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Номер 79, страница 137 - 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
Молекулярная физика и термодинамика. Термодинамика. 18. Тепловые двигатели. Олимпиадные задачи - номер 79, страница 137.
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" => 1319607 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "137" "field_page_end" => "138" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-79" "field_display_title" => "79" "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" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "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" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "104" "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 {#1227 #items: array:4 [ 0 => App\Models\Branch {#1237 #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" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1238 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "104" "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 {#1272 #items: array:1 [ 0 => App\Models\Book {#1239 #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 {#1241 #items: array:1 [ 0 => App\Models\Term {#1240 #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 {#1242 #items: array:2 [ 0 => App\Models\Term {#1243 #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 {#1244 #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 {#1245 #items: array:1 [ 0 => App\Models\Term {#1246 #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 {#1247 #items: array:3 [ 0 => App\Models\Term {#1248 #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 {#1249 #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 {#1250 #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 {#1251 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1252 #items: array:1 [ 0 => App\Models\Term {#1253 #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 {#1254 #items: array:1 [ 0 => App\Models\Term {#1255 #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 {#1256 #items: array:1 [ 0 => App\Models\Term {#1257 #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 {#1258 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1259 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:1 [ 0 => App\Models\Term {#1261 #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 {#1262 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1263 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1264 #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 {#1265 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1266 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1267 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:1 [ 0 => App\Models\Term {#1269 #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 {#1241} "field_class" => Illuminate\Database\Eloquent\Collection {#1242} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1245} "field_author" => Illuminate\Database\Eloquent\Collection {#1247} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1251} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1252} "field_country" => Illuminate\Database\Eloquent\Collection {#1254} "field_city" => Illuminate\Database\Eloquent\Collection {#1256} "field_series" => Illuminate\Database\Eloquent\Collection {#1258} "field_umk" => Illuminate\Database\Eloquent\Collection {#1259} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1260} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1262} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1263} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1264} "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 {#1265} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1266} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1267} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1268} "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 {#1270 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Молекулярная физика и термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1238} "field_page_start" => "104" "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 {#1272} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1270} ] #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 {#1271 #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" => 1319321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1273 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "126" "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 {#1274 #items: array:1 [ 0 => App\Models\Book {#1239} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1275 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1273} "field_page_start" => "126" "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 {#1274} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1275} ] #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 {#1276 #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" => 1319325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "18. Тепловые двигатели" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1277 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "134" "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 {#1278 #items: array:1 [ 0 => App\Models\Book {#1239} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1320 #items: array:1 [ 0 => App\Models\Branch {#1279 #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" => 1319321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1281 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "126" "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 {#1280 #items: array:1 [ 0 => App\Models\Book {#1282 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1313 #items: array:1 [ 0 => App\Models\Branch {#1314 …30} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Термодинамика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1281} "field_page_start" => "126" "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 {#1280} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1313} ] #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" => 1319325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "18. Тепловые двигатели" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1277} "field_page_start" => "134" "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 {#1278} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1320} ] #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 {#1318 #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" => 1319327 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1319 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "135" "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 {#1321 #items: array:1 [ 0 => App\Models\Book {#1239} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1330 #items: array:1 [ 0 => App\Models\Branch {#1329 #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" => 1319325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "18. Тепловые двигатели" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1331 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "134" "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 {#1332 #items: array:1 [ 0 => App\Models\Book {#1239} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1333 #items: array:1 [ 0 => App\Models\Branch {#1279} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "18. Тепловые двигатели" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1331} "field_page_start" => "134" "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 {#1332} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1333} ] #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" => 1319327 