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Номер 8.6, страница 60 - 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 8.6, страница 60.
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" => 1319473 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "60" "field_page_end" => "61" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/08-6" "field_display_title" => "8.6" "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 {#1074 #items: array:1 [ 0 => App\Models\Branch {#1034 #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 {#1039 #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 {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #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 {#1041 #items: array:1 [ 0 => App\Models\Term {#1042 #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: [] …8 } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:2 [ 0 => App\Models\Term {#1044 …30} 1 => App\Models\Term {#1045 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ 0 => App\Models\Term {#1049 …30} 1 => App\Models\Term {#1050 …30} 2 => App\Models\Term {#1051 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Term {#1058 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 …30} ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 #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 {#1066 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1070 …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" => 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 {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_country" => Illuminate\Database\Eloquent\Collection {#1055} "field_city" => Illuminate\Database\Eloquent\Collection {#1057} "field_series" => Illuminate\Database\Eloquent\Collection {#1059} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065} "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 {#1066} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069} "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 {#1071 #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 {#1039} "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 {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071} ] #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 {#1149 #items: array:4 [ 0 => App\Models\Branch {#1159 #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 {#1160 #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 {#1194 #items: array:1 [ 0 => App\Models\Book {#1161 #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 {#1163 #items: array:1 [ 0 => App\Models\Term {#1162 #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 {#1164 #items: array:2 [ 0 => App\Models\Term {#1165 #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 {#1166 #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 {#1167 #items: array:1 [ 0 => App\Models\Term {#1168 #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 {#1169 #items: array:3 [ 0 => App\Models\Term {#1170 #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 {#1171 #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 {#1172 #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 {#1173 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:1 [ 0 => App\Models\Term {#1175 #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 {#1176 #items: array:1 [ 0 => App\Models\Term {#1177 #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 {#1178 #items: array:1 [ 0 => App\Models\Term {#1179 #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 {#1180 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1181 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1182 #items: array:1 [ 0 => App\Models\Term {#1183 #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 {#1184 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1186 #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 {#1187 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Term {#1191 #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 {#1163} "field_class" => Illuminate\Database\Eloquent\Collection {#1164} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1167} "field_author" => Illuminate\Database\Eloquent\Collection {#1169} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1173} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1174} "field_country" => Illuminate\Database\Eloquent\Collection {#1176} "field_city" => Illuminate\Database\Eloquent\Collection {#1178} "field_series" => Illuminate\Database\Eloquent\Collection {#1180} "field_umk" => Illuminate\Database\Eloquent\Collection {#1181} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1182} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1184} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1185} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1186} "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 {#1187} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1188} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1189} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1190} "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 {#1192 #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 {#1160} "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 {#1194} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1192} ] #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 {#1193 #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 {#1195 #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 {#1196 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #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 {#1195} "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 {#1196} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1197} ] #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 {#1198 #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 {#1199 #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 {#1200 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:1 [ 0 => App\Models\Branch {#1201 #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 {#1203 #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 {#1202 #items: array:1 [ 0 => App\Models\Book {#1204 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\Branch {#1236 …30} ] #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 {#1203} "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 {#1202} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1235} ] #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 {#1199} "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 {#1200} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1242} ] #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 {#1240 #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 {#1241 #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 {#1243 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1252 #items: array:1 [ 0 => App\Models\Branch {#1251 #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 {#1253 #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 {#1254 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:1 [ 0 => App\Models\Branch {#1201} ] #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 {#1253} "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 {#1254} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1255} ] #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 {#1241} "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 {#1243} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1252} ] #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 {#1108 #items: array:2 [ 0 => App\Models\Element {#1138 #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" => 1493728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Edition {#1137 #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" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Term {#1136 …30} ] #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 {#1134 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1133 #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 {#1135} "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 {#1134} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1132} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1133} "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" => "1319473" "type" => "task" ] "text" => "<p><strong>8.6.