All You Need to Know About Torque and the Right-Hand Rules

Photo Credit: Public Domain

By Andrew Bennett

Torque can be a tricky concept, particularly when we are asked to think of it as a vector. Since torque is a vector, to fully describe the torque caused by a force, we have to give both the magnitude and direction of the torque.

The torque describes the ability of a force to change the rotational motion of some object or system. If you were to grab the handle of a door and pull it toward you, you could cause the entire door to rotate.  If you pulled harder (with more force), you could make the door rotate more rapidly (or rather, you would make it have a larger angular acceleration). If you tried to pull the door again, but this time pulled on the hinges, you wouldn't get any rotation. Similarly, if you grabbed the handle and used it to pull directly toward or away from the hinges (to the side, instead of toward yourself), the door's rotation wouldn't change.

How Do We Calculate Torque?

From this, we can gather that the amount of force, the location at which that force acts, and the direction of the force are all factors in determining our ability to change the rotation of the door. Mathematically, we'd say that the toque vector is the *cross product* of the position vector (that is, the location at which the force is applied, relative to the hinge or other axis of rotation) and the force vector. In shorthand, we say that torque equals r cross F.

The magnitude of the torque (or of any cross product) can be calculated as the magnitude of the first vector times the magnitude of the component of the second vector that is perpendicular to the first vector. In the door example, if you pull the handle toward you, but also to the right, you could calculate the magnitude of the torque as the distance from the hinges to the handle times the magnitude of that part of the force that acts toward you (perpendicular to the face of the door).

The direction of the torque is tricky and requires you to know some conventions. Because it would be difficult in every rotating system to say the system rotates clockwise or counterclockwise *IF* you look at it from this one particular direction, we have a shorthand for rotation directions (and therefore for torque directions). To name and calculate these directions, we have two different tricks, both of which are called right-hand rules.

So, What Are the Right-Hand Rules?

The first one helps us understand what direction we're talking about for a rotational vector. Picture that door again. Let's say the hinges are on the left, and the door opens toward you. If you were to look down from above, you'd say the door rotates counterclockwise. 

The first right-hand rule helps you find the direction of the rotational vector.

Now take your right hand and relax your fingers (so they curl a bit like you're holding a can of soda), but stick your thumb straight out (like you're giving someone a thumbs up but forgot you're holding a can of soda). Without changing that hand position (but moving the orientation of your whole hand however you need to), position your hand so that your thumb is in line with the hinges, and your fingers curl in the direction the door rotates.

The only way to do this is to flip your hand over (like you're giving someone a thumbs down, but forgot you're holding a can of soda and now it's spilling all over). Try not to injure yourself getting your fingers lined up just right.

Since your thumb points straight down, in math we'd say the direction of rotation is down for this door.  If it swung the other direction, you'd keep your thumb pointing up to get your fingers to match the direction of rotation of the door, so you'd say the direction of rotation is up. If you were looking at a clock on the wall, you'd have to point your thumb toward the clock to get your fingers curling the right way, so clock hands rotate toward the wall.

The Other Right-Hand Rule

Back to the torque vector, now. To determine the direction of the torque vector (which is also the direction of the angular acceleration it causes), you'll need to use another rule involving your right-hand rule. This one, confusingly, is also called "the right-hand rule." I usually call the first one the curly right-hand rule, and this new one will be the pointy right-hand rule.

The pointy right-hand rule helps you find the direction of the torque vector.

Point your index finger straight out, fully extended. Don't let it leave this position. Point that finger in the direction of the r vector (from the axis of rotation to the point at which the force acts). Next, through a combination of changing how far your middle finger is extended and how your hand is oriented, get your middle finger pointed in the same direction as the F vector.  Keep the index finger all the way extended, though!

Once you have your pointer in the direction of r and the middle finger in the direction of F, stick your thumb out straight. Whichever way your thumb points is the direction of the torque vector. If you keep your thumb pointing that way and switch to curly position (holding a soda can), you can see which direction of rotation that force will cause.

If you're reading this via email, click here to view the video.