Force components on a rolling sphere at the top of a loop

[kambrian]

I am having a problem with this question. I can't figure out how to approach it.


A solid sphere of mass m and radius r rolls without slipping along the track. it starts from rest with the lowest point of the sphere at height h above the bottom of the loop of radius R, much larger than r.

What are the force components on the sphere at he point P if h = 3R?

How would I go about solving this problem? I attached a picture of the situation.

 

[marlon]

Well you need to split up this problem. First, the sphere moves down the incline under the influence of gravity (thus, there will also be a normal force)

Then, secondly, it moves along a loop, which is circular. Then, there must be some centripetal force and also gravity right ?

regards
marlon

 

[marlon]

You know, just to elaborate, there is a great classic associated with this figure. Suppose we let a point particle of mass m (the solution is analoguous for a solid object, you only will need to incorporate the rotational inertia I) slide down the incline from heigth h. What will this h be as a function of radius r, in the case that the object must STOP at the top of the loop.

Well, first we study the motion along the incline : energy conservation learns us :

mgh =1/2mv² and thus v² = 2gh

Then, we study the loop : there will be both gravity and centripetal force (mv²/r) on the object. At the top : mv²/r = mg (the centripetal force must eliminate gravity. Well, actually this is the centrifugal force, directed along the opposite direction of the centripetal force) : mg = (mv²)/r and thus v² = gr

Again applying energy conservation between bottom and top of the loop learns us :

1/2mv² (here v² = 2gh) = mg2r + 1/2mv² (here v² = gr)

This eaquation gives : mgh = mg2r + mgr/2 --> h = 2r + r/2 = 5r/2

regards
marlon

 

[kambrian]

Im still not seeing how I would do this. You said to split it up into two parts but what exactly am I trying to find in these two separate parts?

I am assuming the first part, when the ball is rolling down the incline, I am trying to find the velocity, right?