# Torque and Levers

Torque is the equivalent of force in rotational motion. Although we are all more familiar with the idea of forces (like gravity), it is far more natural to think about torques acting on parts of robots because almost all of their motion is rotational. Motors exert torques on things like wheels and arms, and things like gravity and friction exert torques against the motors.

## Contents

## Physics Formalism

A **torque τ** is defined as the vector product of the distance

*r*from a pivot point (around which the object rotates) to where a force acts, and the force

*F*.

*τ=r × F*

It can be thought of more simply as the product of a force with the distance from the pivot point, times the sine of the angle a between the two.

*τ = F * r * sin(a)*

## Intuition: Why are torques different from forces

The force of gravity is always the same on earth for a pencil. But the torque is not.

Think about a pencil that you try to balance upright on your hand, with the eraser touching your palm. It is really hard to get the pencil to balance perfectly, but you can sometimes make the pencil stay put for a few seconds before it slowly starts to fall to the side.

When the pencil falls, it is rotating, which means there must be a torque caused by the force of gravity. But since gravity is in almost exactly the same direction as the eraser (the eraser is almost exactly beneath the center of mass of the pencil), the angle *a* is very small.

When *a* is very small, sin(a) is very close to zero, which means the torque on the object is close to zero. This is why the pencil falls slowly.

Now imagine putting the eraser on your finger, and putting the pencil horizontal hanging out away from your finger. The pencil will fall very quickly! This is because when the force of gravity is at a right angle to the distance from the center of the pencil to the point around which it rotates, *a* is close to 90 degrees. This means sin(a) is close to 1, and the torque is at its greatest value.

So the changing torque of an object has real effects, even though the force of gravity is always the same.

## Using levers: How torques affect a robot

The pencil from the section above is an example of a lever. Keeping the same experiment in place, let's replace the pencil with the arm of a robot. The greater the torque acting by gravity, the greater the torque a motor has to exert to keep the arm steady.

Torques and levers can also be thought of as an extension of Gear Ratios. The radius of a lever can be thought of as the radius of a gear, meaning that:

r_{1}/r_{2}= τ_{2}/τ_{1}, where r is the length of the arm.

So, a motor driving a longer arm will have to put out more torque than a motor driving a short arm.