Suppose the particle of the previous problem has a mass m and a speed ν before the collision and it - Physics -

Question

Suppose the particle of the previous problem has a mass m
and a speed ν before the collision and it sticks to the rod
after the collision The rod has a mass M (a) Find the
velocity of the centre of mass C of the system
constituting "the rod plus the particle" (b) Find the velocity of
the particle with respect to C before the collision (c)
Find the velocity of the rod with respect to C before the
collision (d) Find the angular momentum of the particle and of the
rod about the centre of mass C before the collision (e)
Find the moment of inertia of the system about the vertical axis
through the centre of mass C after the collision (f) Find
the velocity of the centre of mass C and the angular
velocity of the system about the centre of mass after the collision

Global Expert · Accepted Answer

(a) It is given that no external torque and force is applied on the
system
Applying the law of conservation of momentum, we get:
mν=M+m ν'
⇒ν'=mνM+m

(b) Velocity of the particle w r t centre of mass (COM) C
before the collision = vc=v-v'
⇒vc=v-mvM+m=MvM+v

(c) Velocity of the particle w r t COM C before
collision=-MνM+m

(d) Distance of the COM from the particle,
xcm=m1x1+m2x2m1+m2⇒r=M×L2+m×0M+m⇒r=ML2M+m
∴ Angular momentum of body about COM

=mvr=m×MvM+m×ML2M+m=M2 mvL2M+m2

∴ Angular momentum of rod about COM

=M×mvM+m×12mLM+m= Mm2vL2 M+m2

(e) Moment of inertia about COM = I=I1+I2

I=mML2M+M2+ML212+MML2m+M2  =mM2L24m+M2+ML212+Mm2L24M+m2 =3mM2L2+Mm+M2 L2+3Mm2L212m+M2 =MM+4mL212M+m

(f) About COM,
Vcm=mνM+m∴ Iω=mvr=mv×ML2 M+m⇒ ω=mνML2 M+m×12 M+mM M+4mL2       =6mνM+4mL