A problem in rigid body dynamics

[vijayramakrishnan]

Homework Statement

A disc of mass M and radius r is kept on a horizontal,frictional plane and is connected to a horizontal spring at the centre.A particle of mass m strikes the topmost point of the disc,tangentially and sticks to it.Assume that the mass of the particle is m and it's velocity is v.Find out the velocity after the collision.Can you apply the law of conservation of angular momentum?if so write down the equation and find ω of the system.[/B]

Homework Equations

conservation of momentum
conservation of angular momentum[/B]

The Attempt at a Solution

i know that conservation of momentum and conservation of angular momentum should be applied.
initial angular momentum = mvR
let v_f be the final velocity of topmost point and v_cm be the velocity of centre of mass of disc.
conservation of angular momentum about centre(not centre of mass as it changes after particle strikes)
final angular momentum should be moment of inertia about centre of mass of disc (angular velocity) + m v_fR

v_f = v_cm + ω(distance of centre of mass from particle)

conservation of momentum:
mv= Mv_cm + mv_f

but in my book it is written that in momentum conservation mv=(M+m)v_cm
i don't understand it aren't they moving with different velocity,so individual momentum should be added or momentum of centre of mass should be taken.
Also correct any of the other steps which i have done wrong
Please help[/B]

 

[haruspex]

I agree with both of your equations (linear and angular momentum). I cannot explain the book's equation.

 

[vijayramakrishnan]

I agree with both of your equations (linear and angular momentum). I cannot explain the book's equation.

thank you very much sir for replying.

 

[kinemath]

I thought conservation of angular momentum cannot be applied since there is friction?

 

[haruspex]

I thought conservation of angular momentum cannot be applied since there is friction?

The impulse from the collision is assumed to be very brief, i.e. an unlimitedly large force acting for an infinitesimal time. Since the impulse is horizontal, it does not affect the normal force, so that, and hence the corresponding friction, remain strictly limited in magnitude. As a result, the contribution to momentum is negligible.
That said, it does bother me that the question specifies a frictional surface. Either that is a trap, making you think you do need to consider friction, or they should have specified something more like a toothed gear set on a rack.