gdz.top
Номер 5, страница 458 - гдз по физике 10 класс учебник Касьянов
Авторы: Касьянов В. А.
Тип: Учебник
Издательство: Просвещение
Год издания: 2021 - 2026
Уровень обучения: углублённый
Цвет обложки: белый самолет и молнии изображены
ISBN: 978-5-09-103621-3
Популярные ГДЗ в 10 классе
Лабораторные работы. 7. Изучение капиллярных явлений, обусловленных поверхностным натяжением жидкости - номер 5, страница 458.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1390621 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1389652 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Лабораторные работы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "426" "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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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" => 7000 "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 {#1032 #items: array:1 [ 0 => App\Models\Term {#1034 #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 {#1035 #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" => 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 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1040 #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" => 4566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Касьянов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валерий" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 4566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Касьянов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валерий" "field_patronymic" => "Алексеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1041 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\Term {#1043 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 #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 {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #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 {#1048 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1049 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1051 #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" => 39 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "углублённый" "field_cases" => array:6 [ "field_accusative_case" => "углубленный" "field_creative_case" => "углубленным" "field_dative_case" => "углубленному" "field_genitive_case" => "углубленного" "field_nominative_case" => "углубленный" "field_prepositional_case" => "углубленном" ] "field_translit" => "uglublennyy" ] #original: array:6 [ "id" => 39 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "углублённый" "field_cases" => array:6 [ "field_accusative_case" => "углубленный" "field_creative_case" => "углубленным" "field_dative_case" => "углубленному" "field_genitive_case" => "углубленного" "field_nominative_case" => "углубленный" "field_prepositional_case" => "углубленном" ] "field_translit" => "uglublennyy" ] #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 {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "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" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "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" => "stereotipnoe" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "самолет и молнии изображены" "field_publication_year" => "2021" "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" => "984" "field_priority" => null "field_default_folder" => "/fizika_10/kasyanov-u/" "field_isbn" => "978-5-09-103621-3" "field_cover" => array:1 [ 0 => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" "alt" => "" "width" => "1604" "height" => "2397" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/kasyanov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 45 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "белый" "field_cases" => array:6 [ "field_accusative_case" => "белый" "field_creative_case" => "белым" "field_dative_case" => "белому" "field_genitive_case" => "белого" "field_nominative_case" => "белый" "field_prepositional_case" => "белом" ] "field_translit" => "belyy" ] #original: array:6 [ "id" => 45 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "белый" "field_cases" => array:6 [ "field_accusative_case" => "белый" "field_creative_case" => "белым" "field_dative_case" => "белому" "field_genitive_case" => "белого" "field_nominative_case" => "белый" "field_prepositional_case" => "белом" ] "field_translit" => "belyy" ] #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" => 7000 "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 {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1041} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1042} "field_country" => Illuminate\Database\Eloquent\Collection {#1044} "field_city" => Illuminate\Database\Eloquent\Collection {#1046} "field_series" => Illuminate\Database\Eloquent\Collection {#1048} "field_umk" => Illuminate\Database\Eloquent\Collection {#1049} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1050} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056} "field_under_the_edition_degree" => null "field_cover_description" => "самолет и молнии изображены" "field_publication_year" => "2021" "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" => "984" "field_priority" => null "field_default_folder" => "/fizika_10/kasyanov-u/" "field_isbn" => "978-5-09-103621-3" "field_cover" => array:1 [ 0 => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" "alt" => "" "width" => "1604" "height" => "2397" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1057} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059} "field_url" => "/10-klass/fizika/kasyanov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 