этого груза направлено вниз).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1451" "height" => 1298 "path" => "/media/fizika_10/gelfgat-z/0-08/2-1.webp?ts=1756251021" ] 1 => array:5 [ "name" => "2-2.jpg" "alt" => null "width" => "1338" "height" => 1321 "path" => "/media/fizika_10/gelfgat-z/0-08/2-2.webp?ts=1756251021" ] ] ] #original: array:7 [ "id" => 1493724 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "1319469" "type" => "task" ] "text" => "<p><strong>8.2.</strong> К концам шнура, перекинутого через блок, подвешены грузы с массами $m_1 = 100 \text{ г}$ и $m_2 = 150 \text{ г}$. Найдите ускорение грузов, силу натяжения шнура $\text{T}$ и показание $\text{F}$ динамометра, на котором висит блок.</p><p>☑ 2 м/с$^2$; 1,2 Н; 2,4 Н.</p><p><strong>Решение.</strong> Благодаря нерастяжимости шнура перемещения обоих грузов одинаковы по величине; значит, одинаковы по величине и их ускорения: $a_1 = a_2 = a$. Пренебрегая массами шнура и блока, а также трением в блоке, можно считать силу $\text{T}$ натяжения шнура всюду одинаковой. На рисунке а показаны действующие на грузы силы. Запишем уравнения второго закона Ньютона для каждого из грузов в проекции на ось $\text{y}$:</p><p>$\begin{cases} m_1 a = -m_1 g + T \\ -m_2 a = -m_2 g + T \end{cases}$</p><p>Отсюда $a = \frac{m_2 - m_1}{m_2 + m_1} g = 2,0 \text{ (м/с}^2\text{)}$, $T = \frac{2m_1 m_2 g}{m_1 + m_2} = 1,2 \text{ (Н)}$.</p><p>Для нахождения силы $\text{F}$ упругости пружины динамометра надо рассмотреть силы, действующие на блок (см. рис. б).</p><p>Из условия равновесия блока получаем, что $F = 2T = 2,4 \text{ (Н)}$. На примере данной задачи покажем два эффективных метода проверки полученного результата.</p><p><strong>Первый из них — проверка на симметрию.</strong> Очевидно, если поменять грузы местами, натяжение шнура и величина ускорения грузов не должны измениться. Полученные для $\text{a}$ и $\text{T}$ выражения этому требованию удовлетворяют (при замене $m_1 \leftrightarrow m_2$, значение $\text{T}$ не изменяется, а ускорение меняет только знак).</p><p><strong>Второй метод — проверка на частные и предельные случаи.</strong> Например, при $m_1 = m_2 = m$ ускорение должно быть равно нулю, а $T = mg$. При $m_1 \rightarrow 0$ должно быть $a \rightarrow g$, $T \rightarrow 0$ (груз 2 свободно падает и поэтому находится в состоянии невесомости). Полученные формулы удовлетворяют и этим требованиям. Обратим внимание на то, что вес обоих грузов одинаков, как бы сильно ни отличались их массы: ведь оба груза действуют на шнур с одинаковыми силами. Дело в том, что вес меньшего груза увеличивается (ускорение этого груза направлено вверх), а вес большего груза уменьшается (ускорение этого груза направлено вниз).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1451" "height" => 1298 "path" => "/media/fizika_10/gelfgat-z/0-08/2-1.webp?ts=1756251021" ] 1 => array:5 [ "name" => "2-2.jpg" "alt" => null "width" => "1338" "height" => 1321 "path" => "/media/fizika_10/gelfgat-z/0-08/2-2.webp?ts=1756251021" ] ] ] #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 {#1094 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true 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#guarded: array:1 [ 0 => "*" ] } ] #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 {#1091 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090} "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 {#1091} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1085} "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" => "1319469" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$m_1 = 100$ г</p><p>$m_2 = 150$ г</p><p>В системе СИ:</p><p>$m_1 = 0.1$ кг</p><p>$m_2 = 0.15$ кг</p><p>Ускорение свободного падения $g \approx 10 \, \text{м/с}^2$.