Question
(a) Show that if the amplitude of oscillation is small the energy of a simple pendulum is given by 𝐸 = 𝐽𝑣.
(b) Consider a simple pendulum formed by a point weight on a string which passes vertically through a hole in a horizontal board. Suppose the length of the pendulum is decreased by slowly pulling the string up through the hole. The rate of decrease of length is to be extremely slow compared with the frequency of swing, so that one can still speak of a period of oscillation for any given length. Compute the change in the energy of the pendulum, as the length is shortened, from the work done in pulling against the tension of the string. Show as a result that the action variable 𝐽 = 𝐸/𝑣 is constant throughout the process. A change in the external parameters of the system at a rate slow compared to the intrinsic frequencies is known as an adiabatic variation and the action variable of the pendulum is therefore an adiabatic invariant. It is possible to prove in general the adiabatic invariance of the action variables for any system in which degeneracy does not occur. The action variables are thus stable constants of the system under the influence of slowly varying external conditions. Now, the quantum state of a system is also an adiabatic invariant; a slow change of the external parameters does not induce transitions from one state to another. We have here another indication of the suitability of the action variables for describing the quantization of the system states.
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      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheMissed (338μs)
      • Illuminate\Database\Events\StatementPrepared (930μs)
      • Illuminate\Database\Events\QueryExecuted (1.06ms)
      • eloquent.retrieved: App\Models\SubjectCat (91μs)
      • Illuminate\Cache\Events\KeyWritten (376μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (157μs)
      • Illuminate\Cache\Events\CacheHit (361μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (230μs)
      • Illuminate\Cache\Events\CacheHit (293μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (499μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (238μs)
      • Illuminate\Cache\Events\CacheHit (364μs)
      • Illuminate\Cache\Events\CacheHit (4.7ms)
      • Illuminate\Cache\Events\CacheHit (232μs)
      • creating: site.layouts.app (537μs)
      • composing: site.layouts.app (26μs)
      • creating: components.canonical (547μs)
      • composing: components.canonical (118μs)
      • creating: components.open-graph (226μs)
      • composing: components.open-graph (88μs)
      • creating: site.headers.header (444μs)
      • composing: site.headers.header (91μs)
      • Illuminate\Cache\Events\CacheHit (2.52ms)
      • creating: components.footer (81μs)
      • composing: components.footer (103μs)
      • Illuminate\Cache\Events\CacheHit (919μs)
      • Illuminate\Cache\Events\CacheMissed (386μs)
      • Illuminate\Database\Events\StatementPrepared (807μs)
      • Illuminate\Database\Events\QueryExecuted (1.3ms)
      • eloquent.retrieved: App\Models\SubjectCat (97μs)
      • Illuminate\Cache\Events\KeyWritten (555μs)
      • Illuminate\Cache\Events\CacheHit (295μs)
      • Illuminate\Cache\Events\CacheHit (261μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (304μs)
      • Illuminate\Cache\Events\CacheHit (319μs)
      • Illuminate\Cache\Events\CacheHit (344μs)
      • Illuminate\Cache\Events\CacheHit (321μs)
      • Illuminate\Cache\Events\CacheHit (219μs)
      • Illuminate\Cache\Events\CacheHit (205μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • creating: components.forms.contact-us (352μs)
      • composing: components.forms.contact-us (184μs)
      • creating: components.forms.get-started (204μs)
      • composing: components.forms.get-started (89μs)
      • creating: components.forms.free-tool-download (154μs)
      • composing: components.forms.free-tool-download (81μs)
      • creating: components.forms.claim-free-worksheet (202μs)
      • composing: components.forms.claim-free-worksheet (91μs)
      • creating: components.forms.tutor-subscription-waitlist (201μs)
      • composing: components.forms.tutor-subscription-waitlist (167μs)
      • creating: components.forms.tutor-subscription-join (146μs)
      • composing: components.forms.tutor-subscription-join (77μs)
      • creating: components.forms.tutor-support (137μs)
      • composing: components.forms.tutor-support (76μs)
      • 311 x Illuminate\Cache\Events\CacheHit (15.48%)
        103ms
        20 x Illuminate\Database\Events\StatementPrepared (6.48%)
        43.21ms
        20 x Illuminate\Database\Events\QueryExecuted (5.62%)
        37.47ms
        11 x Illuminate\Cache\Events\CacheMissed (2.44%)
        16.28ms
        11 x Illuminate\Cache\Events\KeyWritten (1.05%)
        6.98ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.3%)
        1.98ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.16%)
        1.09ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.13%)
        853μs
        1 x Illuminate\Routing\Events\Routing (0.12%)
        800μs
        1 x creating: components.canonical (0.08%)
        547μs
        1 x eloquent.booting: App\Models\SubjectCat (0.08%)
        544μs
        1 x creating: site.layouts.app (0.08%)
        537μs
        1 x creating: components.breadcrumbs (0.07%)
        452μs
        1 x creating: site.headers.header (0.07%)
        444μs
        1 x Illuminate\Routing\Events\RouteMatched (0.06%)
        415μs
        1 x creating: homework.show (0.05%)
        363μs
        1 x creating: components.forms.contact-us (0.05%)
        352μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        348μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        285μs
        10 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.04%)
        246μs
        1 x creating: components.open-graph (0.03%)
        226μs
        1 x creating: components.forms.get-started (0.03%)
        204μs
        1 x creating: components.forms.claim-free-worksheet (0.03%)
        202μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        201μs
        1 x composing: components.breadcrumbs (0.03%)
        196μs
        2 x eloquent.retrieved: App\Models\Subject (0.03%)
        188μs
        1 x composing: components.forms.contact-us (0.03%)
        184μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.03%)
        167μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.02%)
        163μs
        1 x composing: homework.show (0.02%)
        158μs
        1 x creating: components.forms.free-tool-download (0.02%)
        154μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        146μs
        1 x creating: components.forms.tutor-support (0.02%)
        137μs
        1 x composing: components.canonical (0.02%)
        118μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        105μs
        1 x composing: components.footer (0.02%)
        103μs
        1 x composing: site.headers.header (0.01%)
        91μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        91μs
        1 x composing: components.forms.get-started (0.01%)
        89μs
        1 x composing: components.open-graph (0.01%)
        88μs
        1 x creating: components.footer (0.01%)
        81μs
        1 x composing: components.forms.free-tool-download (0.01%)
        81μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        77μs
        1 x composing: components.forms.tutor-support (0.01%)
        76μs
        1 x eloquent.booting: App\Models\Subject (0.01%)
        72μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        58μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        58μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        54μs
        1 x composing: site.layouts.app (0%)
        26μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated57.73ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '20158' limit 1
        28.12mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 20158
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (401)
        1.02mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (20158)
        910μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        950μstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 34
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 20158 and `subject` = 401 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        3.44mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 20158
        • 1: 401
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (401)
        800μstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 20158 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        2.71mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 20158
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 20158 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        950μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 20158
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 20158 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        720μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 20158
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        3.58mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 34
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.06mstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.02mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        1.94mstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        3.47mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.08mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        980μstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.04mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.64mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.1mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.2mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      10HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          Yhb4pkBCAuq4hm8CEvscyyBrni97ooCUud71NIgX
          utm_source
          direct
          redirectUrl
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          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Physics/Classic...
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          PHPDEBUGBAR_STACK_DATA
          []
          path_info
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          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
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          request_cookies
          []
          
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