Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0000 362584 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0802 1212024 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0802 1212024 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1660 4192760 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1674 4220520 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1682 4238232 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1684 4268752 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1694 4291472 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1694 4291472 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2305 4977296 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2342 5169448 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2441 5190376 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2561 5391712 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2561 5391968 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0000 362584 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0802 1212024 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0802 1212024 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1660 4192760 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1674 4220520 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1682 4238232 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1684 4268752 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1694 4291472 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1694 4291472 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2305 4977296 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2342 5169448 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2441 5190376 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.6395 13814360 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.6397 13814760 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.6398 13814760 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.6400 13823008 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.6402 13824744 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.6408 13857792 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.6408 13858192 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.6409 13858272 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Making Equations Dimentionless

It is often useful to remove the units from an equation. We can see which physical mechanisms are more important and the equation to be solved is simpler. We scale all variables so that the variables become dimensionless. We can solve the equation relative to properties of the system e.g. the period.

To make an equation dimensionless one must do the following:

  1. Identify all the independent and dependent variables, e.g.having dimension

  2. Replace each of them with a quantity scaled relative to a characteristic unit e.g. whereis constant and has dimensionandis variable and dimensionless;.

  3. Rewrite the system of equations in terms of their new dimensionless quantities.

  4. Divide through by the coefficient of the highest order derivative term;.

  5. Choose the scale factors so that as many terms as possible have coefficient 1.

Example: Makedimensionless.

1. The independent variable isand the dependent variable is

2. Letandwhereandare constants to be defined later.

3. Sincewe have

4. The coefficient of the highest order term isand dividing by this leads to

5. Chooseand

Example: Makedimensionless whereis the equilibrium position.

This is the equation of the system below.

1. The independent variable isand the dependent variable is

  1. Makeandthen follow the procedure in 3 above to get

Divide through by the leading coefficientto give

5. Chooseand add 1 to both sides to givewhere

Add comment

Security code
Refresh