A uniform solid sphere of mass \(M\) and radius \(R\) is rolling without sliding along a level plane with a speed \(v=3.00 \mathrm{~m} / \mathrm{s}\) when it encounters a ramp that is at an angle \(\theta=23.0^{\circ}\) above the horizontal. Find the maximum distance that the sphere travels up the ramp in each case: a) The ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height. b) The ramp provides enough friction to prevent the sphere from sliding, so both the linear and rotational motion stop (instantaneously).

Question: A solid sphere of mass m and radius R is rolling without slipping on a flat surface with a linear speed of 3.00 m/s. The sphere then starts to roll up a ramp inclined at an angle θ. Calculate the maximum distance the sphere would travel up the ramp in two cases: a) a frictionless ramp, and b) a ramp with friction. Answer: To calculate the maximum distance the sphere travels up the ramp in both cases, we first determine the initial linear and rotational kinetic energies of the sphere. Next, we use conservation of energy principles to set up equations for each case. The maximum distance in cases a) and b) can be found by dividing their corresponding maximum heights (h_a and h_b) by the sine of the angle θ: distance_a = h_a / sin(θ) distance_b = h_b / sin(θ)