A hollow sphere is released from the top of an inclined plane of inclination . Determine the minimum coefficient of friction between the sphere and the plane required to prevent sliding. Furthermore, find the kinetic energy of the ball as it travels down a length l on the incline if the friction coefficient is half of this calculated minimum value.

Step-by-step solution

For a hollow sphere, the moment of inertia is I = 2 3 mR^2 . Let a be the linear acceleration and be the angular acceleration. For pure rolling, a = R . The equations of motion are: mg - f = ma fR = I = ( 2 3 mR^2 ) ( a R ) f = 2 3 ma Substituting f in the first equation: mg - 2 3 ma = ma a = 3 5 g The required friction for pure rolling is: f = 2 3 m ( 3 5 g ) = 2 5 mg To prevent sliding, f N , where N = mg : 2 5 mg mg 2 5 Thus, the minimum coefficient of friction is _ min = 2 5 . If the friction coefficient is hal