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 {#1062 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1389652 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Лабораторные работы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "426" "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 {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1062} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1064 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1389661 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "7. Изучение капиллярных явлений, обусловленных поверхностным натяжением жидкости" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "457" "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 {#1066 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:1 [ 0 => App\Models\Branch {#1068 #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" => 1389652 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Лабораторные работы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "426" "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 {#1070 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1389652 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Лабораторные работы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_page_start" => "426" "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 {#1070} "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 } ] #original: array:24 [ "id" => 1389661 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "7. Изучение капиллярных явлений, обусловленных поверхностным натяжением жидкости" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1065} "field_page_start" => "457" "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 {#1066} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1067} ] #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 {#1072 #items: array:2 [ 0 => App\Models\Element {#1073 #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" => 1641050 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Edition {#1075 #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" => 5810 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Term {#1077 …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 {#1078 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5810 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1076} "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 {#1078} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1079} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080} "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" => "1390621" "type" => "task" ] "text" => "<p><strong>5.</strong> Рассчитайте абсолютные погрешности $\Delta D_1$ и $\Delta D_2$ при косвенном измерении диаметра капилляров.</p><p>$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$, $\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$.</p><p>Погрешностями $\Delta g$ и $\Delta \rho$ можно пренебречь.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "1707" "height" => 415 "path" => "/media/fizika_10/kasyanov-u/0-lab07/5-1.webp?ts=1757958251" ] ] ] #original: array:7 [ "id" => 1641050 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1074} "task" => array:2 [ "refs" => "1390621" "type" => "task" ] "text" => "<p><strong>5.</strong> Рассчитайте абсолютные погрешности $\Delta D_1$ и $\Delta D_2$ при косвенном измерении диаметра капилляров.</p><p>$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$, $\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$.</p><p>Погрешностями $\Delta g$ и $\Delta \rho$ можно пренебречь.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "1707" "height" => 415 "path" => "/media/fizika_10/kasyanov-u/0-lab07/5-1.webp?ts=1757958251" ] ] ] #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 {#1081 #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" => 1693633 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Edition {#1083 #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" => 6145 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Term {#1085 …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 {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 6145 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1084} "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 {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "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" => "1390621" "type" => "task" ] "text" => "<p><strong>Решение</strong></p><p>Задачей является расчет (в данном контексте — теоретический вывод) формул для абсолютных погрешностей $\Delta D_1$ и $\Delta D_2$ при косвенном измерении диаметров капилляров. Данные формулы выводятся на основе физического закона, описывающего подъем жидкости в капилляре (закон Жюрена), и общих правил расчета погрешностей косвенных измерений.</p><p><strong>1.</strong> <strong>Вывод рабочей формулы для диаметра капилляра.</strong></p><p>Диаметр капилляра $D$ определяется из условия равновесия столбика жидкости высотой $h$, поднявшегося в капилляре. Сила поверхностного натяжения $F_{\sigma}$, действующая вверх по периметру смачивания, уравновешивается силой тяжести $F_{g}$ столбика жидкости.</p><p>Сила поверхностного натяжения: $F_{\sigma} = \pi D \sigma \cos \theta$, где $\sigma$ – коэффициент поверхностного натяжения жидкости, а $\theta$ – краевой угол смачивания. Для полного смачивания (например, вода в чистом стеклянном капилляре) $\cos \theta \approx 1$, и формула упрощается: $F_{\sigma} = \pi D \sigma$.