26. (b) Prove the theorem of parallel axes. (Hint: If the centre of mass is chosen to be the origin ∑ miri = 0). from Physics System of Particles and Rotational Motion Class 11 Manipur Board
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26. (b) Prove the theorem of parallel axes. 

(Hint: If the centre of mass is chosen to be the origin ∑ miri = 0).

The theorem of parallel axes states that the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes. 


                          
Suppose a rigid body is made up of n particles, having masses m1m2m3, … , mn, at perpendicular distances r1,r2r3, … , rn respectively from the centre of mass O of the rigid body. 

The moment of inertia about axis RS passing through the point O, 

straight I space subscript RS space equals space sum from straight i space equals space 1 to straight n of space straight m subscript straight i space straight r subscript straight i squared space

The space perpendicular space distance space of space mass space straight m comma
space from space the space axis space QP space equals space straight a space plus space straight r subscript straight i space

Hence comma space the space moment space of space Inertia space about space axis space QP thin space is comma space

straight I subscript QP space equals space sum from straight i space equals 1 to straight n of space straight m subscript straight i space left parenthesis thin space straight a space plus space straight r subscript straight i right parenthesis squared space

space space space space space space equals space sum from straight i space equals 1 to straight n of space straight m subscript straight i space left parenthesis thin space straight a squared space plus space straight r subscript straight i squared space plus space 2 ar subscript straight i space right parenthesis space squared space

space space space space space space equals space sum from straight i space equals 1 to straight n of space straight m subscript straight i space straight a squared space space plus space sum from straight i space equals 1 to straight n of space space straight m subscript straight i space straight r subscript straight i squared space plus space sum from straight i space equals 1 to straight n of space space straight m subscript straight i space space 2 ar subscript straight i space

space space space space space space equals space space sum from straight i space equals 1 to straight n of space straight m subscript straight i space straight a squared space plus space 2 space sum from straight i space equals 1 to straight n of space space straight m subscript straight i space straight r subscript straight i squared


The moment of Inertia of all the particles about the axis passing through the centre of mass is zero, at the centre of mass. 

That is, 

2 space sum from straight i space equals space 1 to straight n of space straight m subscript straight i space straight a space straight r subscript straight i space equals space 0 space left square bracket space because space straight a space not equal to space 0 space right square bracket

therefore space sum space straight m subscript straight i space straight r subscript straight i space equals space 0 space

Also comma space
sum from straight i space equals space 1 to straight n of space straight m subscript straight i space equals space straight M thin space

where comma space

straight M space is space the space total space mass space of space the space rigid space body. space

Therefore comma space

straight I subscript QP space equals space straight I subscript RS space plus space Ma squared

Thus the theorem is proved. 




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