</strong> В показанной на рисунке системе</p><p>$\alpha = 20^\circ$, $m_1 = 2$ кг, $m_2 = 1$ кг. Коэффициент трения между первым грузом и наклонной плоскостью $\mu = 0,1$. Грузы отпускают без начальной скорости. а) Найдите ускорение $\text{a}$ системы грузов и силу натяжения нити $\text{T}$. б) Как изменится ответ, если коэффициент трения станет равным $0,3$?</p><p>☑ а) $a = 0,42$ м/с$^2$; $T = 9,4$ Н; б) $a = 0$, $T = 9,8$ Н.</p><p><strong>Указание.</strong></p><p>Чтобы определить возможное направление движения, необходим предварительный анализ условия (в этом состоит типичная сложность задач с учетом трения).</p><p>Непосредственный подсчет показывает, что если бы трение исчезло, то при заданных в условии значениях $\alpha$, $m_1$ и $m_2$ груз 1 поднимался бы, а груз 2 опускался. Значит, если грузы движутся, то именно в этих направлениях. Теперь можно расставить действующие на систему силы (см. рисунок).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1445" "height" => 1139 "path" => "/media/fizika_10/gelfgat-z/0-08/6-1.webp?ts=1756251021" ] 1 => array:5 [ "name" => "6-2.jpg" "alt" => null "width" => "1338" "height" => 281 "path" => "/media/fizika_10/gelfgat-z/0-08/6-2.webp?ts=1756251021" ] ] ] #original: array:7 [ "id" => 1493728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1319473" "type" => "task" ] "text" => "<p><strong>8.6.</strong> В показанной на рисунке системе</p><p>$\alpha = 20^\circ$, $m_1 = 2$ кг, $m_2 = 1$ кг. Коэффициент трения между первым грузом и наклонной плоскостью $\mu = 0,1$. Грузы отпускают без начальной скорости. а) Найдите ускорение $\text{a}$ системы грузов и силу натяжения нити $\text{T}$. б) Как изменится ответ, если коэффициент трения станет равным $0,3$?</p><p>☑ а) $a = 0,42$ м/с$^2$; $T = 9,4$ Н; б) $a = 0$, $T = 9,8$ Н.</p><p><strong>Указание.</strong></p><p>Чтобы определить возможное направление движения, необходим предварительный анализ условия (в этом состоит типичная сложность задач с учетом трения).</p><p>Непосредственный подсчет показывает, что если бы трение исчезло, то при заданных в условии значениях $\alpha$, $m_1$ и $m_2$ груз 1 поднимался бы, а груз 2 опускался. Значит, если грузы движутся, то именно в этих направлениях. Теперь можно расставить действующие на систему силы (см. рисунок).</p>" "img" => array:2 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1445" "height" => 1139 "path" => "/media/fizika_10/gelfgat-z/0-08/6-1.webp?ts=1756251021" ] 1 => array:5 [ "name" => "6-2.jpg" "alt" => null "width" => "1338" "height" => 281 "path" => "/media/fizika_10/gelfgat-z/0-08/6-2.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 {#1131 #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" => 1691285 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1130 #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" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1129 …30} ] #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 {#1127 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1125 #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 {#1126} "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 {#1127} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1125} "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" => "1319473" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$\alpha = 20^\circ$</p><p>$m_1 = 2 \text{ кг}$</p><p>$m_2 = 1 \text{ кг}$</p><p>$v_0 = 0 \text{ м/с}$</p><p>а) $\mu_1 = 0,1$</p><p>б) $\mu_2 = 0,3$</p><p>$g \approx 9,8 \text{ м/с}^2$</p><p><em>Все данные уже в системе СИ.</em></p><p><strong>Найти:</strong></p><p>а) $a, T$ при $\mu = 0,1$ - ?</p><p>б) $a, T$ при $\mu = 0,3$ - ?</p><p><strong>Решение</strong></p><p>Сначала определим возможное направление движения системы. Для этого сравним силу тяжести, действующую на груз $m_2$, со скатывающей силой — проекцией силы тяжести груза $m_1$ на наклонную плоскость. Трение пока не учитываем.