A solid sphere rolling without friction on a horizontal surface with a constant speed of 2 ~m / s , rolls up on an inclined ramp which is inclined at 30^ . The maximum distance travelled by the sphere on the inclined ramp is (acceleration due to gravity g=10 ~m / s ^2, 30^ =1 / 2 )

Step-by-step solution

The mechanical energy is conserved since no friction acts on the system. The total kinetic energy of a rolling sphere without slipping comprises translational and rotational components, with moment of inertia I = 2 5 mr^2 for a solid sphere. The initial kinetic energy with speed v = 2 m/s is KE_i = 1 2 mv^2 + 1 2 I ^2 . Substituting = v/r and simplifying yields KE_i = 7 10 mv^2 . At maximum displacement s along the 30^ incline, the sphere comes to rest. The height gained is h = s 30^ = s/2 , so the potential energy