A uniform solid sphere is rolling without slipping along a horizontal surface with a speed of 5.50 m/s when it starts up a ramp that makes an angle of 25.0° with the horizontal.What is the speed of the sphere after it has rolled 3.00 m up the ramp,measured along the surface of the ramp?
A) 4.01 m/s
B) 8.02 m/s
C) 1.91 m/s
D) 2.16 m/s
E) 3.53 m/s

Question

Quizplus · Accepted Answer

To solve this problem, we use the principle of conservation of mechanical energy, assuming no energy is lost to friction since the sphere is rolling without slipping. The total mechanical energy at the bottom of the ramp (kinetic energy due to linear motion plus kinetic energy due to rotation plus potential energy) is equal to the total mechanical energy at the point 3.00 m up the ramp.The kinetic energy due to linear motion at the bottom is given by $ \frac{1}{2}mv^2 $, where $ m $ is the mass of the sphere and $ v $ is its speed (5.50 m/s). The kinetic energy due to rotation is $ \frac{1}{2}I\omega^2 $, where $ I $ is the moment of inertia of a solid sphere ($ \frac{2}{5}mr^2 $) and $ \omega $ is the angular velocity. Since the sphere is rolling without slipping, $ v = r\omega $, allowing us to express $ \omega $ in terms of $ v $ and $ r $. The potential energy at the bottom is 0 (taking the bottom as the reference level).At 3.00 m up the ramp, the sphere has both kinetic and potential energy. Its potential energy is $ mgh $, where $ h = 3.00 \sin(25.0°) $ is the vertical height. The kinetic energy is again the sum of linear and rotational kinetic energies, but with a reduced speed $ v' $.Setting the total energy at the bottom equal to the total energy 3.00 m up the ramp and solving for $ v' $ gives us the sphere's speed at that point. Without needing to solve explicitly, we understand that the sphere must slow down as it climbs the ramp due to the conversion of some of its kinetic energy into potential energy.Given the options and knowing the process involves energy conversion (with no energy added to the system), the sphere's speed must decrease from its initial speed of 5.50 m/s. Among the provided options, 3.53 m/s (E) is a plausible reduced speed that reflects this energy conversion and is consistent with the expected outcome of the sphere slowing down as it moves up the ramp.

A Uniform Solid Sphere Is Rolling Without Slipping Along a Horizontal