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1319} "field_page_start" => "135" "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 {#1321} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1330} ] #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 {#1147 #items: array:2 [ 0 => App\Models\Element {#1185 #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" => 1494057 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:1 [ 0 => App\Models\Edition {#1214 #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 {#1213 #items: array:1 [ 0 => App\Models\Term {#1215 …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 {#1212 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1211 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1210 #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 {#1213} "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 {#1212} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1211} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1210} "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" => "1319607" "type" => "task" ] "text" => "<p><strong>O-79.</strong> Определите отношение $\frac{\eta_1}{\eta_2}$ коэффициентов полезного действия двух циклических процессов, проведенных с идеальным газом (см. рисунок): первый процесс 1-2-3-4, второй процесс 5-6-7-4-5.</p><p>☑ 27/23.</p><p><strong>Решение.</strong> КПД цикла $\eta = \frac{A'}{Q_{\text{н}}}$, где $A'$ — работа газа за цикл; $Q_{\text{н}}$ — количество теплоты, полученное от нагревателя за цикл. Работе газа за цикл на $pV$-диаграмме соответствует площадь, заключенная «внутри» цикла. Из условия задачи следует, что работа газа в первом и во втором процессах одна и та же. Следовательно, $\frac{\eta_1}{\eta_2} = \frac{Q_{\text{н}2}}{Q_{\text{н}1}}$, где $Q_{\text{н}1}$ и $Q_{\text{н}2}$ — количества теплоты, полученные системой соответственно в первом и во втором циклах. Найдем $Q_{\text{н}1}$ и $Q_{\text{н}2}$. Пусть температура газа в точке 1 (см. рисунок) была $T_1$. Воспользовавшись уравнением состояния идеального газа, нетрудно показать, что $T_2 = T_5 = 2T_1$, $T_3 = T_6 = 6T_1$, $T_4 = 3T_1$, $T_7 = 9T_1$. Очевидно, что в процессе 1-2-3-4 газ получил тепло $Q_{\text{н}1}$ на участках 1-2 и 2-3, причем $Q_{\text{н}1} = A_1 + \Delta U_1$, где $A_1 = 4p_0V_0 = 4\nu RT_1$ — работа, совершенная на этих участках газом, $\Delta U_1 = \frac{3}{2}\nu R(T_3 - T_1) = \frac{15}{2}\nu RT_1$ — изменение его внутренней энергии, так что $Q_{\text{н}1} = 4\nu RT_1 + \frac{15}{2}\nu RT_1 = \frac{23}{2}\nu RT_1$.</p><p>В процессе 5-6-7-4-5 газ получил тепло $Q_{\text{н}2}$ на участках 5-6 и 6-7, причем</p><p>$Q_{\text{н}2} = A_2 + \Delta U_2 = 3p_0V_0 + \frac{3}{2}\nu R(T_7 - T_5) = \frac{27}{2}\nu RT_1$.</p><p>Таким образом, $\frac{\eta_1}{\eta_2} = \frac{Q_{\text{н}2}}{Q_{\text{н}1}} = \frac{27}{23}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "79-1.jpg" "alt" => null "width" => "1369" "height" => 1997 "path" => "/media/fizika_10/gelfgat-z/0-olimp/79-1.webp?ts=1756251240" ] 1 => array:5 [ "name" => "79-2.jpg" "alt" => null "width" => "1214" "height" => 372 "path" => "/media/fizika_10/gelfgat-z/0-olimp/79-2.webp?ts=1756251240" ] ] ] #original: array:7 [ "id" => 1494057 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1206} "task" => array:2 [ "refs" => "1319607" "type" => "task" ] "text" => "<p><strong>O-79.</strong> Определите отношение $\frac{\eta_1}{\eta_2}$ коэффициентов полезного действия двух циклических процессов, проведенных с идеальным газом (см. рисунок): первый процесс 1-2-3-4, второй процесс 5-6-7-4-5.</p><p>☑ 27/23.</p><p><strong>Решение.