A solid sphere and a hollow spherical shell of equal mass and radius are released from rest at the top of an inclined ramp. Assuming rolling without slipping, which reaches the bottom first?
AThe hollow shell, because its mass is concentrated at the rim, giving it more rotational momentum
BThe solid sphere, because its smaller moment of inertia means less energy goes into rotation and more into translation
CThey arrive simultaneously, because they have the same mass and experience the same gravitational force
DThe result depends on the ramp angle, not on the shape of the object
By energy conservation, both objects convert the same initial potential energy into total kinetic energy (½mv² + ½Iω²). Using the rolling constraint v = Rω, total KE = ½mv²(1 + I/mR²). The object with larger I/mR² puts more energy into rotation and less into translation, arriving with lower v_CM. For a solid sphere, I/mR² = 2/5; for a hollow shell, I/mR² = 2/3. The sphere's smaller rotational fraction means more translational speed — it arrives first. Same mass is irrelevant; the I/mR² ratio is what matters.
Question 2 Multiple Choice
A wheel rolls without slipping on a flat surface. How much work does the static friction force at the contact point do?
APositive work, because friction is what enables rolling and provides energy to the system
BNegative work, because friction always opposes motion
CZero work, because the contact point has zero instantaneous velocity — the force acts on a point that isn't moving
DZero work, but only because rolling friction is negligible on flat surfaces
Work = force × displacement of the point of application. During rolling without slipping, the contact point has zero instantaneous velocity relative to the ground — it is momentarily stationary, not sliding. Therefore the static friction force acts on a point with zero velocity, doing zero work. This is not an approximation; it is exact for ideal rolling without slipping. However, friction is still essential — it provides the torque that causes angular acceleration. A force can be mechanically necessary without doing work.
Question 3 True / False
During rolling without slipping, the contact point of the wheel has zero instantaneous velocity relative to the ground.
TTrue
FFalse
Answer: True
This is the physical meaning of 'no slipping' and the foundation of the rolling constraint. The contact point's velocity has two contributions: the translational velocity v_CM (forward) and the tangential rim velocity Rω (backward at the contact point due to rotation). For rolling without slipping, these exactly cancel: v_contact = v_CM − Rω = 0. This is why static, not kinetic, friction acts at the contact point — the surfaces are not sliding relative to each other.
Question 4 True / False
If a round object is placed on a frictionless surface, it will still roll without slipping because its shape ensures the contact point stays stationary.
TTrue
FFalse
Answer: False
Friction is essential for rolling without slipping — it is the torque-generating mechanism that couples translational and rotational motion. On a frictionless surface, a force applied to a round object's center accelerates it translationally but does not create a torque, so rotation lags behind. The object slides rather than rolls, and the rolling constraint v_CM = Rω is violated. Without friction, there is no mechanism to enforce the coupling between translation and rotation.
Question 5 Short Answer
Derive the rolling constraint v_CM = Rω by explaining what physical condition at the contact point it expresses.
Think about your answer, then reveal below.
Model answer: The condition is zero relative velocity at the contact point. The contact point's velocity has two contributions: the translational velocity v_CM of the wheel's center (forward), and the tangential velocity Rω due to rotation (backward at the bottom of the wheel). For no slipping, these must cancel exactly: v_CM − Rω = 0, which gives v_CM = Rω. This single equation couples the two degrees of freedom (translation and rotation) into one, so knowing either v_CM or ω immediately determines the other.
Understanding why the constraint takes this form — rather than just memorizing the equation — makes it straightforward to handle inclined planes, curved surfaces, and other rolling geometries. The contact-point velocity condition is the physical fact; v_CM = Rω is just that fact expressed mathematically.