</p><p>Сила тяжести столбика жидкости: $F_{g} = mg = \rho V g$. Объем столбика жидкости (цилиндра) $V = \frac{\pi D^2}{4} h$. Таким образом, сила тяжести: $F_{g} = \rho \left( \frac{\pi D^2}{4} h \right) g$, где $\rho$ – плотность жидкости, а $g$ – ускорение свободного падения.</p><p>Приравнивая силы в состоянии равновесия $F_{\sigma} = F_{g}$:$\pi D \sigma = \rho g \frac{\pi D^2}{4} h$</p><p>Выражая из этого уравнения диаметр $D$, получаем расчетную формулу для косвенного измерения:$D = \frac{4\sigma}{\rho g h}$</p><p><strong>2.</strong> <strong>Вывод формулы для погрешности.</strong></p><p>Поскольку диаметр $D$ вычисляется на основе измерений других величин ($\sigma$, $h$, $\rho$, $g$), его погрешность $\Delta D$ является погрешностью косвенного измерения. Для нахождения максимальной абсолютной погрешности $\Delta D$ сначала найдем максимальную относительную погрешность $\frac{\Delta D}{D}$. Для функции, представляющей собой произведение и частное, относительная погрешность равна сумме относительных погрешностей измеряемых величин:$\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta \rho}{\rho} + \frac{\Delta g}{g} + \frac{\Delta h}{h}$</p><p>В условии задачи указано, что погрешностями $\Delta g$ и $\Delta \rho$ можно пренебречь. Это означает, что мы принимаем их вклад в общую погрешность равным нулю ($\frac{\Delta \rho}{\rho} \approx 0$ и $\frac{\Delta g}{g} \approx 0$).</p><p>С учетом этого упрощения формула для относительной погрешности диаметра капилляра принимает вид:$\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h}$</p><p>Абсолютная погрешность $\Delta D$ связана с относительной погрешностью соотношением $\Delta D = D \cdot \frac{\Delta D}{D}$. Таким образом, общая формула для абсолютной погрешности диаметра:$\Delta D = D \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h} \right)$</p><p><strong>3.</strong> <strong>Применение формулы к двум капиллярам.</strong></p><p>Применяя эту общую формулу для двух разных капилляров (с индексами 1 и 2), мы получаем выражения, приведенные в условии задачи. Для первого капилляра с диаметром $D_1$ и высотой подъема жидкости $h_1$ (с погрешностью $\Delta h_1$):$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$</p><p>Для второго капилляра с диаметром $D_2$ и высотой подъема жидкости $h_2$ (с погрешностью $\Delta h_2$):$\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$</p><p>Таким образом, приведенные в задаче формулы являются следствием применения стандартного метода расчета погрешностей косвенных измерений к формуле для диаметра капилляра, полученной из закона Жюрена, с учетом указанных в условии пренебрежений.</p><p><strong>Ответ:</strong> Вывод формул, приведенных в условии, основан на законе Жюрена для капиллярного подъема жидкости ($D = \frac{4\sigma}{\rho g h}$) и правиле сложения относительных погрешностей для косвенных измерений. При пренебрежении погрешностями $\Delta \rho$ и $\Delta g$, относительная погрешность диаметра равна $\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h}$. Отсюда следуют искомые формулы для абсолютных погрешностей:$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$ и $\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$.</p>" ] #original: array:6 [ "id" => 1693633 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1082} "task" => array:2 [ "refs" => "1390621" "type" => "task" ] "text" => "<p><strong>Решение</strong></p><p>Задачей является расчет (в данном контексте — теоретический вывод) формул для абсолютных погрешностей $\Delta D_1$ и $\Delta D_2$ при косвенном измерении диаметров капилляров. Данные формулы выводятся на основе физического закона, описывающего подъем жидкости в капилляре (закон Жюрена), и общих правил расчета погрешностей косвенных измерений.</p><p><strong>1.</strong> <strong>Вывод рабочей формулы для диаметра капилляра.</strong></p><p>Диаметр капилляра $D$ определяется из условия равновесия столбика жидкости высотой $h$, поднявшегося в капилляре. Сила поверхностного натяжения $F_{\sigma}$, действующая вверх по периметру смачивания, уравновешивается силой тяжести $F_{g}$ столбика жидкости.</p><p>Сила поверхностного натяжения: $F_{\sigma} = \pi D \sigma \cos \theta$, где $\sigma$ – коэффициент поверхностного натяжения жидкости, а $\theta$ – краевой угол смачивания. Для полного смачивания (например, вода в чистом стеклянном капилляре) $\cos \theta \approx 1$, и формула упрощается: $F_{\sigma} = \pi D \sigma$.</p><p>Сила тяжести столбика жидкости: $F_{g} = mg = \rho V g$. Объем столбика жидкости (цилиндра) $V = \frac{\pi D^2}{4} h$. Таким образом, сила тяжести: $F_{g} = \rho \left( \frac{\pi D^2}{4} h \right) g$, где $\rho$ – плотность жидкости, а $g$ – ускорение свободного падения.</p><p>Приравнивая силы в состоянии равновесия $F_{\sigma} = F_{g}$:$\pi D \sigma = \rho g \frac{\pi D^2}{4} h$</p><p>Выражая из этого уравнения диаметр $D$, получаем расчетную формулу для косвенного измерения:$D = \frac{4\sigma}{\rho g h}$</p><p><strong>2.</strong> <strong>Вывод формулы для погрешности.</strong></p><p>Поскольку диаметр $D$ вычисляется на основе измерений других величин ($\sigma$, $h$, $\rho$, $g$), его погрешность $\Delta D$ является погрешностью косвенного измерения. Для нахождения максимальной абсолютной погрешности $\Delta D$ сначала найдем максимальную относительную погрешность $\frac{\Delta D}{D}$. Для функции, представляющей собой произведение и частное, относительная погрешность равна сумме относительных погрешностей измеряемых величин:$\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta \rho}{\rho} + \frac{\Delta g}{g} + \frac{\Delta h}{h}$</p><p>В условии задачи указано, что погрешностями $\Delta g$ и $\Delta \rho$ можно пренебречь. Это означает, что мы принимаем их вклад в общую погрешность равным нулю ($\frac{\Delta \rho}{\rho} \approx 0$ и $\frac{\Delta g}{g} \approx 0$).