Question
Show that motion of body dropped in a tunnel dug along the diameter of the earth is simple harmonic motion. Find the time period of motion.
Solution
Let the earth be a homogenous sphere of radius R and density ρ.
Let the bore be dug along the diameter of the earth.
The acceleration due to gravity at point P at a distance y from the centre of the earth is,
, is the acceleration due to gravity at the surface of the earth.
Since acceleration 'a' at point P is directed towards the centre of the earth, therefore in vector form we can write,
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11040452-3.png)
Therefore the motion is simple harmonic motion.
Substituting R and g, we get,
, is the time period of the motion.
Let the bore be dug along the diameter of the earth.
The acceleration due to gravity at point P at a distance y from the centre of the earth is,
, is the acceleration due to gravity at the surface of the earth. Since acceleration 'a' at point P is directed towards the centre of the earth, therefore in vector form we can write,
Therefore the motion is simple harmonic motion.
Substituting R and g, we get,
, is the time period of the motion. 
