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

0.55x745.7j and a calculator

when two balls of the different masses are dropped from equal height, they reach the ground at the same time .this is because they experience the same acceleration.

the acceleration is independent of mass.

thus when they are dropped they will always maintain the same velocity and travel the same distance (neglect air resistance of course) at the same time.