AP Physics C student seeking help...?

Question

Nonuniform ball. In Fig. 12-2, a ball of mass M and radius R smoothly rolls from rest, at a height of h = 0.34 m, along a ramp and onto a circular loop of radius 0.54 m. At the bottom of the loop, the magnitude of the normal force on the ball is 2.00Mg. The ball consists of an outer spherical shell with a certain uniform density (mass per unit volume) that is glued to a central solid sphere with a different uniform density. The rotational inertia of the ball can be expressed in the general form I = BMR2, but B is not 0.4 as for a ball with a single uniform density. Determine B(beta).
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Answer 1

There is a lot of algebra involved in finding the final answer, but it is not too difficult once you have realized a few key pieces of information in the question and you know the correct formulas to use.

Formulas you will need:
-Gravitational Potential Energy = Mgh
-Translational Kinetic Energy = 1/2 Mv^2
-Rotational Kinetic Energy = 1/2 Iw^2
-Angular velocity = v / R
-Centripetal Force = m * v^2 / r

Where M is the objects mass, g is the gravitational acceleration, h is the vertical distance the ball rolls down the hill, v is the ball’s linear speed, w is the ball’s angular speed, I is the rotational inertia of the ball, R is the radius of the ball, and r is the radius of the curved path the ball follows.

Key concepts for this problem:
-Energy is always conserved
-The gravitational Potential Energy the ball looses while rolling down the hill is converted into Kinetic energy,
-The Kinetic Energy the ball has is of two types, translational KE and rotational KE.
-Normal force at the bottom of the path is equal to the sum of the centripetal acceleration and the object’s weight
-If the ball rolls without slipping, the angular velocity of the ball can be found as a function of the linear velocity of the ball.


To solve this question, I would find the linear speed of the ball at the bottom of the curved path remembering what I said about the Normal force acting on the ball. From this you can find v^2 of the ball which will come in very handy later on.
Set up an equation which relates the PE the ball lost to the KE the ball gained. The KE should be divided up into both a translational KE and a rotational KE term. I would advise plugging in what we know about the rotational inertia, I, at this point. It should be obvious that the mass, M, should cancel out and be irrelevant to the answer at this point. But there is also something else that we do not know that will cancel as well (two thing actually). You will be left with an equation with only 1 unknown (B) we you can solve for in terms of “h” and “r” (and ‘g’, but this will cancel).

Answer 2

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