</strong> КПД цикла $\eta = \frac{A'}{Q_{\text{н}}}$, где $A'$ — работа газа за цикл; $Q_{\text{н}}$ — количество теплоты, полученное от нагревателя за цикл. Работе газа за цикл на $pV$-диаграмме соответствует площадь, заключенная «внутри» цикла. Из условия задачи следует, что работа газа в первом и во втором процессах одна и та же. Следовательно, $\frac{\eta_1}{\eta_2} = \frac{Q_{\text{н}2}}{Q_{\text{н}1}}$, где $Q_{\text{н}1}$ и $Q_{\text{н}2}$ — количества теплоты, полученные системой соответственно в первом и во втором циклах. Найдем $Q_{\text{н}1}$ и $Q_{\text{н}2}$. Пусть температура газа в точке 1 (см. рисунок) была $T_1$. Воспользовавшись уравнением состояния идеального газа, нетрудно показать, что $T_2 = T_5 = 2T_1$, $T_3 = T_6 = 6T_1$, $T_4 = 3T_1$, $T_7 = 9T_1$. Очевидно, что в процессе 1-2-3-4 газ получил тепло $Q_{\text{н}1}$ на участках 1-2 и 2-3, причем $Q_{\text{н}1} = A_1 + \Delta U_1$, где $A_1 = 4p_0V_0 = 4\nu RT_1$ — работа, совершенная на этих участках газом, $\Delta U_1 = \frac{3}{2}\nu R(T_3 - T_1) = \frac{15}{2}\nu RT_1$ — изменение его внутренней энергии, так что $Q_{\text{н}1} = 4\nu RT_1 + \frac{15}{2}\nu RT_1 = \frac{23}{2}\nu RT_1$.</p><p>В процессе 5-6-7-4-5 газ получил тепло $Q_{\text{н}2}$ на участках 5-6 и 6-7, причем</p><p>$Q_{\text{н}2} = A_2 + \Delta U_2 = 3p_0V_0 + \frac{3}{2}\nu R(T_7 - T_5) = \frac{27}{2}\nu RT_1$.</p><p>Таким образом, $\frac{\eta_1}{\eta_2} = \frac{Q_{\text{н}2}}{Q_{\text{н}1}} = \frac{27}{23}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "79-1.jpg" "alt" => null "width" => "1369" "height" => 1997 "path" => "/media/fizika_10/gelfgat-z/0-olimp/79-1.webp?ts=1756251240" ] 1 => array:5 [ "name" => "79-2.jpg" "alt" => null "width" => "1214" "height" => 372 "path" => "/media/fizika_10/gelfgat-z/0-olimp/79-2.webp?ts=1756251240" ] ] ] #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 {#1209 #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" => 1691614 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Edition {#1208 #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 {#1205 #items: array:1 [ 0 => App\Models\Term {#1207 …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 {#1204 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1202 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1203 #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 {#1205} "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 {#1204} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1202} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1203} "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" => "1319607" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Идеальный газ (будем считать одноатомным)</p><p>Цикл 1: 1-2-3-4-1</p><p>Цикл 2: 5-6-7-4-5</p><p>Параметры состояний газа в характерных точках циклов (из pV-диаграммы):</p><p>Точка 1: $p_1 = p_0, V_1 = V_0$</p><p>Точка 2: $p_2 = 2p_0, V_2 = 2V_0$</p><p>Точка 3: $p_3 = 2p_0, V_3 = 3V_0$</p><p>Точка 4: $p_4 = p_0, V_4 = 3V_0$</p><p>Точка 5: $p_5 = p_0, V_5 = 2V_0$</p><p>Точка 6: $p_6 = 3p_0, V_6 = 2V_0$</p><p>Точка 7: $p_7 = 3p_0, V_7 = 3V_0$</p><p><strong>Найти:</strong></p><p>$\frac{\eta_1}{\eta_2}$</p><p><strong>Решение:</strong></p><p>Коэффициент полезного действия (КПД) теплового двигателя определяется по формуле:</p><p>$\eta = \frac{A'}{Q_H}$,</p><p>где $A'$ – работа, совершаемая газом за цикл, а $Q_H$ – количество теплоты, полученное газом от нагревателя за цикл.</p><p>Рассмотрим оба цикла по отдельности.</p><p><strong>Цикл 1 (1-2-3-4-1)</strong></p><p><strong>1.</strong> <em>Работа газа за цикл $A'_1$</em>. Работа за цикл численно равна площади фигуры, ограниченной графиком цикла на pV-диаграмме. Разобьем работу на участки:</p><p>$A'_{1} = A_{12} + A_{23} + A_{34} + A_{41}$</p><p>Работа на участке 1-2 (линейное изменение давления с объемом) равна площади трапеции под графиком:</p><p>$A_{12} = \frac{p_1 + p_2}{2}(V_2 - V_1) = \frac{p_0 + 2p_0}{2}(2V_0 - V_0) = \frac{3p_0}{2}V_0 = 1.5p_0V_0$</p><p>Работа на участке 2-3 (изобарное расширение):</p><p>$A_{23} = p_2(V_3 - V_2) = 2p_0(3V_0 - 2V_0) = 2p_0V_0$</p><p>Работа на участке 3-4 (изохорное охлаждение) равна нулю: $A_{34} = 0$.