Torque & Moment of Inertia

You’ll be able to

Define torque as τ = rF sin θ and identify the lever arm
Define the moment of inertia I = ∫r² dm and compute it for a simple body
Explain how the distribution of mass, not just its amount, sets the rotational inertia

Torque is the turning effect of a force

Torque measures how effectively a force rotates something. It depends on the force, on how far from the axis it is applied, and on its direction: τ = rF sin θ, where θ is the angle between the position vector and the force. Equivalently it is the force times the lever arm (the perpendicular distance from the axis to the line of the force). A force aimed straight at the axis produces no torque.

Torque

τ = rF sin θ = r_⊥ F

r is the distance from the axis to the point of application; sin θ picks out the perpendicular component. Maximum when the force is applied perpendicular to the radius.

Moment of inertia is rotational mass

The moment of inertia I plays the role of mass for rotation — the resistance to angular acceleration. It is not just the amount of mass but how far that mass sits from the axis: I = Σmr² for particles, or I = ∫r² dm for a continuous body. Because r is squared, mass far from the axis counts far more. That is why a hoop (mass at the rim) has a larger I than a disk of the same mass and radius.

Moment of inertia

I = ∫ r² dm (hoop: MR²; disk: ½MR²; rod about center: 1/12 ML²; sphere: 2/5 MR²)

r is the distance of each mass element from the rotation axis. The same object has different I about different axes.

Worked example

Derive the moment of inertia of a uniform thin rod of mass M and length L rotating about one end.

1.The rod is uniform, so its linear density is λ = M/L, and a slice at distance x from the end has mass dm = λ dx.
2.Each slice contributes r² dm = x²(λ dx) to the moment of inertia.
3.Integrate from 0 to L: I = ∫₀ᴸ x² λ dx = λ[x³/3]₀ᴸ = λL³/3.
4.Substitute λ = M/L: I = (M/L)(L³/3) = ML²/3.

Answer: I = ∫₀ᴸ x²(M/L)dx = ML²/3 for a rod about its end

Checkpoint

A force of 10 N is applied perpendicular to a wrench at a distance of 0.2 m from the bolt. What torque does it produce about the bolt?

Checkpoint

A solid disk (I = ½MR²) and a hoop (I = MR²) have the same mass and radius. Which has the greater moment of inertia, and why?

On the exam

Two objects with identical mass can have very different moments of inertia. Always ask where the mass sits relative to the axis: the r² weighting in I = ∫r² dm rewards mass at the rim and penalizes it near the center.

Answer the 2 checkpoints as you read.