Torque

### Torque is a twisting force

Twisting is a unique kind of force. Rather than result in linear acceleration of an object, torque (pronounced "tork") results in rotational motion of that object about one or more axes that pass through its center of mass. One thing you should keep in mind is that any object rotating freely, like a diver doing somersaults, rotates around a line passing through its center of mass.

In this section we'll develop the idea of torque conceptually and mathematically – as we do for any physical quantity.

You are already familiar with the idea of torque if you've ever used a screwdriver or a wrench to turn a screw or bolt, or if you've used a lever to move something. There are many more examples.

The symbol for torque is usually the Greek letter tau, τ .

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#### The Greek alphabet

 alpha Α α beta Β β gamma Γ γ delta Δ δ epsilon Ε ε zeta Ζ ζ eta Η η theta Θ θ iota Ι ι kappa Κ κ lambda Λ λ mu Μ μ nu Ν ν xi Ξ ξ omicron Ο ο pi Π π rho Ρ ρ sigma Σ σ tau Τ τ upsilon Υ υ phi Φ φ chi Χ χ psi Ψ ψ omega Ω ω

### Torque: using a wrench

A good, and probably familiar example of applying a torque force is using a wrench to turn a bolt. The wrench can help us to amplify the twisting force on the bolt.

You might have some experience tightening bolts. You know that they can often be turned just so tight by hand, without a wrench, but soon the wrench is needed. When using a wrench, we also know that we can apply more twisting force by pushing on the far end of the wrench as opposed to the end nearest the bolt.

#### The coordinate system

Here is the coordinate system we use when we work with torque. The distance between the applied force, F, and the pivot point is r. The applied force may be applied perpendicular to the wrench or lever arm, or at some angle. If it is not perpendicular, then we need to find the perpendicular vector component of that force on the wrench, F = F·cos(θ).

### What makes a difference

#### 1. Changing the force

The most obvious way to change torque is to change the magnitude of the applied force. All other variables (r, θ) being constant, if F2 > F1, then τ2 > τ1.

If you want more torque, push harder on the wrench.

#### 2. Changing the distance from the pivot to the force (r)

Another way to change the torque is to move the applied force closer to or further from the pivot point.

A wrench is simply a class-2 lever, so the closer we are to the pivot or fulcrum, the less the resulting twisting force – torque. In the figure, if r1 > r2, then τ1 > τ2, even with the same applied force, F.

Sometimes, when a bolt is particularly stuck, a mechanic might add a "cheater bar," a piece of pipe that slips over the wrench handle to extend it, and thus gain more torque as r is increased by the length of the pipe minus the overlap. Be careful, though. That's a good way to break a bolt or a wrench!

#### 3. Changing the angle of the applied force

The only component of the applied force that is relevant to the torque is that part that is perpendicular to the lever (wrench in this case), F.

The two vector components of F are

\begin{align} F_{\perp} = F \cdot sin(\theta) \\[5pt] F_{\parallel} = F \cdot cos(\theta) \end{align}

We don't care too much about F, the component of the applied force parallel to the wrench because it can't really affect the twisting force.

#### Definition of torque

Putting all of these observations together, we can derive an expression for the torque:

#### $$\tau = F \, r \, sin(\theta)$$

Everything we need is there: The torque increases as F and r increase, and the torque is at a maximum when θ = 90˚.

#### Torque

Torque (τ) is twisting force or a force that causes rotation about one or more axes passing through the center of mass of a free object (or about some other axis if an object is so attached).

#### $$\tau = F \, r \, sin(\theta)$$

where F is the applied force, applied at an angle θ to the lever arm and at a distance r from the pivot axis.

The SI units of torque are Newton·meters (N·m). In the U.S. torque is often reported in foot-pounds (ft.-lbs.)

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### SI units

SI stands for Système international (of units). In 1960, the SI system of units was published as a guide to the preferred units to use for a variety of quantities. Here are some common SI units

 length meter (m) mass Kilogram (Kg) time second (s) force Newton N energy Joule J

### A little deeper into the math

Strictly-speaking, torque is a vector product or cross product of two vectors, the radius, r, and the applied force, F :

#### $$\tau = r \times F$$

where all components are vectors. The magnitude or size of the torque vector (the 'amount' of torque) is

#### $$\tau = |r| \cdot |F| \, cos(\theta)$$

The cross product yields a new vector of the magnitude given above, and that is perpendicular both to vectors r and F. The torque vector, therefore, lies along the axis of rotation of the object to which the torque is being applied.

We use the right-hand rule to decide which of the two possible directions it points:

If the fingers of the right hand curl in the direction of the rotation, then the thumb will point along the direction of the torque vector. We do this by long-standing convention – we have to pick one.

### Example 1

The seat post bolt on a bicycle requires a torque of 35 N·m in order to remain tight. If the mechanic uses a wrench that is 30 cm long, how much force must she apply to it in order to achieve the proper torque?

Solution: The torque equation is

$$\tau = F\cdot r \, sin(\theta)$$

We can rearrange that to find the force (always rearrange first before plugging in numbers!):

$$F = \frac{\tau}{r \, sin(\theta)}$$

If we assume that the force on the torque wrench will be applied at a right angle to it, then sin(90˚) = 1, so we can ignore the sin(θ) term.

Plugging in the torque and radius, we get:

$$\require{cancel} F = \frac{35 \, N\cancel{m}}{0.3 \cancel{m}} = 117 \; N$$

Now it's worth considering a couple of things. First, notice from the torque equation that if we were to double the length of the wrench handle, we'd only need half of that 117 N of force. Second, applying the force at any angle other than 90˚ would increase the applied force needed, as the sine of the angle diminishes on either side of 90˚.

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