A solid sphere rolls without slipping on an inclied plane at an angle . The ratio of total kinetic energy to its rotational kinetic energy is

Step-by-step solution

For a solid sphere rolling without slipping, the velocity of the center of mass v and angular velocity are related by v = R . The translational kinetic energy is K_ trans = 1 2 m v^2 . The rotational kinetic energy is K_ rot = 1 2 I ^2 . Substituting the moment of inertia of a solid sphere I = 2 5 m R^2 and = v R : K_ rot = 1 2 ( 2 5 m R^2 ) ( v R )^2 = 1 5 m v^2 The total kinetic energy is the sum of translational and rotational kinetic energies: K_ total = K_ trans + K_ rot = 1 2 m v^2 + 1 5 m v^2 = 7 10 m v^2 Th