</p><p><strong>Найти:</strong></p><p>$a$ — ускорение грузов;</p><p>$T$ — сила натяжения шнура;</p><p>$F$ — показание динамометра.</p><p><strong>Решение:</strong></p><p>Данная система представляет собой машину Атвуда. Будем считать шнур нерастяжимым и невесомым, а блок — идеальным (невесомым и без трения).</p><p><strong>1.</strong> Ускорение грузов (a)</p><p>Поскольку $m_2 > m_1$, груз $m_2$ будет опускаться, а груз $m_1$ — подниматься. Из-за нерастяжимости шнура ускорения обоих грузов будут одинаковы по модулю. Обозначим модуль ускорения как $a$.</p><p>Запишем второй закон Ньютона для каждого груза в проекции на вертикальную ось $y$, направленную вверх:</p><p>Для груза $m_1$: $m_1a = T - m_1g$</p><p>Для груза $m_2$: $-m_2a = T - m_2g$ (ускорение направлено вниз, поэтому проекция отрицательна)</p><p>Получаем систему уравнений:</p><p>$$ \begin{cases} T = m_1a + m_1g \\ T = m_2g - m_2a \end{cases} $$</p><p>Приравняем правые части уравнений, чтобы найти ускорение $a$:</p><p>$m_1a + m_1g = m_2g - m_2a$</p><p>$a(m_1 + m_2) = g(m_2 - m_1)$</p><p>$a = \frac{m_2 - m_1}{m_1 + m_2}g$</p><p>Подставим числовые значения:</p><p>$a = \frac{0.15 \, \text{кг} - 0.1 \, \text{кг}}{0.15 \, \text{кг} + 0.1 \, \text{кг}} \cdot 10 \, \frac{\text{м}}{\text{с}^2} = \frac{0.05}{0.25} \cdot 10 \, \frac{\text{м}}{\text{с}^2} = 0.2 \cdot 10 \, \frac{\text{м}}{\text{с}^2} = 2 \, \frac{\text{м}}{\text{с}^2}$</p><p><strong>Ответ:</strong> Ускорение грузов равно $2 \, \text{м/с}^2$.</p><p><strong>2.</strong> Сила натяжения шнура (T)</p><p>Для нахождения силы натяжения шнура $T$ подставим найденное значение ускорения $a$ в любое из уравнений системы. Воспользуемся первым:</p><p>$T = m_1(a + g)$</p><p>Подставим числовые значения:</p><p>$T = 0.1 \, \text{кг} \cdot (2 \, \frac{\text{м}}{\text{с}^2} + 10 \, \frac{\text{м}}{\text{с}^2}) = 0.1 \, \text{кг} \cdot 12 \, \frac{\text{м}}{\text{с}^2} = 1.2 \, \text{Н}$</p><p><strong>Ответ:</strong> Сила натяжения шнура равна $1.2 \, \text{Н}$.</p><p><strong>3.</strong> Показание динамометра (F)</p><p>Динамометр измеряет силу, с которой система "блок-грузы" действует на него. Эта сила равна по модулю силе упругости $F$, которая уравновешивает силы, действующие на блок со стороны шнура. На невесомый блок действуют две силы натяжения $T$, направленные вертикально вниз (по одной от каждого конца шнура).</p><p>Из условия равновесия блока следует, что показание динамометра $F$ равно сумме сил натяжения:</p><p>$F = T + T = 2T$</p><p>Подставим найденное значение $T$:</p><p>$F = 2 \cdot 1.2 \, \text{Н} = 2.4 \, \text{Н}$</p><p><strong>Ответ:</strong> Показание динамометра равно $2.4 \, \text{Н}$.</p>" ] #original: array:6 [ "id" => 1691281 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1319469" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$m_1 = 100$ г</p><p>$m_2 = 150$ г</p><p>В системе СИ:</p><p>$m_1 = 0.1$ кг</p><p>$m_2 = 0.15$ кг</p><p>Ускорение свободного падения $g \approx 10 \, \text{м/с}^2$.</p><p><strong>Найти:</strong></p><p>$a$ — ускорение грузов;</p><p>$T$ — сила натяжения шнура;</p><p>$F$ — показание динамометра.</p><p><strong>Решение:</strong></p><p>Данная система представляет собой машину Атвуда. Будем считать шнур нерастяжимым и невесомым, а блок — идеальным (невесомым и без трения).