A sphere of mass m rolls along a plane surface. Determine its kinetic energy at an instant when its centre travels with speed v .

Step-by-step solution

The total kinetic energy of a rolling body is the sum of its translational and rotational kinetic energies. K = K_ trans + K_ rot K = 1 2 mv^2 + 1 2 I ^2 For a solid sphere, the moment of inertia about its centre of mass is I = 2 5 mR^2 . For pure rolling, the relation between linear speed and angular speed is v = R , which implies = v R . Substituting the values of I and into the rotational kinetic energy term: K_ rot = 1 2 ( 2 5 mR^2 ) ( v R )^2 = 1 5 mv^2 Adding both kinetic energies: K = 1 2 mv^2 + 1 5 mv^2 = 7