</p><p>Сила, тянущая систему в сторону опускания груза $m_2$: $F_2 = m_2 g = 1 \text{ кг} \cdot 9,8 \text{ м/с}^2 = 9,8 \text{ Н}$.</p><p>Скатывающая сила для груза $m_1$: $F_{1\parallel} = m_1 g \sin\alpha = 2 \text{ кг} \cdot 9,8 \text{ м/с}^2 \cdot \sin(20^\circ) \approx 19,6 \cdot 0,342 \approx 6,7 \text{ Н}$.</p><p>Поскольку $F_2 > F_{1\parallel}$, система будет стремиться двигаться так, что груз $m_2$ опускается, а груз $m_1$ поднимается по наклонной плоскости. Следовательно, сила трения $F_{тр}$, действующая на груз $m_1$, будет направлена вниз вдоль наклонной плоскости, противодействуя движению.</p><p>Запишем второй закон Ньютона для каждого из грузов в проекциях на оси, связанные с направлением движения. Ускорение $a$ у обоих грузов будет одинаковым по модулю из-за нерастяжимости нити. Сила натяжения нити $T$ также одинакова.</p><p>Для груза $m_2$ (ось направлена вертикально вниз):</p><p>$m_2 g - T = m_2 a \quad (1)$</p><p>Для груза $m_1$ (оси направлены вдоль и перпендикулярно наклонной плоскости):</p><p>Проекция на ось OY, перпендикулярную плоскости: $N - m_1 g \cos\alpha = 0$, откуда сила нормальной реакции $N = m_1 g \cos\alpha$.</p><p>Сила трения скольжения: $F_{тр} = \mu N = \mu m_1 g \cos\alpha$.</p><p>Проекция на ось OX, направленную вверх вдоль плоскости: $T - F_{1\parallel} - F_{тр} = m_1 a$.</p><p>$T - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a \quad (2)$</p><p><strong>а) Найдём ускорение системы грузов и силу натяжения нити T при $\mu = 0,1$</strong></p><p>Сложим уравнения (1) и (2), чтобы исключить силу натяжения нити $T$:</p><p>$(m_2 g - m_2 a) - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a$</p><p>Перегруппируем слагаемые, чтобы выразить ускорение $a$:</p><p>$m_2 g - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a + m_2 a$</p><p>$g(m_2 - m_1 \sin\alpha - \mu m_1 \cos\alpha) = a(m_1 + m_2)$</p><p>$a = \frac{g(m_2 - m_1 \sin\alpha - \mu m_1 \cos\alpha)}{m_1 + m_2}$</p><p>Подставим численные значения:</p><p>$a = \frac{9,8 \cdot (1 - 2 \cdot \sin(20^\circ) - 0,1 \cdot 2 \cdot \cos(20^\circ))}{2 + 1} \approx \frac{9,8 \cdot (1 - 2 \cdot 0,342 - 0,2 \cdot 0,940)}{3} \approx \frac{9,8 \cdot (1 - 0,684 - 0,188)}{3} = \frac{9,8 \cdot 0,128}{3} \approx 0,418 \text{ м/с}^2$.</p><p>Округляя до двух значащих цифр, получаем $a \approx 0,42 \text{ м/с}^2$.</p><p>Теперь найдем силу натяжения нити $T$ из уравнения (1):</p><p>$T = m_2(g - a) = 1 \cdot (9,8 - 0,42) = 9,38 \text{ Н}$.</p><p>Округляя, получаем $T \approx 9,4 \text{ Н}$.</p><p><strong>Ответ:</strong> $a = 0,42 \text{ м/с}^2$, $T = 9,4 \text{ Н}$.</p><p><strong>б) Как изменится ответ, если коэффициент трения станет равным 0,3?</strong></p><p>При новом коэффициенте трения $\mu = 0,3$ необходимо проверить, начнётся ли движение вообще. Движение начнется только в том случае, если движущая сила $F_2$ будет больше максимальной силы сопротивления движению, которая складывается из скатывающей силы $F_{1\parallel}$ и максимальной силы трения покоя $F_{тр.пок.макс} = \mu N$.</p><p>Движущая сила: $F_{движ} = m_2 g = 9,8 \text{ Н}$.</p><p>Максимальная сила сопротивления: $F_{сопр.макс} = m_1 g \sin\alpha + \mu m_1 g \cos\alpha$.</p><p>$F_{сопр.макс} = 2 \cdot 9,8 \cdot \sin(20^\circ) + 0,3 \cdot 2 \cdot 9,8 \cdot \cos(20^\circ) \approx 19,6 \cdot 0,342 + 5,88 \cdot 0,940 \approx 6,70 + 5,53 = 12,23 \text{ Н}$.</p><p>Сравниваем силы:</p><p>$F_{движ} (9,8 \text{ Н}) < F_{сопр.макс} (12,23 \text{ Н})$</p><p>Так как движущая сила меньше максимальной силы сопротивления, система не придет в движение и будет находиться в состоянии покоя. Сила трения покоя примет значение, необходимое для равновесия, и не достигнет своего максимального значения.</p><p>Следовательно, ускорение системы равно нулю: $a = 0$.</p><p>Силу натяжения нити найдем из условия равновесия груза $m_2$:</p><p>$m_2 g - T = m_2 a = 0 \Rightarrow T = m_2 g$</p><p>$T = 1 \text{ кг} \cdot 9,8 \text{ м/с}^2 = 9,8 \text{ Н}$.</p><p><strong>Ответ:</strong> $a = 0$, $T = 9,8 \text{ Н}$.