</p><p><strong>1.</strong> Ускорение грузов (a)</p><p>Поскольку $m_2 > m_1$, груз $m_2$ будет опускаться, а груз $m_1$ — подниматься. Из-за нерастяжимости шнура ускорения обоих грузов будут одинаковы по модулю. Обозначим модуль ускорения как $a$.</p><p>Запишем второй закон Ньютона для каждого груза в проекции на вертикальную ось $y$, направленную вверх:</p><p>Для груза $m_1$: $m_1a = T - m_1g$</p><p>Для груза $m_2$: $-m_2a = T - m_2g$ (ускорение направлено вниз, поэтому проекция отрицательна)</p><p>Получаем систему уравнений:</p><p>$$ \begin{cases} T = m_1a + m_1g \\ T = m_2g - m_2a \end{cases} $$</p><p>Приравняем правые части уравнений, чтобы найти ускорение $a$:</p><p>$m_1a + m_1g = m_2g - m_2a$</p><p>$a(m_1 + m_2) = g(m_2 - m_1)$</p><p>$a = \frac{m_2 - m_1}{m_1 + m_2}g$</p><p>Подставим числовые значения:</p><p>$a = \frac{0.15 \, \text{кг} - 0.1 \, \text{кг}}{0.15 \, \text{кг} + 0.1 \, \text{кг}} \cdot 10 \, \frac{\text{м}}{\text{с}^2} = \frac{0.05}{0.25} \cdot 10 \, \frac{\text{м}}{\text{с}^2} = 0.2 \cdot 10 \, \frac{\text{м}}{\text{с}^2} = 2 \, \frac{\text{м}}{\text{с}^2}$</p><p><strong>Ответ:</strong> Ускорение грузов равно $2 \, \text{м/с}^2$.</p><p><strong>2.</strong> Сила натяжения шнура (T)</p><p>Для нахождения силы натяжения шнура $T$ подставим найденное значение ускорения $a$ в любое из уравнений системы. Воспользуемся первым:</p><p>$T = m_1(a + g)$</p><p>Подставим числовые значения:</p><p>$T = 0.1 \, \text{кг} \cdot (2 \, \frac{\text{м}}{\text{с}^2} + 10 \, \frac{\text{м}}{\text{с}^2}) = 0.1 \, \text{кг} \cdot 12 \, \frac{\text{м}}{\text{с}^2} = 1.2 \, \text{Н}$</p><p><strong>Ответ:</strong> Сила натяжения шнура равна $1.2 \, \text{Н}$.</p><p><strong>3.</strong> Показание динамометра (F)</p><p>Динамометр измеряет силу, с которой система "блок-грузы" действует на него. Эта сила равна по модулю силе упругости $F$, которая уравновешивает силы, действующие на блок со стороны шнура. На невесомый блок действуют две силы натяжения $T$, направленные вертикально вниз (по одной от каждого конца шнура).</p><p>Из условия равновесия блока следует, что показание динамометра $F$ равно сумме сил натяжения:</p><p>$F = T + T = 2T$</p><p>Подставим найденное значение $T$:</p><p>$F = 2 \cdot 1.2 \, \text{Н} = 2.4 \, \text{Н}$</p><p><strong>Ответ:</strong> Показание динамометра равно $2.4 \, \text{Н}$.</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" => "1319470" "type" => "task" ] "previous" => array:2 [ "refs" => "1319468" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\BookPage {#1107 #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" => 1341018 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-57" "field_display_title" => "57" "field_folder" => "1" "field_image_name" => "57" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1087 #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 {#1083 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Element {#1109 #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" => 1519563 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1101} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1341018" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "57-1.jpg" "alt" => null "width" => "1588" "height" => 2442 "path" => "/media/fizika_10/gelfgat-z/0-1/57-1.webp?ts=1756642307" ] ] ] #original: array:6 [ "id" => 1519563 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "book_page" => array:2 [ "refs" => "1341018" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "57-1.jpg" "alt" => null "width" => "1588" "height" => 2442 "path" => "/media/fizika_10/gelfgat-z/0-1/57-1.webp?ts=1756642307" ] ] ] #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" => "1341019" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1341017" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:2 [ 0 => App\Models\Task {#1228 #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" => 1319468 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/10-klass/fizika/gelfgat-zadachnik/08-1" "field_display_title" => "8.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1229 #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 {#1230 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ 0 => App\Models\Branch {#1121} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:4 [ 0 => App\Models\Branch {#1121} 1 => App\Models\Branch {#1125} 2 => App\Models\Branch {#1129} 3 => App\Models\Branch {#1133} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:2 [ 0 => App\Models\Element {#1243 #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" => 1493723 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:1 [ 0 => App\Models\Edition {#1101} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319468" "type" => "task" ] "text" => "<p><strong>8.1.</strong> На тележке, движущейся горизонтально с ускорением $\text{a}$, установлен штатив, на котором подвешен шарик на нити. Найдите угол $\alpha$ отклонения нити от вертикали и силу $\text{T}$ натяжения нити.</p><p>$\alpha = \text{arctg}(a/g); T = m\sqrt{g^2 + a^2}$</p><p><strong>Решение.</strong> На шарик действуют сила тяжести $m\vec{g}$ и сила $\vec{T}$ натяжения нити (см. рисунок). Их равнодействующая $\vec{F}$ сообщает шарику ускорение $\vec{a}$, т. е. $\vec{F} = m\vec{a}$.</p><p><strong>Следовательно,</strong> $tg \alpha = a/g$.</p><p><strong>Воспользовавшись теоремой Пифагора,</strong> находим, что $T = m\sqrt{g^2 + a^2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1420" "height" => 865 "path" => "/media/fizika_10/gelfgat-z/0-08/1-1.webp?ts=1756251021" ] ] ] #original: array:7 [ "id" => 1493723 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244} "task" => array:2 [ "refs" => "1319468" "type" => "task" ] "text" => "<p><strong>8.1.</strong> На тележке, движущейся горизонтально с ускорением $\text{a}$, установлен штатив, на котором подвешен шарик на нити. Найдите угол $\alpha$ отклонения нити от вертикали и силу $\text{T}$ натяжения нити.</p><p>$\alpha = \text{arctg}(a/g); T = m\sqrt{g^2 + a^2}$</p><p><strong>Решение.</strong> На шарик действуют сила тяжести $m\vec{g}$ и сила $\vec{T}$ натяжения нити (см. рисунок). Их равнодействующая $\vec{F}$ сообщает шарику ускорение $\vec{a}$, т. е. $\vec{F} = m\vec{a}$.</p><p><strong>Следовательно,</strong> $tg \alpha = a/g$.</p><p><strong>Воспользовавшись теоремой Пифагора,</strong> находим, что $T = m\sqrt{g^2 + a^2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1420" "height" => 865 "path" => "/media/fizika_10/gelfgat-z/0-08/1-1.webp?ts=1756251021" ] ] ] #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 {#1245 #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" => 1691280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1246 #items: array:1 [ 0 => App\Models\Edition {#1095} ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319468" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$m$ – масса шарика<br>$a$ – ускорение тележки (горизонтальное)<br>$g$ – ускорение свободного падения</p><p>Все величины считаются заданными в системе СИ.