5.3 Investigating Torque and Angular Acceleration Relationships

Newton's Second Law for Rotation

Just as F_net = ma describes linear motion, τ_net = Iα describes rotational motion. Net torque causes angular acceleration (α), and the rotational inertia (I) resists that change. A larger torque produces greater angular acceleration; a larger rotational inertia produces less angular acceleration for the same torque.

Designing Torque Experiments

To investigate the τ = Iα relationship experimentally, apply a known torque (e.g., hanging mass over a pulley on a disk) and measure the resulting angular acceleration using angular position/velocity sensors. Plotting torque vs. angular acceleration yields a straight line through the origin, with slope equal to the object's rotational inertia.

Graphical Analysis

Since τ_net = Iα, a graph of net torque (y-axis) versus angular acceleration (x-axis) is linear with slope = I. Changing the mass distribution (e.g., moving weights outward on a rod) increases I, decreasing the slope's steepness relationship... actually increasing the slope, since more torque is needed for the same α.

Key terms

τ_net = Iα

Rotational form of Newton's Second Law: net torque equals rotational inertia times angular acceleration.

Rotational inertia (I)

A measure of an object's resistance to changes in rotational motion; depends on mass distribution relative to the axis.

Angular acceleration (α)

The rate of change of angular velocity, caused by a net torque acting on a rotating object.

Slope of τ vs. α graph

Equals the rotational inertia (I) of the object being tested.

Effect of mass distribution on I

Moving mass farther from the axis increases rotational inertia, requiring more torque to achieve the same angular acceleration.

Try an AP-style question

In a graph of net torque (y) vs. angular acceleration (x), what is the slope?

1. The mass m
2. The radius r
3. The angular velocity ω
4. The rotational inertia I