</p>" ] #original: array:6 [ "id" => 1691285 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "1319473" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>$\alpha = 20^\circ$</p><p>$m_1 = 2 \text{ кг}$</p><p>$m_2 = 1 \text{ кг}$</p><p>$v_0 = 0 \text{ м/с}$</p><p>а) $\mu_1 = 0,1$</p><p>б) $\mu_2 = 0,3$</p><p>$g \approx 9,8 \text{ м/с}^2$</p><p><em>Все данные уже в системе СИ.</em></p><p><strong>Найти:</strong></p><p>а) $a, T$ при $\mu = 0,1$ - ?</p><p>б) $a, T$ при $\mu = 0,3$ - ?</p><p><strong>Решение</strong></p><p>Сначала определим возможное направление движения системы. Для этого сравним силу тяжести, действующую на груз $m_2$, со скатывающей силой — проекцией силы тяжести груза $m_1$ на наклонную плоскость. Трение пока не учитываем.</p><p>Сила, тянущая систему в сторону опускания груза $m_2$: $F_2 = m_2 g = 1 \text{ кг} \cdot 9,8 \text{ м/с}^2 = 9,8 \text{ Н}$.</p><p>Скатывающая сила для груза $m_1$: $F_{1\parallel} = m_1 g \sin\alpha = 2 \text{ кг} \cdot 9,8 \text{ м/с}^2 \cdot \sin(20^\circ) \approx 19,6 \cdot 0,342 \approx 6,7 \text{ Н}$.</p><p>Поскольку $F_2 > F_{1\parallel}$, система будет стремиться двигаться так, что груз $m_2$ опускается, а груз $m_1$ поднимается по наклонной плоскости. Следовательно, сила трения $F_{тр}$, действующая на груз $m_1$, будет направлена вниз вдоль наклонной плоскости, противодействуя движению.</p><p>Запишем второй закон Ньютона для каждого из грузов в проекциях на оси, связанные с направлением движения. Ускорение $a$ у обоих грузов будет одинаковым по модулю из-за нерастяжимости нити. Сила натяжения нити $T$ также одинакова.</p><p>Для груза $m_2$ (ось направлена вертикально вниз):</p><p>$m_2 g - T = m_2 a \quad (1)$</p><p>Для груза $m_1$ (оси направлены вдоль и перпендикулярно наклонной плоскости):</p><p>Проекция на ось OY, перпендикулярную плоскости: $N - m_1 g \cos\alpha = 0$, откуда сила нормальной реакции $N = m_1 g \cos\alpha$.</p><p>Сила трения скольжения: $F_{тр} = \mu N = \mu m_1 g \cos\alpha$.</p><p>Проекция на ось OX, направленную вверх вдоль плоскости: $T - F_{1\parallel} - F_{тр} = m_1 a$.</p><p>$T - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a \quad (2)$</p><p><strong>а) Найдём ускорение системы грузов и силу натяжения нити T при $\mu = 0,1$</strong></p><p>Сложим уравнения (1) и (2), чтобы исключить силу натяжения нити $T$:</p><p>$(m_2 g - m_2 a) - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a$</p><p>Перегруппируем слагаемые, чтобы выразить ускорение $a$:</p><p>$m_2 g - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a + m_2 a$</p><p>$g(m_2 - m_1 \sin\alpha - \mu m_1 \cos\alpha) = a(m_1 + m_2)$</p><p>$a = \frac{g(m_2 - m_1 \sin\alpha - \mu m_1 \cos\alpha)}{m_1 + m_2}$</p><p>Подставим численные значения:</p><p>$a = \frac{9,8 \cdot (1 - 2 \cdot \sin(20^\circ) - 0,1 \cdot 2 \cdot \cos(20^\circ))}{2 + 1} \approx \frac{9,8 \cdot (1 - 2 \cdot 0,342 - 0,2 \cdot 0,940)}{3} \approx \frac{9,8 \cdot (1 - 0,684 - 0,188)}{3} = \frac{9,8 \cdot 0,128}{3} \approx 0,418 \text{ м/с}^2$.</p><p>Округляя до двух значащих цифр, получаем $a \approx 0,42 \text{ м/с}^2$.</p><p>Теперь найдем силу натяжения нити $T$ из уравнения (1):</p><p>$T = m_2(g - a) = 1 \cdot (9,8 - 0,42) = 9,38 \text{ Н}$.</p><p>Округляя, получаем $T \approx 9,4 \text{ Н}$.</p><p><strong>Ответ:</strong> $a = 0,42 \text{ м/с}^2$, $T = 9,4 \text{ Н}$.</p><p><strong>б) Как изменится ответ, если коэффициент трения станет равным 0,3?</strong></p><p>При новом коэффициенте трения $\mu = 0,3$ необходимо проверить, начнётся ли движение вообще. Движение начнется только в том случае, если движущая сила $F_2$ будет больше максимальной силы сопротивления движению, которая складывается из скатывающей силы $F_{1\parallel}$ и максимальной силы трения покоя $F_{тр.пок.макс} = \mu N$.</p><p>Движущая сила: $F_{движ} = m_2 g = 9,8 \text{ Н}$.</p><p>Максимальная сила сопротивления: $F_{сопр.макс} = m_1 g \sin\alpha + \mu m_1 g \cos\alpha$.</p><p>$F_{сопр.макс} = 2 \cdot 9,8 \cdot \sin(20^\circ) + 0,3 \cdot 2 \cdot 9,8 \cdot \cos(20^\circ) \approx 19,6 \cdot 0,342 + 5,88 \cdot 0,940 \approx 6,70 + 5,53 = 12,23 \text{ Н}$.