A hollow cylinder and a solid sphere of the same mass and radius are released from rest at the top of the same ramp. Which reaches the bottom first, and why?
AThe hollow cylinder, because its mass is concentrated at the rim, giving it more rotational momentum
BThey arrive simultaneously, since they have the same mass and radius and start from the same height
CThe solid sphere, because its moment of inertia is smaller (2/5 MR² vs. MR²), so less of the available potential energy goes into rotation and more into translational speed
DThe hollow cylinder, because a larger moment of inertia means more total kinetic energy at the bottom
Both objects start with the same gravitational potential energy (mgh). At the bottom, that energy is split between translational KE (½mv²) and rotational KE (½Iω²). The hollow cylinder has I = MR² (all mass at the rim); the solid sphere has I = 2/5 MR². The larger I of the hollow cylinder means more energy is channeled into rotation and less into translation — so the cylinder moves along the ramp more slowly. The sphere wins because its mass distribution puts less emphasis on rotation.
Question 2 Multiple Choice
For a ball rolling without slipping down a ramp, which equation correctly expresses the total kinetic energy at any point?
AKE = ½Iω², since rolling is purely rotational
BKE = ½mv², since the relevant velocity is the center-of-mass velocity
CKE = ½mv_CM² + ½I_CM ω², where both translational and rotational terms contribute
DKE = mv_CM² because the factor of ½ cancels when both terms are combined
A rolling object simultaneously translates (center of mass moves) and rotates (spins about the center of mass). Both motions store kinetic energy. The total is the sum: translational KE of the center of mass plus rotational KE about the center of mass. The rolling-without-slipping constraint v_CM = Rω links these terms, but they are distinct contributions to the total energy. Treating rolling as purely translational or purely rotational both give wrong answers.
Question 3 True / False
For a rigid object rolling without slipping, the total kinetic energy equals the translational kinetic energy of the center of mass plus the rotational kinetic energy about the center of mass.
TTrue
FFalse
Answer: True
This additive decomposition is exact for rigid bodies. Rolling without slipping means the object simultaneously translates at v_CM and rotates at ω = v_CM/R. The translational term ½mv_CM² captures the motion of the center of mass through space; the rotational term ½I_CM ω² captures the spin about the center of mass. The two terms are funded by the same energy source (gravity when rolling downhill) and add linearly.
Question 4 True / False
A hollow hoop (I = MR²) and a solid disk (I = ½MR²) of the same mass and radius, both rolling without slipping, will reach the same translational speed at the bottom of any ramp.
TTrue
FFalse
Answer: False
They will not. Because both start with the same potential energy (mgh), and that energy splits between translation and rotation, the disk — with the smaller I — puts less energy into rotation and more into translation, arriving faster. The hoop concentrates all mass at the rim (maximum I), so it puts proportionally more energy into rotation, leaving less for translational speed. The final translational speed depends on the moment of inertia, not just the mass and radius.
Question 5 Short Answer
Explain why a hollow cylinder rolls slower down a ramp than a solid sphere of the same mass and radius, using rotational kinetic energy and moment of inertia.
Think about your answer, then reveal below.
Model answer: Both objects start with the same gravitational potential energy, which converts entirely to kinetic energy at the bottom. For each, total KE = ½mv² + ½Iω². The rolling constraint v = Rω lets us write this as KE = ½mv²(1 + I/MR²). A larger I means a larger fraction of total KE goes into rotation and a smaller fraction goes into translational speed. The hollow cylinder has I = MR² (all mass at radius R), giving a factor of (1 + 1) = 2. The solid sphere has I = 2/5 MR², giving a factor of (1 + 2/5) = 7/5. The sphere's smaller factor means more of the energy is translational — so it moves faster along the ramp.
This is the power of the energy method: without computing torques, angular acceleration, or friction forces, a single energy equation explains why shape matters for rolling. The moment of inertia summarizes how mass is distributed relative to the rotation axis, and that distribution directly controls how the available energy partitions between rotation and translation.