Torque is the rotational equivalent of force.

To get an object to move in a straight-line, or to deflect a moving object, apply a force. Similarly, to get something to spin, or to alter the rotation of a spinning object, apply a torque.

Forces produce torques

If you understand how to open a door, you understand something about torque. To open a door, where do you push? If you exert a force at the hinge, the door will not move; the easiest way to open a door is to exert a force as far from the hinge as possible, and to push or pull with a force perpendicular to the door. This maximizes the torque you exert.

The magnitude of the torque depends on the force, the direction of the force, and where the force is applied.

τ = r × F

The magnitude of the resultant vector is r F sin(θ). r is measured from the axis of rotation to the line of the force, and θ is the angle between r and F.

The direction of the torque is given by the right-hand rule. Using your right hand, point your fingers in the direction of the first vector (r) and curl them into the direction of the second vector (F). Your thumb, sticking out, will point in the direction of the torque.

The torque is ...

• zero when r and F are parallel or in opposite directions
• maximum when r and F are perpendicular to one another

When multiplying vectors via the cross product, the result is a vector perpendicular to the two vectors in the cross product.

Note also that a × b = - b × a.

If c = a × b then the magnitude of c = a b sin(θ)

The direction can be found from the right-hand rule.

c = a × b = (a_y b_z - b_y a _z ) + (a_z b_x - b_z a_x ) + (a_x b_y - b_x a_y )

Consider the torque from one force exerted on a hinged rod, as shown in the diagram.

The torque is counterclockwise, although the rod does not rotate because of the balancing clockwise torque coming from the force of gravity. For now, focus on the counterclockwise torque.

There are three equivalent ways to determine the magnitude of the torque about a rotation axis passing through the hinge:

Method 1 - Measure r from the hinge along the rod to where the force is applied, multiply by the force, and then multiply by the sine of the angle between the rod (the line you measure r along) and the force.

τ = r F sin(θ)

Method 2 - Split the force into components perpendicular to and parallel to the rod. The parallel component produces no torque. The perpendicular component, Fsin(θ), results in a torque with a magnitude of plenty of torque:

τ = r [Fsin(θ)] sin(90) = r F sin(θ)

Method 3 - Extend the line of the force and measure the distance from the rotation axis to the line of the force along a line that is perpendicular to the line of the force. The distance measured along this line is often called the lever arm - we'll use r' for this perpendicular distance. The magnitude of the torque is:

τ = r' F sin(90) = r' F

Using geometry, r' = r sin(θ), so the torque (once again) has a magnitude of:

τ = r F sin(θ)