Rotational Motion & Torque
Angular momentum, moment of inertia, and spinning objects
Torque is the rotational equivalent of force: τ = rF sinθ. The rotational equivalent of Newton's 2nd Law is τ = Iα. Angular momentum L = Iω is conserved when no external torque acts — this is why a spinning skater spins faster when they pull in their arms (reducing I increases ω to keep L constant). The moment of inertia I depends on how mass is distributed relative to the rotation axis.
Why do figure skaters spin faster when they pull their arms in?
No extra push, no extra energy input — yet they visibly accelerate. The answer lies in angular momentum conservation.
What is Rotational Motion & Torque?
A figure skater starts a slow spin with arms wide, then pulls them tight against her chest — and visibly accelerates into a blur. Nobody pushed her. Her muscles only pulled inward, perpendicular to the spin. Yet her angular velocity tripled in a second. The reason is angular momentum conservation: when no external torque acts, the product L = Iω stays constant, so shrinking the moment of inertia I forces the angular velocity ω to grow in lockstep. The same physics governs a diver tucking for a flip, a neutron star spinning up as it collapses, and a wheel resisting being shoved off-axis. This lab lets you torque a disk to study τ = Iα directly, then switch to the skater mode to watch L = Iω hold its line as you pull her arms in and out — angular momentum, moment of inertia, and rotational kinetic energy all on display in real time.
Parameters explained
- Mass (m)(kg)
- Mass sets how much material the rotating object has, from a light 0.1 kg object to a 20 kg object. In every preset, moment of inertia is proportional to mass: I = 1/2 mR² for a disk, I = mR² for a ring, and I = 2/5 mR² for a sphere. Increasing Mass therefore makes the object harder to angularly accelerate under the same Torque. Keep Radius and Torque fixed, then compare how α falls as Mass rises. This separates the effect of total material from the effect of shape.
- Radius (R)(m)
- Radius sets how far the object's mass extends from the rotation axis. It matters strongly because the moment-of-inertia formulas all include R², so doubling Radius makes I four times larger for the same Mass and object type. A larger Radius therefore reduces angular acceleration when Torque is unchanged, while a smaller Radius lets the same torque spin the object up faster. Use the Disk, Ring, and Sphere presets at the same Radius first, then move only Radius to see why mass distribution is more than just the total amount of mass.
- Applied Torque (τ)(N·m)
- Torque is the external twist applied to the object, measured in N·m. It is the rotational analog of net force and controls angular acceleration through τ = Iα. When Mass, Radius, and preset geometry stay fixed, increasing Torque increases α directly, so angular velocity builds faster and rotational kinetic energy rises more quickly. A zero Torque setting lets students see that no new angular acceleration is added; a high Torque setting makes the same inertia respond more dramatically. Compare the three presets at one Torque value to isolate how I changes the outcome.
Common misconceptions
MisconceptionTorque is just rotational force — bigger force always means bigger torque.
CorrectTorque depends on force, lever arm, and angle: τ = rF·sinθ. The same force applied close to the pivot or in line with the lever produces little torque; far from the pivot and perpendicular, it produces a lot. That's why door handles are placed far from the hinge — the geometry is doing as much work as the force is.
MisconceptionIf two objects have the same mass and radius, they must spin up the same way.
CorrectMass and radius are not enough; distribution matters. A disk, ring, and sphere with the same M and R have different moments of inertia: ½MR², MR², and 2/5MR². Under the same torque, the smaller-I object has the larger angular acceleration.
MisconceptionAngular momentum is conserved in any system as long as the total energy stays constant.
CorrectAngular momentum is conserved when the net external TORQUE on the system is zero — that's the key requirement, and it's distinct from energy conservation. In this lab, a nonzero Torque slider means angular momentum is changing. Conversely, energy can be conserved while L is not, if external torques are present.
MisconceptionAll objects with the same mass and radius have the same moment of inertia.
CorrectMoment of inertia depends on how the mass is distributed, not just the totals. A solid disk of mass M and radius R has I = ½MR²; a hollow ring with the same M and R has I = MR² — twice as much. The ring puts all its mass at the maximum radius, so it's harder to spin up. That's why hollow tubes resist twisting more than solid rods of equal mass.