</p><p><strong>Найти:</strong></p><p>$\alpha$ – угол отклонения нити от вертикали<br>$T$ – сила натяжения нити</p><p><strong>Решение:</strong></p><p>Рассмотрим шарик в инерциальной системе отсчета, связанной с землей. На шарик действуют две силы:</p><p><strong>1.</strong> Сила тяжести $\vec{F_g} = m\vec{g}$, направленная вертикально вниз.</p><p><strong>2.</strong> Сила натяжения нити $\vec{T}$, направленная вдоль нити под углом $\alpha$ к вертикали.</p><p>Поскольку шарик движется вместе с тележкой, он имеет горизонтальное ускорение $\vec{a}$. Согласно второму закону Ньютона, равнодействующая всех сил, действующих на шарик, равна произведению его массы на ускорение:</p><p>$\vec{T} + m\vec{g} = m\vec{a}$</p><p>Для решения этого векторного уравнения разложим силы на компоненты. Введем систему координат: ось <em>Ox</em> направим горизонтально в сторону ускорения, а ось <em>Oy</em> – вертикально вверх.</p><p>Запишем проекции второго закона Ньютона на эти оси:</p><p>На ось <em>Ox</em>: $T_x = ma$<br>Из геометрии рисунка видно, что $T_x = T \sin(\alpha)$.<br>Следовательно, $T \sin(\alpha) = ma$ (1)</p><p>На ось <em>Oy</em>: $T_y - mg = 0$ (так как по вертикали шарик не движется, его ускорение в этой проекции равно нулю).<br>Из геометрии рисунка $T_y = T \cos(\alpha)$.<br>Следовательно, $T \cos(\alpha) = mg$ (2)</p><p>Мы получили систему из двух уравнений с двумя неизвестными, $\alpha$ и $T$.</p><p><strong>1.</strong> Найдем угол $\alpha$.</p><p>Разделим уравнение (1) на уравнение (2):</p><p>$\frac{T \sin(\alpha)}{T \cos(\alpha)} = \frac{ma}{mg}$</p><p>Сократив $T$ и $m$, получим:</p><p>$\tan(\alpha) = \frac{a}{g}$</p><p>Отсюда угол отклонения нити равен:</p><p>$\alpha = \arctan(\frac{a}{g})$</p><p><strong>2.</strong> Найдем силу натяжения нити $T$.</p><p>Возведем в квадрат оба уравнения (1) и (2) и сложим их:</p><p>$(T \sin(\alpha))^2 + (T \cos(\alpha))^2 = (ma)^2 + (mg)^2$</p><p>$T^2 \sin^2(\alpha) + T^2 \cos^2(\alpha) = m^2 a^2 + m^2 g^2$</p><p>Вынесем $T^2$ и $m^2$ за скобки:</p><p>$T^2 (\sin^2(\alpha) + \cos^2(\alpha)) = m^2 (a^2 + g^2)$</p><p>Используя основное тригонометрическое тождество $\sin^2(\alpha) + \cos^2(\alpha) = 1$, получаем:</p><p>$T^2 = m^2 (g^2 + a^2)$</p><p>Извлекая квадратный корень, находим модуль силы натяжения нити:</p><p>$T = \sqrt{m^2 (g^2 + a^2)} = m\sqrt{g^2 + a^2}$</p><p><strong>Ответ:</strong> Угол отклонения нити от вертикали $\alpha = \arctan(\frac{a}{g})$, сила натяжения нити $T = m\sqrt{g^2 + a^2}$.</p>" ] #original: array:6 [ "id" => 1691280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1246} "task" => array:2 [ "refs" => "1319468" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$m$ – масса шарика<br>$a$ – ускорение тележки (горизонтальное)<br>$g$ – ускорение свободного падения</p><p>Все величины считаются заданными в системе СИ.</p><p><strong>Найти:</strong></p><p>$\alpha$ – угол отклонения нити от вертикали<br>$T$ – сила натяжения нити</p><p><strong>Решение:</strong></p><p>Рассмотрим шарик в инерциальной системе отсчета, связанной с землей. На шарик действуют две силы:</p><p><strong>1.</strong> Сила тяжести $\vec{F_g} = m\vec{g}$, направленная вертикально вниз.</p><p><strong>2.</strong> Сила натяжения нити $\vec{T}$, направленная вдоль нити под углом $\alpha$ к вертикали.