</p><p>Работа на участке 4-1 (изобарное сжатие):</p><p>$A_{41} = p_4(V_1 - V_4) = p_0(V_0 - 3V_0) = -2p_0V_0$</p><p>Суммарная работа за цикл 1:</p><p>$A'_1 = 1.5p_0V_0 + 2p_0V_0 + 0 - 2p_0V_0 = 1.5p_0V_0$</p><p><strong>2.</strong> <em>Количество теплоты $Q_{H1}$</em>. Газ получает теплоту на тех участках, где его температура растет. Это участки 1-2 и 2-3. Найдем количество теплоты, используя первый закон термодинамики $Q = \Delta U + A$. Для одноатомного идеального газа изменение внутренней энергии $\Delta U = \frac{3}{2}\nu R \Delta T = \frac{3}{2}\Delta(pV)$.</p><p>На участке 1-2:</p><p>$Q_{12} = \Delta U_{12} + A_{12} = \frac{3}{2}(p_2V_2 - p_1V_1) + A_{12} = \frac{3}{2}(2p_0 \cdot 2V_0 - p_0V_0) + 1.5p_0V_0 = \frac{3}{2}(3p_0V_0) + 1.5p_0V_0 = 4.5p_0V_0 + 1.5p_0V_0 = 6p_0V_0$</p><p>На участке 2-3:</p><p>$Q_{23} = \Delta U_{23} + A_{23} = \frac{3}{2}(p_3V_3 - p_2V_2) + A_{23} = \frac{3}{2}(2p_0 \cdot 3V_0 - 2p_0 \cdot 2V_0) + 2p_0V_0 = \frac{3}{2}(2p_0V_0) + 2p_0V_0 = 3p_0V_0 + 2p_0V_0 = 5p_0V_0$</p><p>Суммарное количество полученной теплоты:</p><p>$Q_{H1} = Q_{12} + Q_{23} = 6p_0V_0 + 5p_0V_0 = 11p_0V_0$</p><p><strong>3.</strong> <em>КПД цикла 1</em>:</p><p>$\eta_1 = \frac{A'_1}{Q_{H1}} = \frac{1.5p_0V_0}{11p_0V_0} = \frac{3}{22}$</p><p><strong>Цикл 2 (5-6-7-4-5)</strong></p><p><strong>1.</strong> <em>Работа газа за цикл $A'_2$</em>. Цикл представляет собой прямоугольник на pV-диаграмме, его площадь:</p><p>$A'_2 = (p_6 - p_5)(V_7 - V_6) = (3p_0 - p_0)(3V_0 - 2V_0) = (2p_0)(V_0) = 2p_0V_0$</p><p><strong>2.</strong> <em>Количество теплоты $Q_{H2}$</em>. Газ получает теплоту на участках 5-6 (изохорное нагревание) и 6-7 (изобарное расширение).</p><p>На участке 5-6 ($A_{56} = 0$):</p><p>$Q_{56} = \Delta U_{56} = \frac{3}{2}(p_6V_6 - p_5V_5) = \frac{3}{2}(3p_0 \cdot 2V_0 - p_0 \cdot 2V_0) = \frac{3}{2}(4p_0V_0) = 6p_0V_0$</p><p>На участке 6-7:</p><p>$Q_{67} = \Delta U_{67} + A_{67} = \frac{3}{2}(p_7V_7 - p_6V_6) + p_6(V_7-V_6) = \frac{3}{2}(3p_0 \cdot 3V_0 - 3p_0 \cdot 2V_0) + 3p_0(3V_0 - 2V_0) = \frac{3}{2}(3p_0V_0) + 3p_0V_0 = 4.5p_0V_0 + 3p_0V_0 = 7.5p_0V_0$</p><p>Суммарное количество полученной теплоты:</p><p>$Q_{H2} = Q_{56} + Q_{67} = 6p_0V_0 + 7.5p_0V_0 = 13.5p_0V_0$</p><p><strong>3.</strong> <em>КПД цикла 2</em>:</p><p>$\eta_2 = \frac{A'_2}{Q_{H2}} = \frac{2p_0V_0}{13.5p_0V_0} = \frac{2}{13.5} = \frac{4}{27}$</p><p><strong>Отношение КПД</strong></p><p>Теперь найдем искомое отношение коэффициентов полезного действия:</p><p>$\frac{\eta_1}{\eta_2} = \frac{3/22}{4/27} = \frac{3}{22} \cdot \frac{27}{4} = \frac{81}{88}$</p><p><strong>Ответ:</strong> $\frac{\eta_1}{\eta_2} = \frac{81}{88}$.</p>" ] #original: array:6 [ "id" => 1691614 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1198} "task" => array:2 [ "refs" => "1319607" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Идеальный газ (будем считать одноатомным)</p><p>Цикл 1: 1-2-3-4-1</p><p>Цикл 2: 5-6-7-4-5</p><p>Параметры состояний газа в характерных точках циклов (из pV-диаграммы):</p><p>Точка 1: $p_1 = p_0, V_1 = V_0$</p><p>Точка 2: $p_2 = 2p_0, V_2 = 2V_0$</p><p>Точка 3: $p_3 = 2p_0, V_3 = 3V_0$</p><p>Точка 4: $p_4 = p_0, V_4 = 3V_0$</p><p>Точка 5: $p_5 = p_0, V_5 = 2V_0$</p><p>Точка 6: $p_6 = 3p_0, V_6 = 2V_0$</p><p>Точка 7: $p_7 = 3p_0, V_7 = 3V_0$</p><p><strong>Найти:</strong></p><p>$\frac{\eta_1}{\eta_2}$</p><p><strong>Решение:</strong></p><p>Коэффициент полезного действия (КПД) теплового двигателя определяется по формуле:</p><p>$\eta = \frac{A'}{Q_H}$,</p><p>где $A'$ – работа, совершаемая газом за цикл, а $Q_H$ – количество теплоты, полученное газом от нагревателя за цикл.</p><p>Рассмотрим оба цикла по отдельности.</p><p><strong>Цикл 1 (1-2-3-4-1)</strong></p><p><strong>1.