Question

a sphere of mass 0.5kg and diameter 1m rolls without slipping with a constant velocity of 5ms-1. calculate its total energy

Answer 1

Express its kinetic energy of linear motion of its center of mass:

$$E_L=\frac{mv^2}{2}.$$

Express its angular velocity in terms of linear velocity and radius:

$$\omega=v/r.$$

Express the moment of inertia:

$$I=\frac{2}{5}mr^2.$$

Express the kinetic energy of rotational motion:

$$E_R=\frac{I\omega^2}{2.}$$

The total energy of motion is

$$E=E_L+E_R,\\\space\\ E=\frac{mv^2}{2}+\frac{mr^2v^2}{5r^2}=\frac{7}{10}mv^2=8.75\text{ J}.$$