</p><p>С учетом этого упрощения формула для относительной погрешности диаметра капилляра принимает вид:$\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h}$</p><p>Абсолютная погрешность $\Delta D$ связана с относительной погрешностью соотношением $\Delta D = D \cdot \frac{\Delta D}{D}$. Таким образом, общая формула для абсолютной погрешности диаметра:$\Delta D = D \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h} \right)$</p><p><strong>3.</strong> <strong>Применение формулы к двум капиллярам.</strong></p><p>Применяя эту общую формулу для двух разных капилляров (с индексами 1 и 2), мы получаем выражения, приведенные в условии задачи. Для первого капилляра с диаметром $D_1$ и высотой подъема жидкости $h_1$ (с погрешностью $\Delta h_1$):$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$</p><p>Для второго капилляра с диаметром $D_2$ и высотой подъема жидкости $h_2$ (с погрешностью $\Delta h_2$):$\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$</p><p>Таким образом, приведенные в задаче формулы являются следствием применения стандартного метода расчета погрешностей косвенных измерений к формуле для диаметра капилляра, полученной из закона Жюрена, с учетом указанных в условии пренебрежений.</p><p><strong>Ответ:</strong> Вывод формул, приведенных в условии, основан на законе Жюрена для капиллярного подъема жидкости ($D = \frac{4\sigma}{\rho g h}$) и правиле сложения относительных погрешностей для косвенных измерений. При пренебрежении погрешностями $\Delta \rho$ и $\Delta g$, относительная погрешность диаметра равна $\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h}$. Отсюда следуют искомые формулы для абсолютных погрешностей:$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$ и $\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$.</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" => "1390622" "type" => "task" ] "previous" => array:2 [ "refs" => "1390620" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\BookPage {#1094 #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" => 1391106 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_url" => "/10-klass/fizika/kasyanov-uchebnik/page-458" "field_display_title" => "458" "field_folder" => "1" "field_image_name" => "458" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Book {#1098 #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" => 7000 "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 {#1099 #items: array:1 [ 0 => App\Models\Term {#1100 …30} ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Term {#1102 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Term {#1104 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Term {#1106 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Term {#1109 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Term {#1111 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Term {#1113 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1115 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Term {#1117 …30} ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Term {#1119 …30} ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Term {#1121 …30} ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "самолет и молнии изображены" "field_publication_year" => "2021" "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" => "984" "field_priority" => null "field_default_folder" => "/fizika_10/kasyanov-u/" "field_isbn" => "978-5-09-103621-3" "field_cover" => array:1 [ 0 => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" "alt" => "" "width" => "1604" "height" => "2397" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1124 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/fizika/kasyanov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 556 "class_subject" => 494 ] ] #original: array:50 [ "id" => 7000 "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 {#1099} "field_class" => Illuminate\Database\Eloquent\Collection {#1101} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103} "field_author" => Illuminate\Database\Eloquent\Collection {#1105} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1107} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1108} "field_country" => Illuminate\Database\Eloquent\Collection {#1110} "field_city" => Illuminate\Database\Eloquent\Collection {#1112} "field_series" => Illuminate\Database\Eloquent\Collection {#1114} "field_umk" => Illuminate\Database\Eloquent\Collection {#1115} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1116} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1118} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1120} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1122} "field_under_the_edition_degree" => null "field_cover_description" => "самолет и молнии изображены" "field_publication_year" => "2021" "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" => "984" "field_priority" => null "field_default_folder" => "/fizika_10/kasyanov-u/" "field_isbn" => "978-5-09-103621-3" "field_cover" => array:1 [ 0 => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/fizika_10/kasyanov-u/covers/cover1.webp?ts=1757588862" "alt" => "" "width" => "1604" "height" => "2397" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1123} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1124} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1125} "field_url" => "/10-klass/fizika/kasyanov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1126} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 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 } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ 0 => App\Models\Element {#1130 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1641606 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Edition {#1132 …30} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "458-1.jpg" "alt" => null "width" => "1868" "height" => 2533 "path" => "/media/fizika_10/kasyanov-u/0-1/458-1.webp?ts=1757958810" ] ] ] #original: array:6 [ "id" => 1641606 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131} "book_page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "458-1.jpg" "alt" => null "width" => "1868" "height" => 2533 "path" => "/media/fizika_10/kasyanov-u/0-1/458-1.webp?ts=1757958810" ] ] ] #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" => "1391107" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1391105" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1138 #items: array:6 [ 0 => App\Models\Task {#1139 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1390617 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Branch {#1144 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:2 [ 0 => App\Models\Branch {#1149 …30} 1 => App\Models\Branch {#1153 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:2 [ 0 => App\Models\Element {#1158 …30} 1 => App\Models\Element {#1160 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1390618" "type" => "task" ] "previous" => array:2 [ "refs" => "1390616" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Book {#1098} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #original: array:24 [ "id" => 1390617 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1140} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1142} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1143} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1148} "content" => Illuminate\Database\Eloquent\Collection {#1157} "next" => array:2 [ "refs" => "1390618" "type" => "task" ] "previous" => array:2 [ "refs" => "1390616" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1168} "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1169 #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" => 1390618 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1171 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Branch {#1149 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:2 [ 0 => App\Models\Branch {#1149 …30} 1 => App\Models\Branch {#1153 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:2 [ 0 => App\Models\Element {#1175 …30} 1 => App\Models\Element {#1177 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1390619" "type" => "task" ] "previous" => array:2 [ "refs" => "1390617" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Book {#1098} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #original: array:24 [ "id" => 1390618 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1170} "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 {#1171} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1172} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1173} "content" => Illuminate\Database\Eloquent\Collection {#1174} "next" => array:2 [ "refs" => "1390619" "type" => "task" ] "previous" => array:2 [ "refs" => "1390617" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1179} "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1180 #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" => 1390619 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Branch {#1149 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1184 #items: array:2 [ 0 => App\Models\Branch {#1149 …30} 1 => App\Models\Branch {#1153 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:2 [ 0 => App\Models\Element {#1186 …30} 1 => App\Models\Element {#1188 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1390620" "type" => "task" ] "previous" => array:2 [ "refs" => "1390618" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Book {#1098} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #original: array:24 [ "id" => 1390619 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1181} "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 {#1182} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1183} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1184} "content" => Illuminate\Database\Eloquent\Collection {#1185} "next" => array:2 [ "refs" => "1390620" "type" => "task" ] "previous" => array:2 [ "refs" => "1390618" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1190} "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1191 #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" => 1390620 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1193 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Branch {#1149 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:2 [ 0 => App\Models\Branch {#1149 …30} 1 => App\Models\Branch {#1153 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:2 [ 0 => App\Models\Element {#1197 …30} 1 => App\Models\Element {#1199 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1390621" "type" => "task" ] "previous" => array:2 [ "refs" => "1390619" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ 0 => App\Models\Book {#1098} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #original: array:24 [ "id" => 1390620 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1192} "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 {#1193} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1194} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1195} "content" => Illuminate\Database\Eloquent\Collection {#1196} "next" => array:2 [ "refs" => "1390621" "type" => "task" ] "previous" => array:2 [ "refs" => "1390619" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1201} "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1202 #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" => 1390621 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Term {#1141 …30} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1204 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ 0 => App\Models\Branch {#1149 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:2 [ 0 => App\Models\Branch {#1149 …30} 1 => App\Models\Branch {#1153 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:2 [ 0 => App\Models\Element {#1208 …30} 1 => App\Models\Element {#1210 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1390622" "type" => "task" ] "previous" => array:2 [ "refs" => "1390620" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ 0 => App\Models\Book {#1098} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #original: array:24 [ "id" => 1390621 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1203} "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 {#1204} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1205} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1206} "content" => Illuminate\Database\Eloquent\Collection {#1207} "next" => array:2 [ "refs" => "1390622" "type" => "task" ] "previous" => array:2 [ "refs" => "1390620" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1212} "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1213 #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" => 1390622 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ …1] #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 {#1215 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:2 [ …2] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1390623" "type" => "task" ] "previous" => array:2 [ "refs" => "1390621" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #original: array:24 [ "id" => 1390622 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1214} "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 {#1215} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1216} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1217} "content" => Illuminate\Database\Eloquent\Collection {#1218} "next" => array:2 [ "refs" => "1390623" "type" => "task" ] "previous" => array:2 [ "refs" => "1390621" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1223} "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1391106 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_url" => "/10-klass/fizika/kasyanov-uchebnik/page-458" "field_display_title" => "458" "field_folder" => "1" "field_image_name" => "458" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1097} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1096} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1128} "content" => Illuminate\Database\Eloquent\Collection {#1129} "next" => array:2 [ "refs" => "1391107" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1391105" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1138} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1390621 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "458" "field_page_end" => null "field_url" => "/10-klass/fizika/kasyanov-uchebnik/lab07-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1063} "content" => Illuminate\Database\Eloquent\Collection {#1072} "next" => array:2 [ "refs" => "1390622" "type" => "task" ] "previous" => array:2 [ "refs" => "1390620" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1089} "page" => array:2 [ "refs" => "1391106" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№5 (с. 458)
Условие. №5 (с. 458)
скриншот условия
5. Рассчитайте абсолютные погрешности $\Delta D_1$ и $\Delta D_2$ при косвенном измерении диаметра капилляров.
$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$, $\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$.
Погрешностями $\Delta g$ и $\Delta \rho$ можно пренебречь.
Решение. №5 (с. 458)
Решение
Задачей является расчет (в данном контексте — теоретический вывод) формул для абсолютных погрешностей $\Delta D_1$ и $\Delta D_2$ при косвенном измерении диаметров капилляров. Данные формулы выводятся на основе физического закона, описывающего подъем жидкости в капилляре (закон Жюрена), и общих правил расчета погрешностей косвенных измерений.
1. Вывод рабочей формулы для диаметра капилляра.
Диаметр капилляра $D$ определяется из условия равновесия столбика жидкости высотой $h$, поднявшегося в капилляре. Сила поверхностного натяжения $F_{\sigma}$, действующая вверх по периметру смачивания, уравновешивается силой тяжести $F_{g}$ столбика жидкости.
Сила поверхностного натяжения: $F_{\sigma} = \pi D \sigma \cos \theta$, где $\sigma$ – коэффициент поверхностного натяжения жидкости, а $\theta$ – краевой угол смачивания. Для полного смачивания (например, вода в чистом стеклянном капилляре) $\cos \theta \approx 1$, и формула упрощается: $F_{\sigma} = \pi D \sigma$.
