Rotational inertia lab | Mistake Master Studio

The situation

A disk sits on a low-friction vertical axle. A string wraps around a spool of radius R at the center of the disk, runs over a pulley at the table edge, and hangs down to a mass m. Release the mass: it falls, unwinding the string, and spins the disk up. A photogate times how long the mass takes to drop through a set height H. Below you will get a working rig with three knobs: the hanging mass m, the spool radius R, and the drop height H.

First, commit to a prediction. Then the apparatus unlocks.

Predict first

Commit before you touch the sliders

Which changes make the disk speed up faster (a bigger acceleration for the falling mass)? Check all you expect to. Hanging a heavier mass Winding the string on a wider spool Releasing the mass from a greater height

Plot the acceleration a straight against the hanging mass m. What shape do you expect? A straight line sloping up (each added kilogram buys the same extra acceleration) A curve that climbs quickly at first and then levels off toward a ceiling (the mass can never fall faster than free fall) A flat line (the hanging mass has no effect)

Explore

Test your prediction on the apparatus

Drag the sliders and time the drop at different settings. Or pick a test chip to lock everything but one variable and watch its relationship draw itself. The m slider swaps in a lighter or heavier hanging mass; the R slider winds the string on a narrower or wider spool; the H slider raises or lowers the release point. The disk is labeled I = 0.0020 kg m², the axle is low-friction, the string does not slip on the spool, and the rig sits on Earth, g = 10 N/kg. Re-timing at identical settings repeats exactly, so any change you see is real.

Hanging mass mm = 0.150 kg

Spool radius RR = 0.040 m

Drop height HH = 0.80 m

Mystery disk: I = ?

Inverse acceleration vs inverse hanging mass

Drag the unlocked slider. Every position you visit drops a measured point.

Which fit matches your points?

Measure and straighten

What is the rotational inertia of the sealed disk?

The disk has been sealed and its rotational inertia is hidden. So the slope stays honest, the spool is fixed at R = 0.040 m and the release height at H = 0.80 m (both stated on the rig), the axle stays low-friction, and g = 10 N/kg. Choose at least 5 different hanging masses, time the drop at each, turn each drop time into an acceleration with a = 2 H / t², and enter each (m, a) pair below. Then straighten the plot, and work through Read the slope: the walkthrough hands you the rotational inertia only once you can say why the slope carries it, and that opens the seal.

Collect your data, then work through the Read the slope steps: they walk you to the transform that straightens the plot and the slope that carries the rotational inertia I. Finish them and the Record button turns on and checks your I against the sealed disk.

Transfer

One changed variable

Back at the open rig, a friend swaps the disk for a different one of exactly the same total mass, but with that mass packed close to the axle instead of spread out near the rim. Same disk mass, same 0.150 kg hanging weight, same spool, they argue, so the disk should speed up at the same rate. What actually happens?

Rotational inertia is not set by the mass alone. It is set by how far that mass sits from the axle: I adds up each bit of mass times the square of its distance from the axle. Packing the same mass close to the axle makes I smaller, so I / (m R²) shrinks and a = g / (1 + I / (m R²)) climbs closer to free fall. The disk speeds up faster and the weight drops in less time, even though the disk mass never changed. Two disks of equal mass can hand back two different slopes, which is exactly why the sealed disk hid its rotational inertia and not its mass.