Moment of inertia and torque problem

[bargain]

The Question : A cylinder of mass M , and moment of inertia around its center Io, is initially rotating around its central axis with angular velocity wo. İt is gently placed on a homogeneous thin plate of mass m which rests on a smooth horizontal surface. Assume that the plate is sufficiently long that eventually the cylinder starts rolling on the plate without slipping due to the kinetic friction. What is the final velocity of the plate?
I could not write much for the solution since i am stucked in energy conservation.

1/2 Io wo2 = 1/2 mv2 + 1/2 M Vcm2

torque : 2/5 MR2 = RF(friction)

does anyone has the answer or have any ideas in which way should i go with this problem?

 

[SteamKing]

It's 'torque', not 'torgue', with a Q instead of a G.

 

[bargain]

It's 'torque', not 'torgue', with a Q instead of a G.

ok i changed it. do you have any help other than spelling?

 

[Vibhor]

I could not write much for the solution since i am stucked in energy conservation.

1/2 Io wo2 = 1/2 mv2 + 1/2 M Vcm2

torque : 2/5 MR2 = RF(friction)

does anyone has the answer or have any ideas in which way should i go with this problem?

Energy is not conserved in this problem . Think of some other conservation law.

First , find the speed of the cylinder v in terms of ω₀ ,when it starts to roll without slipping .

Can you do that ?

 

[bargain]

v= wo.R then what?

but how about the torque equality of cylinder: Force of friction x Radius = Moment of inertia x alpha

 

[vela]

You could try using Newton's second law (F = ma and $\tau = I\alpha$), but you're probably making things hard for yourself. The problem with using those equations is that you'll likely have to integrate with respect to time to get to the answer you want. The appeal of using conservation laws is that you don't have to worry how the system evolves from one state to another. You just look at the system in its initial and final states, and the conservation laws let you relate the relevant quantities to each other.

With this problem, you need to be careful. If you say V is the speed of the ball, are you talking about the speed of the ball relative to an observer at rest or are you talking about the speed of the ball relative to the plate? If you say V = ωR, which of those two speeds appears on the lefthand side of the equation?

Let's use the convention that all velocities are with respect to the observer at rest. Use $V$ to denote the velocity of the ball's center of mass and $v$ to denote the velocity of the plate.

What is $V_i$, the initial velocity of the ball?
What is $v_i$, the initial velocity of the plate?

What quantities are conserved? You've already been told energy isn't conserved. Why isn't it? What about momentum? How do you know? What about angular momentum? Again, how do you know?

 

[Vibhor]

Vela ,

You have raised a fine point regarding whether the speed of the cylinder is with respect to the observer or plate .That made me rethink .

Are you getting the same answer under the assumption that the cylinder is hollow and is rotating clockwise ?

Spoiler

$$\frac{I^\frac{1}{2}M^\frac{3}{2}}{m^2+mM-M^2}ω_0$$