</strong> <em>Работа газа за цикл $A'_1$</em>. Работа за цикл численно равна площади фигуры, ограниченной графиком цикла на pV-диаграмме. Разобьем работу на участки:</p><p>$A'_{1} = A_{12} + A_{23} + A_{34} + A_{41}$</p><p>Работа на участке 1-2 (линейное изменение давления с объемом) равна площади трапеции под графиком:</p><p>$A_{12} = \frac{p_1 + p_2}{2}(V_2 - V_1) = \frac{p_0 + 2p_0}{2}(2V_0 - V_0) = \frac{3p_0}{2}V_0 = 1.5p_0V_0$</p><p>Работа на участке 2-3 (изобарное расширение):</p><p>$A_{23} = p_2(V_3 - V_2) = 2p_0(3V_0 - 2V_0) = 2p_0V_0$</p><p>Работа на участке 3-4 (изохорное охлаждение) равна нулю: $A_{34} = 0$.</p><p>Работа на участке 4-1 (изобарное сжатие):</p><p>$A_{41} = p_4(V_1 - V_4) = p_0(V_0 - 3V_0) = -2p_0V_0$</p><p>Суммарная работа за цикл 1:</p><p>$A'_1 = 1.5p_0V_0 + 2p_0V_0 + 0 - 2p_0V_0 = 1.5p_0V_0$</p><p><strong>2.</strong> <em>Количество теплоты $Q_{H1}$</em>. Газ получает теплоту на тех участках, где его температура растет. Это участки 1-2 и 2-3. Найдем количество теплоты, используя первый закон термодинамики $Q = \Delta U + A$. Для одноатомного идеального газа изменение внутренней энергии $\Delta U = \frac{3}{2}\nu R \Delta T = \frac{3}{2}\Delta(pV)$.</p><p>На участке 1-2:</p><p>$Q_{12} = \Delta U_{12} + A_{12} = \frac{3}{2}(p_2V_2 - p_1V_1) + A_{12} = \frac{3}{2}(2p_0 \cdot 2V_0 - p_0V_0) + 1.5p_0V_0 = \frac{3}{2}(3p_0V_0) + 1.5p_0V_0 = 4.5p_0V_0 + 1.5p_0V_0 = 6p_0V_0$</p><p>На участке 2-3:</p><p>$Q_{23} = \Delta U_{23} + A_{23} = \frac{3}{2}(p_3V_3 - p_2V_2) + A_{23} = \frac{3}{2}(2p_0 \cdot 3V_0 - 2p_0 \cdot 2V_0) + 2p_0V_0 = \frac{3}{2}(2p_0V_0) + 2p_0V_0 = 3p_0V_0 + 2p_0V_0 = 5p_0V_0$</p><p>Суммарное количество полученной теплоты:</p><p>$Q_{H1} = Q_{12} + Q_{23} = 6p_0V_0 + 5p_0V_0 = 11p_0V_0$</p><p><strong>3.</strong> <em>КПД цикла 1</em>:</p><p>$\eta_1 = \frac{A'_1}{Q_{H1}} = \frac{1.5p_0V_0}{11p_0V_0} = \frac{3}{22}$</p><p><strong>Цикл 2 (5-6-7-4-5)</strong></p><p><strong>1.</strong> <em>Работа газа за цикл $A'_2$</em>. Цикл представляет собой прямоугольник на pV-диаграмме, его площадь:</p><p>$A'_2 = (p_6 - p_5)(V_7 - V_6) = (3p_0 - p_0)(3V_0 - 2V_0) = (2p_0)(V_0) = 2p_0V_0$</p><p><strong>2.</strong> <em>Количество теплоты $Q_{H2}$</em>. Газ получает теплоту на участках 5-6 (изохорное нагревание) и 6-7 (изобарное расширение).</p><p>На участке 5-6 ($A_{56} = 0$):</p><p>$Q_{56} = \Delta U_{56} = \frac{3}{2}(p_6V_6 - p_5V_5) = \frac{3}{2}(3p_0 \cdot 2V_0 - p_0 \cdot 2V_0) = \frac{3}{2}(4p_0V_0) = 6p_0V_0$</p><p>На участке 6-7:</p><p>$Q_{67} = \Delta U_{67} + A_{67} = \frac{3}{2}(p_7V_7 - p_6V_6) + p_6(V_7-V_6) = \frac{3}{2}(3p_0 \cdot 3V_0 - 3p_0 \cdot 2V_0) + 3p_0(3V_0 - 2V_0) = \frac{3}{2}(3p_0V_0) + 3p_0V_0 = 4.5p_0V_0 + 3p_0V_0 = 7.5p_0V_0$</p><p>Суммарное количество полученной теплоты:</p><p>$Q_{H2} = Q_{56} + Q_{67} = 6p_0V_0 + 7.5p_0V_0 = 13.5p_0V_0$</p><p><strong>3.</strong> <em>КПД цикла 2</em>:</p><p>$\eta_2 = \frac{A'_2}{Q_{H2}} = \frac{2p_0V_0}{13.5p_0V_0} = \frac{2}{13.5} = \frac{4}{27}$</p><p><strong>Отношение КПД</strong></p><p>Теперь найдем искомое отношение коэффициентов полезного действия:</p><p>$\frac{\eta_1}{\eta_2} = \frac{3/22}{4/27} = \frac{3}{22} \cdot \frac{27}{4} = \frac{81}{88}$</p><p><strong>Ответ:</strong> $\frac{\eta_1}{\eta_2} = \frac{81}{88}$.</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" => "1319608" "type" => "task" ] "previous" => array:2 [ "refs" => "1319606" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:1 [ 0 => App\Models\Book {#1239} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\BookPage {#1220 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false 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№79 (с. 137)
Условие. №79 (с. 137)
скриншот условия
O-79. Определите отношение $\frac{\eta_1}{\eta_2}$ коэффициентов полезного действия двух циклических процессов, проведенных с идеальным газом (см. рисунок): первый процесс 1-2-3-4, второй процесс 5-6-7-4-5.
☑ 27/23.