</p><p>Поскольку шарик движется вместе с тележкой, он имеет горизонтальное ускорение $\vec{a}$. Согласно второму закону Ньютона, равнодействующая всех сил, действующих на шарик, равна произведению его массы на ускорение:</p><p>$\vec{T} + m\vec{g} = m\vec{a}$</p><p>Для решения этого векторного уравнения разложим силы на компоненты. Введем систему координат: ось <em>Ox</em> направим горизонтально в сторону ускорения, а ось <em>Oy</em> – вертикально вверх.</p><p>Запишем проекции второго закона Ньютона на эти оси:</p><p>На ось <em>Ox</em>: $T_x = ma$<br>Из геометрии рисунка видно, что $T_x = T \sin(\alpha)$.<br>Следовательно, $T \sin(\alpha) = ma$ (1)</p><p>На ось <em>Oy</em>: $T_y - mg = 0$ (так как по вертикали шарик не движется, его ускорение в этой проекции равно нулю).<br>Из геометрии рисунка $T_y = T \cos(\alpha)$.<br>Следовательно, $T \cos(\alpha) = mg$ (2)</p><p>Мы получили систему из двух уравнений с двумя неизвестными, $\alpha$ и $T$.</p><p><strong>1.</strong> Найдем угол $\alpha$.</p><p>Разделим уравнение (1) на уравнение (2):</p><p>$\frac{T \sin(\alpha)}{T \cos(\alpha)} = \frac{ma}{mg}$</p><p>Сократив $T$ и $m$, получим:</p><p>$\tan(\alpha) = \frac{a}{g}$</p><p>Отсюда угол отклонения нити равен:</p><p>$\alpha = \arctan(\frac{a}{g})$</p><p><strong>2.</strong> Найдем силу натяжения нити $T$.</p><p>Возведем в квадрат оба уравнения (1) и (2) и сложим их:</p><p>$(T \sin(\alpha))^2 + (T \cos(\alpha))^2 = (ma)^2 + (mg)^2$</p><p>$T^2 \sin^2(\alpha) + T^2 \cos^2(\alpha) = m^2 a^2 + m^2 g^2$</p><p>Вынесем $T^2$ и $m^2$ за скобки:</p><p>$T^2 (\sin^2(\alpha) + \cos^2(\alpha)) = m^2 (a^2 + g^2)$</p><p>Используя основное тригонометрическое тождество $\sin^2(\alpha) + \cos^2(\alpha) = 1$, получаем:</p><p>$T^2 = m^2 (g^2 + a^2)$</p><p>Извлекая квадратный корень, находим модуль силы натяжения нити:</p><p>$T = \sqrt{m^2 (g^2 + a^2)} = m\sqrt{g^2 + a^2}$</p><p><strong>Ответ:</strong> Угол отклонения нити от вертикали $\alpha = \arctan(\frac{a}{g})$, сила натяжения нити $T = m\sqrt{g^2 + a^2}$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: 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№8.2 (с. 57)
Условие. №8.2 (с. 57)
скриншот условия
8.2. К концам шнура, перекинутого через блок, подвешены грузы с массами $m_1 = 100 \text{ г}$ и $m_2 = 150 \text{ г}$. Найдите ускорение грузов, силу натяжения шнура $\text{T}$ и показание $\text{F}$ динамометра, на котором висит блок.
☑ 2 м/с$^2$; 1,2 Н; 2,4 Н.
Решение. Благодаря нерастяжимости шнура перемещения обоих грузов одинаковы по величине; значит, одинаковы по величине и их ускорения: $a_1 = a_2 = a$. Пренебрегая массами шнура и блока, а также трением в блоке, можно считать силу $\text{T}$ натяжения шнура всюду одинаковой. На рисунке а показаны действующие на грузы силы. Запишем уравнения второго закона Ньютона для каждого из грузов в проекции на ось $\text{y}$:
$\begin{cases} m_1 a = -m_1 g + T \\ -m_2 a = -m_2 g + T \end{cases}$
Отсюда $a = \frac{m_2 - m_1}{m_2 + m_1} g = 2,0 \text{ (м/с}^2\text{)}$, $T = \frac{2m_1 m_2 g}{m_1 + m_2} = 1,2 \text{ (Н)}$.
Для нахождения силы $\text{F}$ упругости пружины динамометра надо рассмотреть силы, действующие на блок (см. рис. б).
Из условия равновесия блока получаем, что $F = 2T = 2,4 \text{ (Н)}$. На примере данной задачи покажем два эффективных метода проверки полученного результата.
Первый из них — проверка на симметрию. Очевидно, если поменять грузы местами, натяжение шнура и величина ускорения грузов не должны измениться. Полученные для $\text{a}$ и $\text{T}$ выражения этому требованию удовлетворяют (при замене $m_1 \leftrightarrow m_2$, значение $\text{T}$ не изменяется, а ускорение меняет только знак).
