Question

in this Q pls tell.. how to find the distance of the from the centre mass..
but if the axis passing through centre of then it should be in centre means the distance should be equal for both masses but why we taking here different??​

Answer 1

Solution:-Since we have to find moment of inertia about the centre of mass,so let us first the position of centre of mass

If C will be the centre mass,,

then it should follow the condition

$m1x1 = m2x2$

$here \: x2 = l - x \: \: \\ so \: putting \: the \: value \: we \: get \\ m1x1 = m2(l - x1) \\ (m1 + m2)x1 = m2l \\ x1 = \frac{m2l}{m1 + m2}$

this is the distance of m1 from the centre of mass

Distance of m2 from the centre of mass=

$l - x1 = l - \frac{m2l}{m1 + m2} \\ = \frac{m1l}{m1 + m2}$

now , moment of inertia about centre of mass =$m1 {(x1)}^{2} + m2( {x2)}^{2} \\ = m1 \times { (\frac{m2l}{m1 + m2}) }^{2} + m2 \times { (\frac{m1l}{m1 + m2} )}^{2} \\ = \frac{m1 \times {(m2l)}^{2} + m2 \times {(m1l)}^{2} }{ {(m1 + m2)}^{2} } \\ = \frac{m1m2 {l}^{2} (m1 + m2)}{ {(m1 + m2)}^{2} } = \frac{m1m2 {l}^{2} }{m1 + m2}$

So optionB) is right one

Ur doubt:-It is not said in the question that the axis is passing through centre of rod

it is said that the perpendicular axis is passing through centre of mass

centre of mass will not be at the centre of rod... because masses at the two ends are different.......

the axis will only pass through the centre if m1=m2,, because in that case centre of mass will pass through the centre of rod