Topic 5.3 Notes – Torque

1. What Torque Is

If force measures how strongly something translates, torque measures how strongly something rotates about a chosen pivot.

Later you’ll use $$\sum \vec{\tau} = I\vec{\alpha}$$ but right now the focus is identifying and describing individual torques.

Vector definition

$$\vec{\tau} = \vec{r} \times \vec{F}$$

• $\vec{r}$ is the position vector from the pivot to where the force is applied
• $\vec{F}$ is the applied force
• The pivot must be specified. Change the pivot and the torques change.

Magnitude

$$|\vec{\tau}| = rF\sin\theta$$

• $r$ is the distance from pivot to force application point
• $\theta$ is the angle between $\vec{r}$ and $\vec{F}$

Only the perpendicular component of the force contributes.

Special cases:

• $\theta = 90^\circ$ → maximum torque → $\tau = rF$
• $\theta = 0^\circ$ or $180^\circ$ → zero torque

Units are $\text{N}\cdot\text{m}$. Same units as energy, but torque is a vector, not a scalar.

2. Components of the Torque Formula

Position Vector $\vec{r}$

This is where students mess up.

• It always starts at the pivot
• It ends at the point where the force is applied
• Direction matters for the cross product

If you draw it backward, your torque direction will be wrong.

Perpendicular Force Component

From $rF\sin\theta$:

$$\tau = rF_\perp$$

where $F_\perp = F\sin\theta$.

So you can either:

• Break the force into components and use the perpendicular one, or
• Use $rF\sin\theta$ directly.

Same physics either way.

Lever Arm (Moment Arm)

There’s an equivalent and often easier method.

The lever arm $r_\perp$ is the perpendicular distance from the pivot to the line of action of the force.

$$\tau = r_\perp F$$

And since $r_\perp = r\sin\theta$, this matches the earlier formula.

Here’s what that geometry looks like in a typical setup with an angled force:

Torque geometry: $r$, $\theta$, and lever arm $r_\perp$

Notice how the dashed perpendicular from the pivot to the dashed line of action gives $r_\perp$. That distance alone determines how effective the force is at causing rotation.

Important consequences:

• A small force far from the pivot can equal a large force close in.
• If the force acts through the pivot, the lever arm is zero → no torque.
• If the force is directed straight toward or away from the pivot, $r_\perp = 0$.

AP problems often hide this in geometry. Always look for that perpendicular distance.

3. Direction of Torque and the Cross Product

Torque direction comes from the cross product.

For any $\vec{A} \times \vec{B}$:

• Magnitude is $AB\sin\theta$
• Direction is perpendicular to the plane containing them

So $\vec{\tau}$ is perpendicular to both $\vec{r}$ and $\vec{F}$.

Right-hand rule

The diagram below shows the cross product $\vec{A} \times \vec{B}$ and the hand motion that determines its direction.

Right-hand rule for the cross product

How to apply it for torque:

1. Point fingers along $\vec{r}$
2. Curl toward $\vec{F}$
3. Your thumb gives the direction of $\vec{\tau}$

In 2D:

• Out of the page → counterclockwise → usually positive
• Into the page → clockwise → usually negative

On most quizzes and FRQs, torque becomes a signed scalar. You decide a sign convention and stay consistent.

4. Drawing Force Diagrams for Torque Problems

A torque diagram is basically a free-body diagram with locations included.

You must show:

• The rigid object
• The pivot point
• Every force
• Where each force is applied

If you leave out location, you can’t determine torque.

For each force, ask:

• Does it pass through the pivot? → zero torque
• Is it parallel to $\vec{r}$? → zero torque
• Does it try to rotate CW or CCW?

Example thinking:

• Weight at the end of a horizontal beam about a wall hinge → clockwise.
• Normal force at the hinge → zero torque about that hinge.

The exam loves pivots chosen to eliminate unknown forces. If you pick the pivot at a hinge, hinge forces produce zero torque automatically.

5. Strategy for Identifying Torques

When analyzing a rigid system:

1. Choose a pivot.
2. Draw $\vec{r}$ to each force.
3. Determine sign (CW or CCW).
4. Compute with $rF\sin\theta$ or $r_\perp F$.

Changing pivots changes individual torques, but the physical rotation of the object doesn’t change.

Your goal in this topic is speed and clarity. You should be able to look at a diagram and immediately classify every force as positive torque, negative torque, or zero torque.