Torque & Couples (DP IB Physics) : Revision Note

Torque & Couples

Moment of a Force

• A moment is the turning effect of a force around a particular point

• Moments occur when forces cause objects to rotate about some pivot

• The moment of a force is given by

Moment (N m) = Force (N) × perpendicular distance from the pivot (m)

• The SI unit for the moment is newton metres (N m)

Moments of perpendicular & non-perpendicular forces

The force might not always be perpendicular to the distance. In this case, use trigonometry to resolve in the perpendicular direction

Couples

• A couple is a pair of equal and opposite coplanar forces that act to produce rotation only

• A couple consists of a pair of forces that are:

  • Equal in magnitude

  • Opposite in direction

  • Perpendicular to the distance between them

Force couple diagram

A couple must consist of two equal and opposite forces separated by a perpendicular distance

• Unlike moments of a single force, the moment of a couple doesn’t depend on a pivot

• The moment of a couple is equal to:

  Moment (N m) = Force (N) × Perpendicular distance between the lines of action of the forces (m)

• A couple does not produce a net linear force

  • However, it does produce a turning effect called a torque

Torque

• The change in rotational motion due to a turning force is called torque

• The torque of a force F about an axis is given by

Torque of a non-perpendicular force

The torque applied by a cyclist on a bicycle pedal can be determined using the magnitude of the applied force and the perpendicular component of the distance between the line of action of the force and the axis of rotation

• For scenarios where the forces are perpendicular (θ = 90°) to one another, the equation simplifies to

• Where:

  • τ = torque (N m)

  • F = applied force (N)

  • r = perpendicular distance between the axis of rotation and the line of action of the force (m)

  • θ = angle between the force and the axis of rotation (°)

Torque of a couple on a steering wheel

The steering wheel is acted on by a couple; it has zero resultant force but a net torque. This causes angular acceleration.

• When applied to a couple, torque can be described as:

The sum of the moments produced by each of the forces in the couple

• For example, the net torque provided by a couple on a steering wheel of radius is:

Since , then

• Therefore, the net torque of a couple is equal to double the magnitude of the torque of the individual forces

• The forces in a couple are equal in magnitude and act in opposite directions, which means:

  • the resultant force is zero

  • the wheel does not have a linear acceleration

  • the wheel is in translational equilibrium

• However, since the couple produces a net torque, the wheel experiences an angular acceleration

  • In other words, when the force is applied, the steering wheel rotates and remains in the same location, but its angular speed increases

The effect of angle on torque

The force produces a maximum torque when applied perpendicular to the wrench. The same force is less effective when applied at non-perpendicular angles