Question

Consider a race between the following three objects: object 1, a disk; object 2, a solid sphere; and object 3, a hollow spherical shell. All objects have the same mass and radius.

Required:
a. Rank the three objects in the order in which they finish the race. To rank objects that tie, overlap them.
b. Rank the objects in order of increasing kinetic energy at the bottom of the ramp. Rank objects from largest to smallest. To rank items as equivalent, overlap them.

Answer 1

Answer:

Since the angular acceleration of the objects will be proportional to the torque (due to gravity) acting on them and they will all experience the same torque their accelerations will be inversely proportional to their moments of inertia:

I disk = 1/2 M R^2

I sphere = 2/5 M R^2

I shell = 2/3 M R^2

Thus the sphere will experience the greatest angular acceleration and reach the bottom first, and then be followed by the disk and the shell.

By conservation of energy they will all have the same kinetic energy when they reach the bottom of the ramp.

Answer 2

(a) The ranking of the objects in order of how they will finish the race is

solid sphere > disk > hollow spherical shell

(b) The ranking of the objects in order of kinetic energy is

solid sphere > disk > hollow spherical shell

The moment of inertia of each object is calculated as follows;

• disk: $I = \frac{1} {2} MR^2$
• solid sphere: I = $\frac{2}{5} MR^2$
• hollow spherical shell: I =  $\frac{2}{3} MR^2$

The angular momentum of the objects is calculated as follows;

$L =I \omega \\\\\omega = \frac{L}{I}$

The object with the least moment of inertia is will have the highest speed.

The ranking of the objects in order of how they will finish the race;

solid sphere > disk > hollow

The kinetic energy of the objects is calculated as follows;

$K.E = \frac{1}{2} I \omega ^2$

The ranking of the objects in order of kinetic energy;

solid sphere > disk > hollow

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