Решение. КПД цикла $\eta = \frac{A'}{Q_{\text{н}}}$, где $A'$ — работа газа за цикл; $Q_{\text{н}}$ — количество теплоты, полученное от нагревателя за цикл. Работе газа за цикл на $pV$-диаграмме соответствует площадь, заключенная «внутри» цикла. Из условия задачи следует, что работа газа в первом и во втором процессах одна и та же. Следовательно, $\frac{\eta_1}{\eta_2} = \frac{Q_{\text{н}2}}{Q_{\text{н}1}}$, где $Q_{\text{н}1}$ и $Q_{\text{н}2}$ — количества теплоты, полученные системой соответственно в первом и во втором циклах. Найдем $Q_{\text{н}1}$ и $Q_{\text{н}2}$. Пусть температура газа в точке 1 (см. рисунок) была $T_1$. Воспользовавшись уравнением состояния идеального газа, нетрудно показать, что $T_2 = T_5 = 2T_1$, $T_3 = T_6 = 6T_1$, $T_4 = 3T_1$, $T_7 = 9T_1$. Очевидно, что в процессе 1-2-3-4 газ получил тепло $Q_{\text{н}1}$ на участках 1-2 и 2-3, причем $Q_{\text{н}1} = A_1 + \Delta U_1$, где $A_1 = 4p_0V_0 = 4\nu RT_1$ — работа, совершенная на этих участках газом, $\Delta U_1 = \frac{3}{2}\nu R(T_3 - T_1) = \frac{15}{2}\nu RT_1$ — изменение его внутренней энергии, так что $Q_{\text{н}1} = 4\nu RT_1 + \frac{15}{2}\nu RT_1 = \frac{23}{2}\nu RT_1$.
В процессе 5-6-7-4-5 газ получил тепло $Q_{\text{н}2}$ на участках 5-6 и 6-7, причем
$Q_{\text{н}2} = A_2 + \Delta U_2 = 3p_0V_0 + \frac{3}{2}\nu R(T_7 - T_5) = \frac{27}{2}\nu RT_1$.
Таким образом, $\frac{\eta_1}{\eta_2} = \frac{Q_{\text{н}2}}{Q_{\text{н}1}} = \frac{27}{23}$.
Решение. №79 (с. 137)
Дано:
Идеальный газ (будем считать одноатомным)
Цикл 1: 1-2-3-4-1
Цикл 2: 5-6-7-4-5
Параметры состояний газа в характерных точках циклов (из pV-диаграммы):
Точка 1: $p_1 = p_0, V_1 = V_0$
Точка 2: $p_2 = 2p_0, V_2 = 2V_0$
Точка 3: $p_3 = 2p_0, V_3 = 3V_0$
Точка 4: $p_4 = p_0, V_4 = 3V_0$
Точка 5: $p_5 = p_0, V_5 = 2V_0$
Точка 6: $p_6 = 3p_0, V_6 = 2V_0$
Точка 7: $p_7 = 3p_0, V_7 = 3V_0$
Найти:
$\frac{\eta_1}{\eta_2}$
Решение:
Коэффициент полезного действия (КПД) теплового двигателя определяется по формуле:
$\eta = \frac{A'}{Q_H}$,
где $A'$ – работа, совершаемая газом за цикл, а $Q_H$ – количество теплоты, полученное газом от нагревателя за цикл.
Рассмотрим оба цикла по отдельности.
Цикл 1 (1-2-3-4-1)
1. Работа газа за цикл $A'_1$. Работа за цикл численно равна площади фигуры, ограниченной графиком цикла на pV-диаграмме. Разобьем работу на участки:
$A'_{1} = A_{12} + A_{23} + A_{34} + A_{41}$
Работа на участке 1-2 (линейное изменение давления с объемом) равна площади трапеции под графиком:
$A_{12} = \frac{p_1 + p_2}{2}(V_2 - V_1) = \frac{p_0 + 2p_0}{2}(2V_0 - V_0) = \frac{3p_0}{2}V_0 = 1.5p_0V_0$
Работа на участке 2-3 (изобарное расширение):
$A_{23} = p_2(V_3 - V_2) = 2p_0(3V_0 - 2V_0) = 2p_0V_0$
Работа на участке 3-4 (изохорное охлаждение) равна нулю: $A_{34} = 0$.
