Question

A uniform solid marble, of mass m = 20.0 g and diameter 1.00 cm, rolls without sliding down a large symmetric steel bowl, starting from rest at point A, at the top of the left(no-slip) side. The top of each side is a distance h = 15.0 cm above the bottom of the bowl. The left half of the bowl is rough enough to cause the marble to roll without slipping, but the right half of the bowl is frictionless because it is coated with oil. Assume that there is no air resistance and no loss of energy due to kinetic friction.

(a) Use energy techniques to determine the translational speed vcm of the marble when it is at point B at the bottom of the bowl.

(b) The transition from the rough to a smooth surface just after point B changes the marble’s motion – it is not rolling anymore. Thus, its rotational motion and translational motion are no longer constrained to change together. However, both motions will still continue on into the right side of the bowl where there is no friction to stop either one. Use energy techniques to show that the maximum height that the marble reaches on the right side of the bowl is 2 max 5 2 7 vcm h h g = = .

c) If there were no oil on the right side of the bowl, how high would the marble get then? Why does it go higher if the marble is always rolling? What energy transformations are occurring?

d) Find the max height reached on the right side if there were oil on the right side of the bowl, but the marble was a hollow shell instead of solid sphere. Why is it different than for the solid marble? What energy transformations that are occurring

Answer 1

DATA

PART A)

The angular speed is related to the translational speed by the equation,

so, equation (1) yields

the moment of inertia of the marble is

inserting in equation (2),

isolating the speed at point B,

or

PART B)

Let C be the point where the marble reach its maximum height, then

isolating the maximum height of the marble

or

PART C)

again, remember that moment of inertia of the marble is

inserting in eq. (3),

isolating the maximum height

or

Why does it go higher if the marble is always rolling?

When the marble is always rolling, the marble has energy extra (kinetic rotational energy).

What energy transformations are occurring?

The energy is being transformed from kinetic to potential.

PART D)

We expect a different result because the moment of inertia a hollow shell is different of a solid sphere. This time,

inserting in eq. (2),

isolating the speed of the howllow shell at point B,

or

Now, let's consider the section from point B to C,

isolating the maximum height of the marble

or