This question is from HC verma. Still, There is a rod of length L and mass M, a particle traveling with velocity v of mass m, strikes tangentially at the end of the rod and sticks to it. Find the moment of inertia of the system after collision about the vertical axis through the center of mass of the system.

yo man see when a particle of velocity v strikes the rod at the edge then a torque is produced =(vl)/2

we know torque =I(alpha) ok

alpha the angular acceleration will change and the moment of inertia will remain same for the rod through the centre of mass ie=(ml²)/12

then too please check

First of all moment of inertia asked is about a vertical axis through the center of mass of the system(Rod + Ball). And the answer is supposed to be (M(M+4m)L*L)/12(M+m)

let the rod of length L is placed along x axis one end is at O.....

particle strikes at x=0 & sticks to it ...

center of  mass of system = X_com = ML/2 + m(0)/m+M

                                                = ML/2(m+M)

moment of inertial of rod(I_r) = ML²/12  + MX²_com ..........................1

moment of inertia of mass (I_m)= mX²com        .....................2

total momen of inertia system =  I_m + I_r

Approved