Rotational Dynamics and Rolling Without Slipping · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Rolling without slipping, and why shape wins races

Science · Forces & Motion · ages 20-22
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  1. A homogeneous hoop and a solid cylinder of the same radius start from rest together at the top of the same stationary slope, roll without slipping, and feel no rolling resistance. Which reaches the bottom first?

    • The solid cylinder
    • The hoop
    • Both at the same time

  2. A wheel of radius R rolls without slipping. How is the centre speed v linked to the spin rate omega?

    • v equals omega divided by R
    • v equals R times omega
    • v equals R divided by omega

  3. For ideal rigid rolling on a stationary surface, static friction between the wheel and the ground does no work.

    Circle one:   True   False

  4. For a passive axisymmetric round body rolling down a straight incline, which equation links the friction force to the change in spin?

    • Friction equals mass times centre acceleration
    • Friction times radius equals I times alpha
    • Friction times mass equals I times alpha

  5. A homogeneous solid cylinder rolls down a slope of angle theta without slipping, starting from rest. What is its centre acceleration?

    • g sin theta
    • one half of g sin theta
    • two thirds g sin theta

  6. A driver floors the pedal and the tires spin while the car stays still. What happened to the rolling link?

    • The contact slides, so v no longer equals R omega
    • The grip holds, so centre speed still equals R omega
    • The spin stops mattering, so centre speed equals omega alone

  7. Lee watches a solid cylinder rolling without slip down a stationary straight incline in the ideal rigid model. Lee says static friction slows the rolling cylinder and wastes its energy as heat. What is the error?

    • Static friction enables rolling with no work at the contact
    • Static friction resists the roll and burns energy as heat away
    • Static friction only appears once slipping has started

  8. Sam races a homogeneous hoop and a solid cylinder of the same radius from rest down the same slope without slipping. Sam says the hoop wins because it is lighter. What is the mistake?

    • Mass decides the race, so the lighter hoop must win
    • The slope angle decides it, so both shapes tie
    • Shape, not mass: the hoop parks more in spin
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Answer key

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Rolling without slipping, and why shape wins races W1-mt_1mV6ukwDxK-s1

  1. The solid cylinder · The hoop stores more energy in spin, so less remains for forward motion.
  2. v equals R times omega · Rolling without slipping locks forward motion to spin: v equals R omega.
  3. True · The contact point is at rest, so the force acts through no distance.
  4. Friction times radius equals I times alpha · Friction supplies the torque, and torque equals I times alpha.
  5. two thirds g sin theta · With I equal to half m R squared, the rolling formula gives two thirds g sin theta.
  6. The contact slides, so v no longer equals R omega · Spinning in place is slipping, so the contact slides and the link breaks.
  7. Static friction enables rolling with no work at the contact · Static friction acts uphill along the slope, so it does reduce the centre acceleration to two thirds g sin theta for a solid cylinder, but at the instantaneously stationary contact it does no work, so no energy is lost as heat in the ideal model.
  8. Shape, not mass: the hoop parks more in spin · Mass cancels inside the shape factor I over m R squared, which is what decides the race.
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