</p><p>Сравниваем силы:</p><p>$F_{движ} (9,8 \text{ Н}) < F_{сопр.макс} (12,23 \text{ Н})$</p><p>Так как движущая сила меньше максимальной силы сопротивления, система не придет в движение и будет находиться в состоянии покоя. Сила трения покоя примет значение, необходимое для равновесия, и не достигнет своего максимального значения.</p><p>Следовательно, ускорение системы равно нулю: $a = 0$.</p><p>Силу натяжения нити найдем из условия равновесия груза $m_2$:</p><p>$m_2 g - T = m_2 a = 0 \Rightarrow T = m_2 g$</p><p>$T = 1 \text{ кг} \cdot 9,8 \text{ м/с}^2 = 9,8 \text{ Н}$.</p><p><strong>Ответ:</strong> $a = 0$, $T = 9,8 \text{ Н}$.</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" => "1319474" "type" => "task" ] "previous" => array:2 [ "refs" => "1319472" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Book {#1161} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\BookPage {#1111 #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" => 1341021 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "60" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/page-60" "field_display_title" => "60" "field_folder" => "1" "field_image_name" => "60" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Book {#1161} ] #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 {#1121 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Element {#1144 #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" => 1519566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Edition {#1137} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1341021" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "60-1.jpg" "alt" => null "width" => "1569" "height" => 2452 "path" => 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№8.6 (с. 60)
Условие. №8.6 (с. 60)
скриншот условия
8.6. В показанной на рисунке системе
$\alpha = 20^\circ$, $m_1 = 2$ кг, $m_2 = 1$ кг. Коэффициент трения между первым грузом и наклонной плоскостью $\mu = 0,1$. Грузы отпускают без начальной скорости. а) Найдите ускорение $\text{a}$ системы грузов и силу натяжения нити $\text{T}$. б) Как изменится ответ, если коэффициент трения станет равным $0,3$?
☑ а) $a = 0,42$ м/с$^2$; $T = 9,4$ Н; б) $a = 0$, $T = 9,8$ Н.
Указание.
Чтобы определить возможное направление движения, необходим предварительный анализ условия (в этом состоит типичная сложность задач с учетом трения).
Непосредственный подсчет показывает, что если бы трение исчезло, то при заданных в условии значениях $\alpha$, $m_1$ и $m_2$ груз 1 поднимался бы, а груз 2 опускался. Значит, если грузы движутся, то именно в этих направлениях. Теперь можно расставить действующие на систему силы (см. рисунок).
Решение. №8.6 (с. 60)
Дано:
$\alpha = 20^\circ$
$m_1 = 2 \text{ кг}$
$m_2 = 1 \text{ кг}$
$v_0 = 0 \text{ м/с}$
а) $\mu_1 = 0,1$
б) $\mu_2 = 0,3$
$g \approx 9,8 \text{ м/с}^2$
Все данные уже в системе СИ.
Найти:
а) $a, T$ при $\mu = 0,1$ - ?
б) $a, T$ при $\mu = 0,3$ - ?
Решение
Сначала определим возможное направление движения системы. Для этого сравним силу тяжести, действующую на груз $m_2$, со скатывающей силой — проекцией силы тяжести груза $m_1$ на наклонную плоскость. Трение пока не учитываем.
Сила, тянущая систему в сторону опускания груза $m_2$: $F_2 = m_2 g = 1 \text{ кг} \cdot 9,8 \text{ м/с}^2 = 9,8 \text{ Н}$.
Скатывающая сила для груза $m_1$: $F_{1\parallel} = m_1 g \sin\alpha = 2 \text{ кг} \cdot 9,8 \text{ м/с}^2 \cdot \sin(20^\circ) \approx 19,6 \cdot 0,342 \approx 6,7 \text{ Н}$.
Поскольку $F_2 > F_{1\parallel}$, система будет стремиться двигаться так, что груз $m_2$ опускается, а груз $m_1$ поднимается по наклонной плоскости. Следовательно, сила трения $F_{тр}$, действующая на груз $m_1$, будет направлена вниз вдоль наклонной плоскости, противодействуя движению.
Запишем второй закон Ньютона для каждого из грузов в проекциях на оси, связанные с направлением движения. Ускорение $a$ у обоих грузов будет одинаковым по модулю из-за нерастяжимости нити. Сила натяжения нити $T$ также одинакова.
