12.5 Torque

Consider the common experience of pushing open a door. FIGURE 12.17 is a top view of a door hinged on the left. Four pushing forces are shown, all of equal strength. Which of these will be most effective at opening the door?

The figure illustrates the top view of a door with a hinge on the left side. The top view of the door is a horizontal rectangle.

FIGURE 12.17 Full Alternative Text

Force $\vec{F}_{1}$ will open the door, but force $\vec{F}_{2}$ , which pushes straight at the hinge, will not. Force $\vec{F}_{3}$ will open the door, but not as easily as $\vec{F}_{1}$ . What about $\vec{F}_{4} ?$ It is perpendicular to the door, it has the same magnitude as $\vec{F}_{1}$ , but you know from experience that pushing close to the hinge is not as effective as pushing at the outer edge of the door.

The ability of a force to cause a rotation depends on three factors:

The magnitude F of the force.
The distance r from the point of application to the pivot.
The angle at which the force is applied.

We can incorporate these three factors into a single quantity called the torque. FIGURE 12.18 shows a force $\vec{F}$ trying to rotate the wrench and nut about a pivot point—the axis about which the nut will rotate. We say that this force exerts a torque $τ$ (Greek tau), which we define as

τ \equiv r F sin ϕ

(12.20)

Torque depends on the three properties we just listed: the magnitude of the force, its distance from the pivot, and its angle. Loosely speaking, $τ$ measures the “effectiveness” of the force at causing an object to rotate about a pivot. Torque is the rotational equivalent of force.

The figure illustrates a wrench tightening a bolt. The bolt is at the origin of an x y coordinate plane.

FIGURE 12.18 Full Alternative Text

NOTE Angle $ϕ$ is measured counterclockwise from the dashed line that extends outward along the radial line. This is consistent with our sign convention for the angular position $θ$ .

The SI units of torque are newton-meters, abbreviated $N m$ . Although we defined $1 N m = 1 J$ during our study of energy, torque is not an energy-related quantity and so we do not use joules as a measure of torque.

Torque, like force, has a sign. A torque that tries to rotate the object in a ccw direction is positive while a negative torque gives a cw rotation. FIGURE 12.19 summarizes the signs. Notice that a force pushing straight toward the pivot or pulling straight out from the pivot exerts no torque.

The figure illustrates the Signs and strengths of the torque using a wrench tightening a bolt.

FIGURE 12.19 Full Alternative Text

NOTE Torque differs from force in a very important way. Torque is calculated or measured about a pivot point. To say that a torque is $20 N m$ is meaningless. You need to say that the torque is $20 N m$ about a particular point. Torque can be calculated about any pivot point, but its value depends on the point chosen because this choice determines r and $ϕ$ .

A photo of a cyclist peddling. The photo is zoomed in on the crank arm of the cycle. — Your foot exerts a torque that rotates the crank.

Returning to the door of Figure 12.17, you can see that $\vec{F}_{1}$ is most effective at opening the door because $\vec{F}_{1}$ exerts the largest torque about the pivot point. $\vec{F}_{3}$ has equal magnitude, but it is applied at an angle less than $90 °$ and thus exerts less torque. $\vec{F}_{2}$ , pushing straight at the hinge with $ϕ = 180 °$ , exerts no torque at all. And $\vec{F}_{4}$ , with a smaller value for r, exerts less torque than $\vec{F}_{1}$ .

12.5 Torque

FIGURE 12.17 The four forces have different effects on the swinging door.

FIGURE 12.18 Force $\vec{F}$ exerts a torque about the pivot point.

FIGURE 12.19 Signs and strengths of the torque.