Второй метод — проверка на частные и предельные случаи. Например, при $m_1 = m_2 = m$ ускорение должно быть равно нулю, а $T = mg$. При $m_1 \rightarrow 0$ должно быть $a \rightarrow g$, $T \rightarrow 0$ (груз 2 свободно падает и поэтому находится в состоянии невесомости). Полученные формулы удовлетворяют и этим требованиям. Обратим внимание на то, что вес обоих грузов одинаков, как бы сильно ни отличались их массы: ведь оба груза действуют на шнур с одинаковыми силами. Дело в том, что вес меньшего груза увеличивается (ускорение этого груза направлено вверх), а вес большего груза уменьшается (ускорение этого груза направлено вниз).
Решение. №8.2 (с. 57)
Дано:
$m_1 = 100$ г
$m_2 = 150$ г
В системе СИ:
$m_1 = 0.1$ кг
$m_2 = 0.15$ кг
Ускорение свободного падения $g \approx 10 \, \text{м/с}^2$.
Найти:
$a$ — ускорение грузов;
$T$ — сила натяжения шнура;
$F$ — показание динамометра.
Решение:
Данная система представляет собой машину Атвуда. Будем считать шнур нерастяжимым и невесомым, а блок — идеальным (невесомым и без трения).
1. Ускорение грузов (a)
Поскольку $m_2 > m_1$, груз $m_2$ будет опускаться, а груз $m_1$ — подниматься. Из-за нерастяжимости шнура ускорения обоих грузов будут одинаковы по модулю. Обозначим модуль ускорения как $a$.
Запишем второй закон Ньютона для каждого груза в проекции на вертикальную ось $y$, направленную вверх:
Для груза $m_1$: $m_1a = T - m_1g$
Для груза $m_2$: $-m_2a = T - m_2g$ (ускорение направлено вниз, поэтому проекция отрицательна)
Получаем систему уравнений:
$$ \begin{cases} T = m_1a + m_1g \\ T = m_2g - m_2a \end{cases} $$
Приравняем правые части уравнений, чтобы найти ускорение $a$:
$m_1a + m_1g = m_2g - m_2a$
$a(m_1 + m_2) = g(m_2 - m_1)$
$a = \frac{m_2 - m_1}{m_1 + m_2}g$
Подставим числовые значения:
$a = \frac{0.15 \, \text{кг} - 0.1 \, \text{кг}}{0.15 \, \text{кг} + 0.1 \, \text{кг}} \cdot 10 \, \frac{\text{м}}{\text{с}^2} = \frac{0.05}{0.25} \cdot 10 \, \frac{\text{м}}{\text{с}^2} = 0.2 \cdot 10 \, \frac{\text{м}}{\text{с}^2} = 2 \, \frac{\text{м}}{\text{с}^2}$
Ответ: Ускорение грузов равно $2 \, \text{м/с}^2$.
2. Сила натяжения шнура (T)
Для нахождения силы натяжения шнура $T$ подставим найденное значение ускорения $a$ в любое из уравнений системы. Воспользуемся первым:
$T = m_1(a + g)$
Подставим числовые значения:
$T = 0.1 \, \text{кг} \cdot (2 \, \frac{\text{м}}{\text{с}^2} + 10 \, \frac{\text{м}}{\text{с}^2}) = 0.1 \, \text{кг} \cdot 12 \, \frac{\text{м}}{\text{с}^2} = 1.2 \, \text{Н}$
Ответ: Сила натяжения шнура равна $1.2 \, \text{Н}$.
3. Показание динамометра (F)
Динамометр измеряет силу, с которой система "блок-грузы" действует на него. Эта сила равна по модулю силе упругости $F$, которая уравновешивает силы, действующие на блок со стороны шнура. На невесомый блок действуют две силы натяжения $T$, направленные вертикально вниз (по одной от каждого конца шнура).
Из условия равновесия блока следует, что показание динамометра $F$ равно сумме сил натяжения:
$F = T + T = 2T$
Подставим найденное значение $T$:
$F = 2 \cdot 1.2 \, \text{Н} = 2.4 \, \text{Н}$
Ответ: Показание динамометра равно $2.4 \, \text{Н}$.
Другие задания:
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