2.5.2.2. Lever Arm

Table of Contents

The idea of the lever arm

When a force acts on an object, the turning effect of that force depends not only on how large the force is, but also on where and how it is applied. The lever arm is the distance that tells us how effective the force is at producing rotation about a chosen pivot or axis.

If you push on a door near its hinges, the door is hard to turn. If you push at the handle, the same force turns it much more easily. The reason is that the lever arm is larger at the handle.

Definition

The lever arm, sometimes called the moment arm, is the perpendicular distance from the axis of rotation to the line of action of the force.

The line of action of a force is the straight line extending in the direction of the force through the point where the force is applied. The lever arm is not usually the direct distance from the pivot to the point of application. It is the shortest distance from the axis to that line.

Rendering scientific sketch…

Lever arm as perpendicular distance

The lever arm is the perpendicular distance from the rotation axis to the line of action of the force.

Lever arm and torque

Torque depends on the force and the lever arm. If the perpendicular distance is larger, the torque is larger. If the distance is zero, the force does not produce any turning effect about that axis.

For a force of magnitude $F$ and lever arm $r_\perp$, the magnitude of the torque is

$$
\tau = F r_\perp
$$

If the force is applied at a distance $r$ from the pivot and makes an angle $\theta$ with the position vector, then

$$
r_\perp = r \sin\theta
$$

so the torque magnitude can also be written as

$$
\tau = rF\sin\theta
$$

Important relation:
$$
\tau = F r_\perp = rF\sin\theta
$$
The lever arm is
$$
r_\perp = r\sin\theta
$$

Understanding the geometry

The largest torque occurs when the force is perpendicular to the radius from the pivot to the point of application. In that case, $\sin\theta = 1$, so

$$
r_\perp = r
$$

and the full distance from pivot to application point acts as the lever arm.

If the force points directly toward or away from the pivot, then $\theta = 0$ or $\theta = 180^\circ$, so

$$
r_\perp = 0
$$

and the torque is zero. In that case, the force may push or pull, but it does not turn the object about that axis.

Rendering scientific sketch…

How angle changes the lever arm

Everyday examples

A wrench works best when you apply force far from the bolt. This increases the lever arm and therefore increases the torque.

A door handle is placed far from the hinges for the same reason. A small force with a large lever arm can produce enough torque to open the door.

When using a seesaw, a child sitting farther from the center has a larger lever arm and can balance a heavier child sitting closer to the center.

Comparing different situations

The table below shows how the lever arm changes with force direction.

Situation	Angle $\theta$	Lever arm $r_\perp$	Torque
Force perpendicular to rod	$90^\circ$	$r$	Maximum
Force at an angle	Between $0^\circ$ and $90^\circ$	$r\sin\theta$	Intermediate
Force along the rod	$0^\circ$	$0$	Zero

A simple numerical example

Suppose a force of $10 \, \text{N}$ is applied to a wrench at a point $0.30 \, \text{m}$ from the bolt, and the force is perpendicular to the wrench.

Then the lever arm is

$$
r_\perp = 0.30 \, \text{m}
$$

and the torque magnitude is

$$
\tau = F r_\perp = (10)(0.30) = 3.0 \, \text{N} \cdot \text{m}
$$

Now suppose the same force is applied at the same point, but at an angle of $30^\circ$ to the wrench. Then

$$
r_\perp = r\sin\theta = 0.30 \sin 30^\circ = 0.30 \times 0.5 = 0.15 \, \text{m}
$$

So the torque becomes

$$
\tau = Fr_\perp = 10 \times 0.15 = 1.5 \, \text{N} \cdot \text{m}
$$

The same force produces only half as much torque because the lever arm is smaller.

Common mistake

A very common mistake is to use the full distance from the pivot to the point where the force is applied, even when the force is not perpendicular. That is not always correct. You must use the perpendicular distance to the line of action of the force.

Do not confuse these two quantities:
The distance to the point of application is $r$.
The lever arm is the perpendicular distance $r_\perp$.
Only $r_\perp$ goes directly into
$$
\tau = F r_\perp
$$

Summary

The lever arm tells how effectively a force can cause rotation. It is the perpendicular distance from the axis of rotation to the force's line of action. A larger lever arm gives a larger torque for the same force. When the line of action passes through the pivot, the lever arm is zero, so the torque is zero.

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