Для груза $m_2$ (ось направлена вертикально вниз):
$m_2 g - T = m_2 a \quad (1)$
Для груза $m_1$ (оси направлены вдоль и перпендикулярно наклонной плоскости):
Проекция на ось OY, перпендикулярную плоскости: $N - m_1 g \cos\alpha = 0$, откуда сила нормальной реакции $N = m_1 g \cos\alpha$.
Сила трения скольжения: $F_{тр} = \mu N = \mu m_1 g \cos\alpha$.
Проекция на ось OX, направленную вверх вдоль плоскости: $T - F_{1\parallel} - F_{тр} = m_1 a$.
$T - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a \quad (2)$
а) Найдём ускорение системы грузов и силу натяжения нити T при $\mu = 0,1$
Сложим уравнения (1) и (2), чтобы исключить силу натяжения нити $T$:
$(m_2 g - m_2 a) - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a$
Перегруппируем слагаемые, чтобы выразить ускорение $a$:
$m_2 g - m_1 g \sin\alpha - \mu m_1 g \cos\alpha = m_1 a + m_2 a$
$g(m_2 - m_1 \sin\alpha - \mu m_1 \cos\alpha) = a(m_1 + m_2)$
$a = \frac{g(m_2 - m_1 \sin\alpha - \mu m_1 \cos\alpha)}{m_1 + m_2}$
Подставим численные значения:
$a = \frac{9,8 \cdot (1 - 2 \cdot \sin(20^\circ) - 0,1 \cdot 2 \cdot \cos(20^\circ))}{2 + 1} \approx \frac{9,8 \cdot (1 - 2 \cdot 0,342 - 0,2 \cdot 0,940)}{3} \approx \frac{9,8 \cdot (1 - 0,684 - 0,188)}{3} = \frac{9,8 \cdot 0,128}{3} \approx 0,418 \text{ м/с}^2$.
Округляя до двух значащих цифр, получаем $a \approx 0,42 \text{ м/с}^2$.
Теперь найдем силу натяжения нити $T$ из уравнения (1):
$T = m_2(g - a) = 1 \cdot (9,8 - 0,42) = 9,38 \text{ Н}$.
Округляя, получаем $T \approx 9,4 \text{ Н}$.
Ответ: $a = 0,42 \text{ м/с}^2$, $T = 9,4 \text{ Н}$.
б) Как изменится ответ, если коэффициент трения станет равным 0,3?
При новом коэффициенте трения $\mu = 0,3$ необходимо проверить, начнётся ли движение вообще. Движение начнется только в том случае, если движущая сила $F_2$ будет больше максимальной силы сопротивления движению, которая складывается из скатывающей силы $F_{1\parallel}$ и максимальной силы трения покоя $F_{тр.пок.макс} = \mu N$.
Движущая сила: $F_{движ} = m_2 g = 9,8 \text{ Н}$.
Максимальная сила сопротивления: $F_{сопр.макс} = m_1 g \sin\alpha + \mu m_1 g \cos\alpha$.
$F_{сопр.макс} = 2 \cdot 9,8 \cdot \sin(20^\circ) + 0,3 \cdot 2 \cdot 9,8 \cdot \cos(20^\circ) \approx 19,6 \cdot 0,342 + 5,88 \cdot 0,940 \approx 6,70 + 5,53 = 12,23 \text{ Н}$.
Сравниваем силы:
$F_{движ} (9,8 \text{ Н}) < F_{сопр.макс} (12,23 \text{ Н})$
Так как движущая сила меньше максимальной силы сопротивления, система не придет в движение и будет находиться в состоянии покоя. Сила трения покоя примет значение, необходимое для равновесия, и не достигнет своего максимального значения.
Следовательно, ускорение системы равно нулю: $a = 0$.
Силу натяжения нити найдем из условия равновесия груза $m_2$:
$m_2 g - T = m_2 a = 0 \Rightarrow T = m_2 g$
$T = 1 \text{ кг} \cdot 9,8 \text{ м/с}^2 = 9,8 \text{ Н}$.
Ответ: $a = 0$, $T = 9,8 \text{ Н}$.
Другие задания:
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