# Q 7.26 (b)     Prove the theorem of parallel axes.           (Hint: If the centre of mass of a system of n particles is chosen to be the origin $\sum m_{i}r_{i}=0$ ).

Consider the figure given below :

The moment of inertia about RS axis:-

$I_{RS}\ =\ \sum m_ir_i^2$

Now the moment of inertia about QP axis:-

$I_{QP}\ =\ \sum m_i(a+r_i)^2$

or                                        $=\ \sum m_i(a^2 +r_i^2 + 2ar_i)$

or                                        $=\ I_{RS} + \sum m_i a^2 + 2 \sum m_iar_i^2$

Thus                         $I_{QP} =\ I_{RS} + M a^2$                                                           $\left ( \because 2\sum m_iar_i^2 \right )\ =\ 0$

Hence proved.

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