Rotational Dynamics: Min Initial Velocity of Inelastic Sphere on Cubical Plate

[Mandeep Deka]

Homework Statement

An inelastic uniform solid sphere of radius 'R' is sliding without rolling over a frictionless ground with a uniform velocity 'v'. A cubical plate of height r (r<R) is fixed on the ground and the sphere impinges upon the cubical plate and rolls over it. What is the minimum initial velocity 'v' of the sphere, for it to be able to impinge upon the plate.

Homework Equations

The Attempt at a Solution

What is basically confusing me is:
i. whether the sphere will show pure rolling after impinging upon the plate?
ii. will it stop its linear motion after striking? (because the sphere is said to be inelastic), but then it can't roll over the plate!
iii. if the sphere rolls then without showing pure rolling, how am i supposed to find the answer coz there will be more variables than equations?

please help me understand what actually happens!

 

[Delzac]

In this question, the inelastic ball, you can think of it as a clay mold, will have the a point on the ball getting stuck on the plate. And it will pivot about that point. The reason why it can pivot is because r is smaller than R, so the centre of mass of ball, which is carrying momentum, will cause a torque.

If r=R, this is no torque, since there is no distance from centre of mass to pivot. Anything after the rotation is over is not required by the question. And cannot be determined accurately anyway from the information given.

 

[Mandeep Deka]

i understand that since r<R, the sphere would pivot about that point. But what will happen after that?? How will we find out the min velocity for the ball to be able to just roll over the plate??

 

[Delzac]

Thats a energy conservation problem. It must have enough kinetic energy to overcome the increase of gravitation potential energy which will result in a lost of kinetic energy.

 

[Mandeep Deka]

IF you say so, what velocity shall i consider the sphere to move after bumping up??
i mean, i equate the kinetic energy of the sphere initially, with the change in the gravitational potential energy plus some kinetic energy( either linear or rotational or both). What would that be then?