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Units And Measurement

Question
CBSEENPH11017453
Wired Faculty App

How would you reduce the two-body problem to a single body problem?

or

How would you reduce the problem of two body system to a problem of single body system?

Solution

Consider two particles of masses m1 and m2 having position vectors r with rightwards harpoon with barb upwards on top subscript 1 space and space r with rightwards harpoon with barb upwards on top subscript 2 respectively.
Let F with rightwards harpoon with barb upwards on top subscript 12 be the force experienced by m2 due to m1.
Therefore, 
                     F with rightwards harpoon with barb upwards on top subscript 12 equals negative space F with rightwards harpoon with barb upwards on top subscript 21
 
Now the equation of motion of m1 is,  
space space space space space space space space straight m subscript 1 space fraction numerator straight d squared straight r with rightwards harpoon with barb upwards on top subscript 1 over denominator dt squared end fraction equals space straight F with rightwards harpoon with barb upwards on top subscript 21 space equals space minus space straight F with rightwards harpoon with barb upwards on top subscript 12 space rightwards double arrow space space space space space space space fraction numerator straight d squared straight r with rightwards harpoon with barb upwards on top subscript 1 over denominator dt squared end fraction space equals fraction numerator negative space straight F with rightwards harpoon with barb upwards on top subscript 12 space over denominator straight m subscript 1 end fraction space space space space space space space space space space space space space... left parenthesis 1 right parenthesis 

Similarly, the equation of motion of m2 is given by
          fraction numerator straight d squared straight r with rightwards harpoon with barb upwards on top subscript 1 over denominator dt squared end fraction space equals fraction numerator negative space straight F with rightwards harpoon with barb upwards on top subscript 12 space over denominator straight m subscript 2 end fraction space space
Subtracting (1) from (2), we get 
space space space space space space space space space fraction numerator straight d squared left parenthesis straight r with rightwards harpoon with barb upwards on top subscript 2 space minus space straight r with rightwards harpoon with barb upwards on top subscript 1 right parenthesis over denominator dt squared end fraction space equals space open square brackets 1 over straight m subscript 1 space plus space 1 over straight m subscript 2 close square brackets space straight F with rightwards harpoon with barb upwards on top subscript 12 space rightwards double arrow space fraction numerator straight d squared straight r with rightwards harpoon with barb upwards on top over denominator dt squared end fraction space space equals space 1 over straight mu space straight F with rightwards harpoon with barb upwards on top subscript 12 rightwards double arrow space straight F with rightwards harpoon with barb upwards on top subscript 12 space equals space straight mu space fraction numerator straight d squared straight r with rightwards harpoon with barb upwards on top over denominator dt squared end fraction space space space space space space space space space space space space space space space space space space space space... space left parenthesis 3 right parenthesis space
where, <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> is called reduced mass of
the system.
Since equation (3) signifies the equation of motion of single particles, thus the two body problem is reduced to single-body problem.

Some More Questions From Units and Measurement Chapter

Give four examples of physical quantities.

Define unit.

What are the two main required characteristics of a unit?

Can we use same units for different physical quantities?

Can we use different units to measure same physical quantity?

To measure a physical quantity X, the unit U is repeatedly used for n times. What is the magnitude of physical quantity?

Does the numerical factor in the quantitative measurement depend on the system of unit used?

Is one unit sufficient to measure a physical quantity?

Can a physical quantity have two or more units? Give example.

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