Calculate the moment of inertia and the rotational kinetic energy for the following objects spinning about a central axis (in units of Joules) : a solid sphere with a mass of 200 grams and a radius of 5.0 cm rotating with an angular speed of 2.5 rad/sec. a thin spherical shell with a mass of 200 grams and a radius of 5.0 cm rotating with an angular speed of 2.5 rad/sec. a solid cylinder with a mass of 200 grams and a radius of 5.0 cm rotating with an angular speed of 2.5 rad/sec. a hoop with a mass of 200 grams and a radius of 5.0 cm rotating with an angular speed of 2.5 rad/sec.

a) K = 6.25 10⁻⁴ J, b) K = 10.41 10⁻⁴ J, c) K = 7.81 10⁻⁴ J, d) K = 15.625 10⁻⁴ J

Explanation:

The moment of inertia is

I = ∫ r² dm

For high symmetry bodies it is tabulated.

The rotational kinetic energy is

K = ½ I w²

Let's calculate the values of the given bodies

a) solid sphere

I = 2/5 M R²

I = 2/5 0.200 0.05²

I = 2 10⁻⁴ kg m²

K = ½ 2 10⁻⁴ 2.5²

K = 6.25 10⁻⁴ J

b) spherical shell

I = 2/3 M R²

I = 2/3 0.200 0.05²

I = 3.33 10⁻⁴ kg m²

K = ½ 3.33 10-4 2.5²

K = 10.41 10⁻⁴ J

c) solid cylinder

I = ½ M R²

I = ½ 0.200 0.05²

I = 2.5 10⁻⁴ kg m²

K = ½ 2.5 10-4 2.5²

K = 7.81 10⁻⁴ J

d) hoop

I = M R²

I = 0.200 0.05²

I = 5 10⁻⁴ kg m²

K = ½ 5 10⁻⁴ 2.5²

K = 15.625 10⁻⁴ J