Torque, Equilibrium, and Angular Acceleration

Driving question

How do force location and mass distribution affect rotational change?

Before learning: door prediction

Imagine pushing a door with the same force in three ways: near the hinge, far from the hinge, and directly toward the hinge. Rank the resulting rotational effects and explain what matters besides force magnitude.

Hook: the strongest push may produce no rotation

A large force directed through a pivot produces zero torque about that pivot. A smaller perpendicular force farther away can rotate the object. Rotational effect depends on where, how, and about which axis a force acts.

Torque about a chosen axis

The torque of a force about an axis is

Its magnitude is

where $\theta$ is the angle between $\vec r$ and $\vec F$, and $\ell=r\sin\theta$ is the perpendicular lever arm. Torque units are $\text{N}\cdot\text{m}$, not joules, because torque is not energy.

In planar problems, choose a sign convention—commonly counterclockwise positive and clockwise negative—and state it.

Worked example: angled wrench

A $40.0\text{ N}$ force acts $0.250\text{ m}$ from a bolt at $60.0^\circ$ to the wrench handle:

The sign depends on the rotation tendency. Only the force component perpendicular to the handle contributes.

Rotational equilibrium

Static equilibrium requires both

and

Zero net force alone prevents translational acceleration but not angular acceleration. Zero net torque alone prevents angular acceleration but not translational acceleration.

Worked example: balanced beam

A negligible-mass horizontal beam pivots at its center. A $150\text{ N}$ load hangs $0.80\text{ m}$ left of the pivot. Where must a $240\text{ N}$ load hang on the right for equilibrium?

Taking counterclockwise positive,

so $x=0.50\text{ m}$. The pivot force produces zero torque about the pivot because its lever arm is zero.

Choosing the pivot strategically

Net torque may be computed about any axis, but the torque values depend on that choice. In equilibrium, choose a pivot through an unknown support force to eliminate that force from the torque equation. Then use force balance to find the remaining support components.

For an extended object, gravitational force acts effectively at its center of mass in a uniform gravitational field.

Rotational dynamics

For a rigid body rotating about a fixed axis,

where $I$ is rotational inertia and $\alpha$ is angular acceleration. This resembles $\sum F=ma$, but $I$ depends on how mass is distributed relative to the axis:

Moving the same mass farther from the axis increases $I$ and reduces $\alpha$ for the same net torque.

Worked comparison

Two wheels receive the same constant net torque $6.0\text{ N}\cdot\text{m}$. Wheel A has $I_A=2.0\text{ kg}\cdot\text{m}^2$ and wheel B has $I_B=3.0\text{ kg}\cdot\text{m}^2$:

The smaller rotational inertia yields the larger angular acceleration.

During learning: representation checks

For each scenario:

1. Mark the chosen axis and draw $\vec r$ from it to the force application point.
2. Identify the perpendicular force component or lever arm.
3. Assign a rotation sign before calculating.
4. Decide whether the object also needs a translational force equation.
5. Identify whether $I$ is given, derivable, or only comparable.

Connected translation and rotation

A string wrapped around a pulley applies torque $Tr$. If the string does not slip, tangential and angular accelerations satisfy

The tension on two sides of a massive accelerating pulley need not be equal; their difference supplies net torque.

Misconception clinic

“Any force produces rotation.” A force through the axis has zero lever arm and zero torque about that axis.

“Torque is a force.” Torque is a rotational effect calculated from force and lever arm.

“Moment of inertia depends only on mass.” It depends on mass distribution and axis location.

“If net force is zero, the object is fully balanced.” Rotational equilibrium also requires zero net torque.

“The longest radius always means the greatest torque.” Force angle also matters through $\sin\theta$.

After learning: retrieval and transfer

1. Why does pushing directly toward a hinge produce no torque?
2. What two equations are required for static equilibrium?
3. Why can a support force disappear from a torque equation but remain physically present?
4. How does moving mass outward change $I$ and $\alpha$ under fixed torque?
5. Explain why torque and energy share units but are different quantities.

AP-style evidence routine

1. Define the rigid body and axis.
2. Draw all external forces and centers of application.
3. Choose signs and calculate perpendicular lever arms.
4. Write $\sum F=ma$ and $\sum\tau=I\alpha$ as needed.
5. Apply geometric or no-slip constraints.
6. Check units, signs, limiting cases, and whether the fixed-axis model applies.

Key takeaway

Rotation is controlled by net torque and rotational inertia. A defensible model always names the axis, locates each force, and accounts for mass distribution.

Further learning and alignment

• College Board, “AP Physics 1: Algebra-Based Course,” https://apcentral.collegeboard.org/courses/ap-physics-1 (retrieved 2026-08-18). Alignment reference only; no endorsement implied.
• OpenStax, “10.6 Torque” and “10.7 Newton's Second Law for Rotation,” https://openstax.org/books/physics/pages/10-6-torque and https://openstax.org/books/physics/pages/10-7-newtons-second-law-for-rotation (retrieved 2026-08-18). CC BY concept verification; no exercises copied.
• OpenStax, “9.3 Stability,” https://openstax.org/books/physics/pages/9-3-stability (retrieved 2026-08-18). CC BY center-of-mass and equilibrium reference.