The center of mass of a pitched baseball or radius 2.11 cm moves at 23.6 m/s. The ball spins about an axis through its center of mass with an angular speed of 139 rad/s. Calculate the ratio of the rotational energy to the translational kinetic energy. Treat the ball as a uniform sphere.

The ratio of the rotational energy to the translational kinetic energy is 0.00618.

Explanation:

The rotational kinetic energy of a solid sphere is given by

where m is its mass, r is its radius and ω is its angular speed.

The translational kinetic energy is given by

where v is its linear speed.

The ratio of the rotational energy to the translational kinetic energy is

From the question,

r = 2.11 cm = 0.0211 m

ω = 139 rad/s

v = 23.6 m/s

The ratio of the rotational energy to the translational kinetic energy is 0.00618

Explanation:

If m is the mass of the baseball and v its speed, then the translational kinetic energy is

Kt = 1/2 mv²

If I is its rotational inertia and ω is its angular speed, then the rotational kinetic energy is

Kr = 1/2 Iω²

But the rotational inertia of a uniform sphere is I = 2mr²/5

Therefore, the ratio of the rotational energy to the translational kinetic energy is 0.00618.

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