Работа на участке 4-1 (изобарное сжатие):
$A_{41} = p_4(V_1 - V_4) = p_0(V_0 - 3V_0) = -2p_0V_0$
Суммарная работа за цикл 1:
$A'_1 = 1.5p_0V_0 + 2p_0V_0 + 0 - 2p_0V_0 = 1.5p_0V_0$
2. Количество теплоты $Q_{H1}$. Газ получает теплоту на тех участках, где его температура растет. Это участки 1-2 и 2-3. Найдем количество теплоты, используя первый закон термодинамики $Q = \Delta U + A$. Для одноатомного идеального газа изменение внутренней энергии $\Delta U = \frac{3}{2}\nu R \Delta T = \frac{3}{2}\Delta(pV)$.
На участке 1-2:
$Q_{12} = \Delta U_{12} + A_{12} = \frac{3}{2}(p_2V_2 - p_1V_1) + A_{12} = \frac{3}{2}(2p_0 \cdot 2V_0 - p_0V_0) + 1.5p_0V_0 = \frac{3}{2}(3p_0V_0) + 1.5p_0V_0 = 4.5p_0V_0 + 1.5p_0V_0 = 6p_0V_0$
На участке 2-3:
$Q_{23} = \Delta U_{23} + A_{23} = \frac{3}{2}(p_3V_3 - p_2V_2) + A_{23} = \frac{3}{2}(2p_0 \cdot 3V_0 - 2p_0 \cdot 2V_0) + 2p_0V_0 = \frac{3}{2}(2p_0V_0) + 2p_0V_0 = 3p_0V_0 + 2p_0V_0 = 5p_0V_0$
Суммарное количество полученной теплоты:
$Q_{H1} = Q_{12} + Q_{23} = 6p_0V_0 + 5p_0V_0 = 11p_0V_0$
3. КПД цикла 1:
$\eta_1 = \frac{A'_1}{Q_{H1}} = \frac{1.5p_0V_0}{11p_0V_0} = \frac{3}{22}$
Цикл 2 (5-6-7-4-5)
1. Работа газа за цикл $A'_2$. Цикл представляет собой прямоугольник на pV-диаграмме, его площадь:
$A'_2 = (p_6 - p_5)(V_7 - V_6) = (3p_0 - p_0)(3V_0 - 2V_0) = (2p_0)(V_0) = 2p_0V_0$
2. Количество теплоты $Q_{H2}$. Газ получает теплоту на участках 5-6 (изохорное нагревание) и 6-7 (изобарное расширение).
На участке 5-6 ($A_{56} = 0$):
$Q_{56} = \Delta U_{56} = \frac{3}{2}(p_6V_6 - p_5V_5) = \frac{3}{2}(3p_0 \cdot 2V_0 - p_0 \cdot 2V_0) = \frac{3}{2}(4p_0V_0) = 6p_0V_0$
На участке 6-7:
$Q_{67} = \Delta U_{67} + A_{67} = \frac{3}{2}(p_7V_7 - p_6V_6) + p_6(V_7-V_6) = \frac{3}{2}(3p_0 \cdot 3V_0 - 3p_0 \cdot 2V_0) + 3p_0(3V_0 - 2V_0) = \frac{3}{2}(3p_0V_0) + 3p_0V_0 = 4.5p_0V_0 + 3p_0V_0 = 7.5p_0V_0$
Суммарное количество полученной теплоты:
$Q_{H2} = Q_{56} + Q_{67} = 6p_0V_0 + 7.5p_0V_0 = 13.5p_0V_0$
3. КПД цикла 2:
$\eta_2 = \frac{A'_2}{Q_{H2}} = \frac{2p_0V_0}{13.5p_0V_0} = \frac{2}{13.5} = \frac{4}{27}$
Отношение КПД
Теперь найдем искомое отношение коэффициентов полезного действия:
$\frac{\eta_1}{\eta_2} = \frac{3/22}{4/27} = \frac{3}{22} \cdot \frac{27}{4} = \frac{81}{88}$
Ответ: $\frac{\eta_1}{\eta_2} = \frac{81}{88}$.
Другие задания:
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