Another rotational motion problem

[naznad]

A spherical ball of radius r and mass m colides with a plank og mass M on a smooth horizontal surface.Before impact, the centre of the ball has a velocity vo( v knot) and angular velocity wo( omega knot) as I've shown in the figure.The norm velocity is reversed with same magnitude and the ball stops rotating after impact.Find the distance on the plank between the first two impacts of the ball.The coefficient of friction between the ball and the plank is µ.Assume that the plank is large enough.

I tried it out this way:
since the ball stops rotating, all of its rotational energy is imparted as translational energy to the plank in the left direction.Also frictional force will act at the point of contact in the same direction.
Therefore, rotational energy + the kinetic energy due to friction = kinetic energy of the plank.

or in terms of forces,
torque/mass m + µmg = Ma (where a is the acc. of the plank)
but i got stuck in writing the eqns.

 

[StatusX]

I don't understand the problem. If the plank moves horizontally, the ball needs to be kicked back in the opposite direction, but it doesn't sound like this is what happens. Also, there will be loss from friction, not gain.

 

[thepatientmental]

Your approach is correct so far. Let's break down the problem and see how we can solve it.

First, let's consider the conservation of energy. Before impact, the ball has both translational and rotational kinetic energy. After impact, the ball stops rotating and only has translational kinetic energy. This energy is then transferred to the plank. Therefore, we can write the following equation:

1/2mv_o^2 + 1/2Iw_o^2 = 1/2Mv^2

Where v_o is the initial velocity of the ball, w_o is the initial angular velocity, v is the velocity of the plank after impact, m is the mass of the ball, I is the moment of inertia of the ball, and M is the mass of the plank.

Next, we need to consider the forces acting on the ball and the plank. The only external force acting on the system is the frictional force between the ball and the plank. This force will cause the ball to decelerate and the plank to accelerate. We can write the following equation using Newton's Second Law:

µmg = Mdv/dt

Where µ is the coefficient of friction, g is the acceleration due to gravity, M is the mass of the plank, and v is the velocity of the plank.

Now, we can combine these two equations to solve for the distance between the first two impacts of the ball on the plank. We can rearrange the first equation to solve for v, and then substitute it into the second equation. This will give us an equation with only one unknown, d, the distance between the two impacts. Solving for d will give us the answer we are looking for.

I hope this helps and good luck with your problem!