Question

A sphere of radius 20 cm and a mass of 3 kg rolls down an incline of 14.6o and 14 cm high. a. Calculate its translational and rotational kinetic energy when it reaches the bottom. b. What is the ratio of the translational kinetic energy to the rotational kinetic energy? c. What quantities does this ratio depend on?

Answer 1

(A) First let's figure out how high the sphere is at the top of the ramp. We know the ramp is 0.14 m long and inclined at 39.0º

height = (.14 m)sin14.6º = 0.03529 m

Now we can calculate the potential energy of the sphere with respect to the ground.

Ug = mgh = (3 kg)(9.81 m/s²)(0.03529 m) = 1.0385 J

Assuming no loss of mechanical energy, the sphere's translational and rotational kinetic energy at the bottom of the ramp should add up to 1.0385 J

K(t) + K(r) = 1.0385 J

K(t) = 1/2 mv²

K(r) = 1/2 I ω²

So...

K = 1/2 mv² + 1/2 I ω²

For a sphere of mass m and radius r, the moment of inertia around the center of mass is:

I = 2/5 mr²

And we can say that ω = v/r, with v being the tangential velocity of the equator of the sphere. So...

K = 1/2 mv² + 1/2 (2/5 mr²) (v/r)²

Simplifying a bit:

K = 1/2 mv² + (2/10) mv²

K = (5/10) mv² + (2/10) mv²

K = (7/10) mv²

Keeping in mind that the total kinetic energy is 1.0385 J...

1.0385 J = (7/10) mv²
=> 1.0385 J = (7/10) (3)v²

v = 0.7032 m/s

(B) You could go back and solve for the rotational kinetic energy, then use that to find the ball's rotational speed, but it's not necessary. We know the ball is moving at 0.7032 m/s, so that's the tangential velocity of its equator. Knowing its radius (20.0 cm or 0.200 m) we can calculate its angular velocity:

v = rω

0.7032 m/s = (0.200 m)ω

ω = 3.516 rad/s

(C) The total kinetic energy was 1.0385 J. The translational kinetic energy is:

K(t) = 1/2 mv² = 1/2(3 kg)(0.7032 m/s)² = 0.7417 J

Which leaves 1.0385 J - 0.7417 J = 0.2968 J for rotational kinetic energy.

Ratio of K(t) to K(r) = 0.7417 J / 0.2968 J = 2.498 to 1 (In reality, it should be exactly 2.50 to 1, but we're dealing with some uncertainty, so 2.498 to 1 is pretty good).