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3.4.2Forces and Motion

Couples and Torque

5 min read 8 parts Lesson 19 of 66 1 quiz

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Lesson summary

3.4.2 Couples and Torque — OCR A Level Physics A notes: Define a couple and its torque, and calculate the moment of a force and the torque of a couple.A force can turn an object about a pivot.

  • perpendicular distance
  • Worked example 1
  • Worked example 2
  • Worked example 3
  • Line of action
  • Using the total of both forces in the torque of a couple. Use the size of one force times the separation

Section 1 of 8Moment of a force

A force can turn an object about a pivot.

Moment
The turning effect is the moment of the force: moment = F × d,
What the symbols mean: moment (N m) · F = force (N) · d = perpendicular distance from the pivot to the line of the force (m)
  • where F is the force in N and d is the perpendicular distance from the pivot to the line of action of the force, in m.
The unit is N m.
Moments are clockwise or anticlockwise; choose one as positive and keep to it.
If the force acts at an angle θ to the line joining the pivot and the point of application, the perpendicular distance is r sin θ,
  • so moment = Fr sin θ.

A longer spanner makes a nut easier to turn because, for the same moment, a larger distance needs a smaller force.

Section 2 of 8Couples

A couple is a pair of forces that are equal in size, opposite in direction and whose lines of action do not coincide (they are parallel and separated by a distance).

Couples
Because the two forces are equal and opposite, the resultant force is zero,
  • so a couple produces rotation only and no translation of the centre of mass.
Examples include turning a steering wheel with both hands, twisting a tap, and the pair of forces on a compass needle in a magnetic field.

The torque of a couple is the moment produced by the couple: torque = F × d, where F is the size of one of the forces and d is the perpendicular distance between the lines of action of the two forces.

Couples
Torque has the unit N m.
A key result is that the torque is the same about any point, so you do not need to specify a pivot.
To see this, let the forces be F up at x and F down at x + d relative to any point: the moments about the point add to Fd regardless of where it lies.

Section 3 of 8Torque and rotation

The word torque is used for the turning effect of a couple or of a force about an axis, such as a motor shaft.

Torque and rotation
If a wheel of radius r is turned by two opposing tangential forces F at opposite ends of a diameter, the separation is d = 2r and the torque is 2Fr.
In a motor, the greater the torque, the greater the rotational acceleration of a given load.
Power transmitted by a rotating shaft equals torque × angular speed, P = Tω, although this is beyond the OCR core.

Do not confuse torque of a couple with the moment of a single force: a single force about a pivot also produces a net force on the pivot, which means the body might be pushed as well as turned.

Section 4 of 8Using couples in problems

If a body is acted on by two parallel equal forces in opposite directions and it is in equilibrium, then another couple of equal and opposite torque must act.

Using couples in problems
For instance a uniform rod held by an upward force at one end,
  • and weight at its centre, is not a couple because the forces are unequal in position and size.
The method for problems: identify the two forces, find their separation perpendicular to the forces, multiply.

For a ladder, a plank, or a door, use the moment of each force about a convenient pivot; for any pure couple, torque is simply F × d with no pivot choice.

Section 5 of 8Applications and links

Couples are everywherea tap, a screwdriver, a bottle top, the two hands on a wheel, and electric motors, where the magnetic field exerts equal and opposite forces on the two sides of a coil, giving a torque that turns the coil.

Applications and links
In each case the resultant force is zero, so the centre of mass does not accelerate, but the object rotates.
The idea links to the principle of moments in 3.4.4: a body with a couple acting on it is not in equilibrium unless a second couple of equal and opposite torque also acts.
When solving problems, keep the units consistent: the distance must be in metres to obtain N m.

Section 6 of 8Worked examples

  1. A force of 25 N is applied at the end of a 0.30 m spanner at 90° to the spanner. Find the moment about the nut. What force is needed with a spanner 0.45 m long for the same moment?

  2. A steering wheel of diameter 0.36 m is turned by two hands each exerting 12 N tangentially in opposite directions at opposite ends of a diameter. Find the torque.

  3. A force of 40 N acts on a lever arm of length 0.50 m at 30° to the arm. Find the moment about the pivot.

Section 7 of 8Key terms

TermMeaning
MomentForce × perpendicular distance from the pivot, in N m.
CoupleTwo equal, opposite, parallel forces with different lines of action.
Torque of a coupleOne force × perpendicular separation of the two forces.
PivotPoint about which a body turns.
Line of actionThe straight line along which a force acts.
Perpendicular distanceShortest distance from a point to the line of action.

Section 8 of 8Exam technique

1
Using the total of both forces in the torque of a couple. Use the size of one force times the separation.
2
Measuring the distance along the force line to the pivot instead of the perpendicular distance.
3
Thinking a couple produces both rotation and a resultant force. The resultant force is zero.
4
Giving torque in N or J. The unit is N m (not a joule here).
5
Saying the torque of a couple depends on the pivot chosen. It is the same about any point.
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