A.4 · IB Physics
Rolling race: moment of inertia
Race a hoop, a disc, a solid sphere and a hollow sphere down the same incline. See how the shape factor k in I = kMR² sets the acceleration, the finishing times and how the kinetic energy is shared between translation and rotation.
I = kMR²
a = g sinθ / (1 + k)
t = √(2L / a)
v = √(2gh / (1 + k)), h = L sinθ
E_rot / E_k = k / (1 + k)
g = 9.8 m s⁻², rolling without slipping
a = g sinθ / (1 + k)
t = √(2L / a)
v = √(2gh / (1 + k)), h = L sinθ
E_rot / E_k = k / (1 + k)
g = 9.8 m s⁻², rolling without slipping
| Object | k | I / kg m² | a / m s⁻² | t / s | v / m s⁻¹ | E_trans | E_rot |
|---|
Try this
- Predict the finishing order before pressing Play. Then change θ: does the order ever change?
- Make M five times larger and R ten times smaller. Check the a and t columns, and explain using a = g sinθ/(1 + k).
- Compare the energy bars of the hoop and the solid sphere at the bottom: same total energy Mgh, so why is the hoop slower?
Common error: assuming a heavier or wider object rolls down faster; for rolling without slipping M and R cancel, so cancel them algebraically and compare only the shape factor k.