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Torque & Couples (HL) (HL IB Physics)

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Katie M

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Katie M

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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

Moment of a Force Scenarios, for IB HL Physics Revision Notes

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

Couple of a Force Diagram, for IB HL Physics Revision Notes

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

tau space equals space F r space sin space theta

Torque of a non-perpendicular force

1-4-1-torque-of-a-non-perpendicular-force-1

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

tau space equals space F r

  • 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

Torque of a Couple Steering Wheel Example, for IB HL Physics Revision Notes

The steering wheel is in rotational equilibrium since the resultant force and resultant torque are zero. This means it does not have linear or 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 torque provided by a couple on a steering wheel of radius r is

tau space equals space open parentheses F space cross times space r space sin space theta close parentheses space plus thin space open parentheses F space cross times space r space sin space theta close parentheses space equals space 2 F r space sin space theta

Since theta space equals space 90 degree, then tau space equals space 2 F r

  • Therefore, the torque of a couple is equal to double the magnitude of the torque of the individual forces
  • The forces are equal and act in opposite directions
    • Therefore, couples produce a resultant force of zero
  • Due to Newton’s Second law (F = ma), the steering wheel does not accelerate
    • In other words, when the force is applied, the steering wheel rotates with a constant angular speed but remains in the same location

The effect of angle on torque

Torque and Angle Wrench Diagram, for IB HL Physics Revision Notes

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

Worked example

Which pair of forces act as a couple on the circular object?

WE - Couples question image, downloadable AS & A Level Physics revision notes

Answer:  A

  • In diagram A, the forces are:
    • Equal in size
    • In opposite directions
    • Perpendicular to the distance between them

  • B is incorrect as the forces are in the same direction
  • C is incorrect as the forces are different in size
  • D is incorrect as the distance between the forces is not perpendicular

Worked example

A ruler of length 0.3 m is pivoted at its centre.

Two equal and opposite forces of magnitude 4 N are applied to the ends of the ruler, creating a couple as shown in the diagram.

Determine the magnitude of the torque of the couple on the ruler.

Answer:

Worked example

The forces acting on a bicycle pedal at different positions during a ride are shown in the diagram below.

The distance from the pedal to the axis of rotation is 24 cm.

1-4-1-torque-worked-example-mcq1-4-1-torque-worked-example-mcq

At which position is the magnitude of the torque the greatest?

Answer:  B

  • In position A:  When θ = 180°, sin θ = 0, so τ = 0 i.e. the force exerts no torque on the pedal. When the pedal is at the bottom, no amount of pushing down will produce any torque on the pedal. 
  • In position B:  When the angle θ is 90° and the force F is 90 N, the torque τ = 0.24 × 90 × sin 90° = 21.6 N m
  • In position C:  When the angle θ is 60° and the force F is 75 N, the torque τ = 0.24 × 75 × sin 60° = 15.6 N m
  • In position D:  When the angle θ is 30° and the force F is 150 N, the torque τ = 0.24 × 150 × sin 30° = 18.0 N m

Examiner Tip

The terminology in this section can get confusing.  For example, a moment is not a 'turning force' - the turning force is only part of the moment, the moment is the effect that the turning force has on the system when applied at a distance from a turning point, or pivot.

Also, torque is another term for 'moment', but in mechanics, torque can be used to describe a couple as a specific case of a moment.

Ultimately, when you carry out calculations, make sure you can identify

  • The magnitude of the applied force
  • The perpendicular distance between the force and the turning point (along the line of action)

Make sure you understand that the forces that make up a couple cannot share the same line of action (the line through the point at which the force is applied). An example of this is shown in the diagram below:Forces with same line of action, downloadable AS & A Level Physics revision notes

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    Katie M

    Author: Katie M

    Expertise: Physics

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.