KAHIBARO
Table of Contents
Turning Effect of a Force
In rotational motion, the quantity that plays a role similar to force in linear motion is torque. Torque tells us how strongly a force tends to make an object rotate about a chosen point or axis. Pushing a door near its handle makes it turn easily, while pushing near the hinges does not. The force may be the same, but the turning effect is different. That difference is torque.
Torque depends on three things, the size of the force, where the force is applied, and the direction of the force relative to the object. A larger force usually gives a larger torque. A force applied farther from the pivot usually gives a larger torque. A force that acts more sideways to the object produces more turning than a force aimed directly toward the pivot.
Choosing a Pivot
To talk about torque, we must first choose a reference point, often called the pivot, origin, or axis of rotation. Torque is always defined relative to this chosen point. The same force can produce different torques about different points.
Imagine a rigid bar attached at one end. If you push on the free end, the bar tends to rotate about the attached end. If you choose that attached end as the pivot, the torque is easy to describe. If you choose another point, the numerical value of the torque changes.
Basic Idea of Torque
Suppose a force $\vec{F}$ acts on an object at a point whose position relative to the pivot is $\vec{r}$. Then the torque measures the turning effect of that force about the pivot.
The magnitude of torque is
$$
\tau = rF\sin\theta
$$
where $r$ is the distance from the pivot to the point of application of the force, $F$ is the magnitude of the force, and $\theta$ is the angle between $\vec{r}$ and $\vec{F}$.
If the force is perpendicular to the position vector, then $\sin\theta = 1$, and the torque is maximum:
$$
\tau = rF
$$
If the force points directly toward or away from the pivot, then $\theta = 0$ or $\pi$, so $\sin\theta = 0$, and the torque is zero. In that case the force does not tend to rotate the object about that point.
Important torque formula:
$$
\tau = rF\sin\theta
$$
Maximum torque occurs when the force is perpendicular to the radius.
Zero torque occurs when the force acts along the radius.
Direction of Torque
Torque is not only a size, it also has a direction. In simple flat problems, we often describe torque as clockwise or counterclockwise.
A counterclockwise torque is usually taken as positive.
A clockwise torque is usually taken as negative.
This sign choice is a convention, but once chosen it must be used consistently.
For example, if a force tends to rotate a wheel counterclockwise, we may write its torque as positive. If another force tends to rotate it clockwise, we may write its torque as negative.
Everyday Examples
Opening a door is a clear example of torque. The handle is placed far from the hinges so that a small force can create a useful turning effect. A wrench is another example. A longer wrench allows the same force to produce greater torque on a bolt.
A seesaw also involves torque. A lighter person can balance a heavier person by sitting farther from the pivot. The heavier person creates a larger force, but the lighter person can compensate with a larger distance.
Perpendicular Force Matters
Only the component of force that is perpendicular to the line from the pivot contributes to torque. If we split the force into components, the perpendicular component is
$$
F_\perp = F\sin\theta
$$
Then torque can be written as
$$
\tau = rF_\perp
$$
This form is often easier to understand. The part of the force that points toward the pivot does not help rotate the object. Only the sideways part matters.
A useful way to think about torque is:
$$
\tau = rF_\perp
$$
Only the perpendicular component of force produces rotation.
Unit of Torque
The SI unit of torque is the newton meter, written as $\text{N} \cdot \text{m}$. This comes directly from force times distance.
Although torque and work both have units of $\text{N} \cdot \text{m}$, they are not the same physical quantity. Torque describes a turning effect, while work describes energy transfer.
Simple Comparison Table
| Situation | Angle $\theta$ | Torque magnitude |
|---|---|---|
| Force perpendicular to radius | $90^\circ$ | Maximum, $\tau = rF$ |
| Force at an angle | between $0^\circ$ and $90^\circ$ | $\tau = rF\sin\theta$ |
| Force along the radius | $0^\circ$ or $180^\circ$ | Zero |
Visualizing Torque
The diagram below shows a force applied to a bar at an angle to the radius from the pivot.
In this picture, the bar extends from the pivot to the point where the force acts. The angle between $\vec{r}$ and $\vec{F}$ is $\theta$, so the torque magnitude is $rF\sin\theta$.
When Torque is Large or Small
Torque becomes larger when the force increases. It also becomes larger when the point of application is farther from the pivot. This is why long handles and long tools are useful.
Torque becomes smaller when the force acts closer to the pivot, or when the force direction points more toward the pivot instead of across the object.
This gives a practical rule. To maximize torque, apply a large force as far as possible from the pivot, and make the force as perpendicular as possible to the line from the pivot.
To increase torque, increase $F$, increase $r$, or make $\theta$ closer to $90^\circ$.
$$
\tau = rF\sin\theta
$$
A Simple Numerical Example
Suppose a force of $10 \, \text{N}$ is applied at a distance of $0.50 \, \text{m}$ from a pivot, and the angle between the force and the radius is $90^\circ$.
Then
$$
\tau = rF\sin\theta
$$
$$
\tau = (0.50)(10)\sin 90^\circ
$$
$$
\tau = 5.0 \, \text{N} \cdot \text{m}
$$
Now suppose the same force acts at the same point, but at an angle of $30^\circ$.
$$
\tau = (0.50)(10)\sin 30^\circ
$$
$$
\tau = 2.5 \, \text{N} \cdot \text{m}
$$
So the same force can produce very different torques depending on direction.
Net Torque
If several forces act on an object, each force may produce its own torque about the chosen pivot. The total turning effect is the net torque, found by adding the torques with their signs.
If clockwise is negative and counterclockwise is positive, then
$$
\tau_{\text{net}} = \sum \tau
$$
A positive net torque means a tendency toward counterclockwise rotation. A negative net torque means a tendency toward clockwise rotation.
Detailed applications of net torque belong to rotational dynamics and equilibrium, but the central idea starts here, torque from each force adds to produce the total rotational effect.
Summary
Torque is the turning effect of a force about a chosen pivot. It depends on force, distance from the pivot, and direction. Its magnitude is
$$
\tau = rF\sin\theta
$$
and only the perpendicular part of the force causes rotation. In simple two dimensional problems, torque is often treated as positive for counterclockwise rotation and negative for clockwise rotation.
Core facts about torque:
$$
\tau = rF\sin\theta = rF_\perp
$$
Torque increases with force, distance from the pivot, and perpendicular direction of the force.
The SI unit of torque is $\text{N} \cdot \text{m}$.