Some curious students hold a rolling race by rolling four items down a steep hil

Question

Some curious students hold a rolling race by rolling four items down a steep hill. The four items are a solid homogeneous sphere, a thin spherical shell, a solid homogeneous cylinder and a hoop with all its mass concentrated on the hoop's perimeter. All of the objects have the same mass and start from rest. Assume that the objects roll without slipping and that air resistance and rolling resistance are negligible. For each statement below, select True or False.

1) Upon reaching the bottom of the hill, the hoop will have a larger rotational kinetic energy than any of the other objects will when they reach the bottom of the hill.

2) The hollow sphere will reach the bottom before the solid sphere.

Explanation / Answer

1)

Using conservation of energy

Potential energy at the Top = Kinetic energy at Bottom + Rotational KE

mgh = (0.5) m V2 + (0.5) I w2

mgh = (0.5) m r2 w2 + (0.5) I w2 eq-1

for sphere , I = (0.4) m r2

multiplying eq-1 by 0.4 both side

(0.4) mgh = (0.5) (0.4) m r2 w2 + (0.5) (0.4) I w2

(0.4) mgh = (0.5) I w2 + (0.5) (0.4) I w2

(0.4) mgh = RE + (0.4) RE

RE = 0.286 mgh

for spherical shell : , I = (0.67) m r2

multiplying eq-1 by 0.67 both side

(0.67) mgh = (0.5) (0.67) m r2 w2 + (0.5) (0.67) I w2

(0.67) mgh = (0.5) I w2 + (0.5) (0.67) I w2

(0.67) mgh = RE + (0.67) RE

RE = (0.4) mgh

for Solid cylinder : , I = (0.5) m r2

multiplying eq-1 by 0.5 both side

(0.5) mgh = (0.5) (0.5) m r2 w2 + (0.5) (0.5) I w2

(0.5) mgh = (0.5) I w2 + (0.5) (0.5) I w2

(0.5) mgh = RE + (0.5) RE

RE = (0.33) mgh

for Hoop : , I = m r2

using eq-1

mgh = (0.5) m r2 w2 + (0.5) I w2

mgh = (0.5) I w2 + (0.5) I w2

mgh = RE + RE

RE = (0.5) mgh

Hence it is true that the hoop will have a larger rotational kinetic energy than any of the other objects will when they reach the bottom of the hill.

2)

False

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