Сила тяжести столбика жидкости: $F_{g} = mg = \rho V g$. Объем столбика жидкости (цилиндра) $V = \frac{\pi D^2}{4} h$. Таким образом, сила тяжести: $F_{g} = \rho \left( \frac{\pi D^2}{4} h \right) g$, где $\rho$ – плотность жидкости, а $g$ – ускорение свободного падения.
Приравнивая силы в состоянии равновесия $F_{\sigma} = F_{g}$:$\pi D \sigma = \rho g \frac{\pi D^2}{4} h$
Выражая из этого уравнения диаметр $D$, получаем расчетную формулу для косвенного измерения:$D = \frac{4\sigma}{\rho g h}$
2. Вывод формулы для погрешности.
Поскольку диаметр $D$ вычисляется на основе измерений других величин ($\sigma$, $h$, $\rho$, $g$), его погрешность $\Delta D$ является погрешностью косвенного измерения. Для нахождения максимальной абсолютной погрешности $\Delta D$ сначала найдем максимальную относительную погрешность $\frac{\Delta D}{D}$. Для функции, представляющей собой произведение и частное, относительная погрешность равна сумме относительных погрешностей измеряемых величин:$\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta \rho}{\rho} + \frac{\Delta g}{g} + \frac{\Delta h}{h}$
В условии задачи указано, что погрешностями $\Delta g$ и $\Delta \rho$ можно пренебречь. Это означает, что мы принимаем их вклад в общую погрешность равным нулю ($\frac{\Delta \rho}{\rho} \approx 0$ и $\frac{\Delta g}{g} \approx 0$).
С учетом этого упрощения формула для относительной погрешности диаметра капилляра принимает вид:$\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h}$
Абсолютная погрешность $\Delta D$ связана с относительной погрешностью соотношением $\Delta D = D \cdot \frac{\Delta D}{D}$. Таким образом, общая формула для абсолютной погрешности диаметра:$\Delta D = D \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h} \right)$
3. Применение формулы к двум капиллярам.
Применяя эту общую формулу для двух разных капилляров (с индексами 1 и 2), мы получаем выражения, приведенные в условии задачи. Для первого капилляра с диаметром $D_1$ и высотой подъема жидкости $h_1$ (с погрешностью $\Delta h_1$):$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$
Для второго капилляра с диаметром $D_2$ и высотой подъема жидкости $h_2$ (с погрешностью $\Delta h_2$):$\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$
Таким образом, приведенные в задаче формулы являются следствием применения стандартного метода расчета погрешностей косвенных измерений к формуле для диаметра капилляра, полученной из закона Жюрена, с учетом указанных в условии пренебрежений.
Ответ: Вывод формул, приведенных в условии, основан на законе Жюрена для капиллярного подъема жидкости ($D = \frac{4\sigma}{\rho g h}$) и правиле сложения относительных погрешностей для косвенных измерений. При пренебрежении погрешностями $\Delta \rho$ и $\Delta g$, относительная погрешность диаметра равна $\frac{\Delta D}{D} = \frac{\Delta \sigma}{\sigma} + \frac{\Delta h}{h}$. Отсюда следуют искомые формулы для абсолютных погрешностей:$\Delta D_1 = D_1 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_1}{h_1} \right)$ и $\Delta D_2 = D_2 \left( \frac{\Delta \sigma}{\sigma} + \frac{\Delta h_2}{h_2} \right)$.
Другие задания:
Помогло решение? Оставьте отзыв в комментариях ниже.
Присоединяйтесь к Телеграм-группе @top_gdz
ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по физике за 10 класс, для упражнения номер 5 расположенного на странице 458 к учебнику 2021 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по физике к упражнению №5 (с. 458), автора: Касьянов (Валерий Алексеевич), ФГОС (старый) углублённый уровень обучения учебного пособия издательства Просвещение.