MisconceptionA solid disk and a hollow ring of equal mass and radius will tie when rolling down an incline.
CorrectThe solid disk wins. Both convert the same gravitational PE, but the ring puts a larger fraction of its KE into rotation (because of its higher I) and less into translation. Less translational KE means less linear speed at the bottom. Lower I → faster roll. This is one of the cleanest demonstrations that geometry, not just mass, governs rotational behavior.
How teachers use this lab
- Preset comparison: set Mass, Radius, and Torque to the same values, then compare Disk, Ring, and Sphere. Students rank angular acceleration and justify the order from I = 1/2 mR², I = mR², and I = 2/5 mR².
- Radius-squared investigation: keep the Disk preset, Mass, and Torque fixed while students double Radius. Have them predict and then verify that moment of inertia grows by about four times and angular acceleration drops accordingly.
- Torque proportionality plot: keep Mass, Radius, and preset geometry fixed while students collect α at several Torque values. Graph α versus τ to verify the direct relationship in τ = Iα.
- Mass isolation task: hold Radius and Torque fixed, then move only the Mass slider. Students explain why changing mass changes I and α without changing the object's geometric formula.
- Rolling-race extension: after comparing Disk, Ring, and Sphere presets, ask students to predict which object would roll down an incline fastest and connect the answer to how much energy goes into rotation.
Frequently asked questions
What is angular momentum and why is it conserved?
Angular momentum L = Iω is the rotational analog of linear momentum. It's conserved when the net external torque on a system is zero, just as linear momentum is conserved when the net external force is zero. The reason is symmetry: the laws of physics don't change if you rotate the universe, and that rotational symmetry mathematically guarantees angular momentum conservation. In this lab, the active focus is torque-driven change: when you apply a nonzero τ, angular momentum changes; when τ is zero, the object keeps its current rotational state.
Why does the ring spin up more slowly than the disk?
A thin ring puts essentially all of its mass at radius R, so its moment of inertia is I = mR². A solid disk spreads mass from the center out to the rim, giving I = 1/2 mR². With equal mass, radius, and torque, the ring has the larger I, so α = τ/I is smaller. The result is a slower increase in angular velocity even though the applied torque is the same.
What is moment of inertia and how is it different from mass?
Moment of inertia is the rotational analog of mass — it tells you how hard it is to change an object's rotational state. Unlike linear mass, it depends on how that mass is arranged relative to the rotation axis. The same 5 kg gathered near the axis has a small I; spread out at a large radius, it has a much larger I. The math is I = Σ m_i·r_i². For continuous bodies, geometry sets the formula: ½MR² for a solid disk, MR² for a hoop, ⅖MR² for a solid sphere.
What happens if I increase radius while mass stays fixed?
Moment of inertia rises with the square of radius in the disk, ring, and sphere formulas. If Radius doubles while Mass and object type stay fixed, I becomes four times larger. Since angular acceleration follows α = τ/I, the same Torque then produces only one quarter of the angular acceleration. This is why moving mass farther from the axis has such a strong effect on rotational motion.
Why does a longer wrench make it easier to loosen a stuck bolt?
Because torque depends on the lever arm, not just the force you apply. The relationship is τ = rF·sinθ. Doubling the wrench length doubles r, and at the same hand force you produce twice the torque on the bolt. The bolt only knows the torque, not the force, so the doubled torque is twice as effective at overcoming whatever holds it in place. The same logic explains why doors have handles far from the hinge and why steering wheels are wider than the steering column.
How does this lab connect to AP Physics 1 standards 4.D.1 through 4.D.3?
AP Physics 1 standards 4.D.1 through 4.D.3 expect students to analyze rotational motion using torque, moment of inertia, and angular momentum, and to apply conservation of angular momentum to systems with no net external torque. This lab gives students known torques, measured angular acceleration, and three preset geometries so they can verify τ = Iα and compare how mass distribution changes I. NGSS HS-PS2-1 supports the same conceptual goals.