OCR Specification focus:
‘Define a couple; calculate torque using torque = Fd for equal and opposite forces.’
A couple is a pair of equal and opposite forces acting on a body that produce rotation without translation. Understanding torque and couples is key to analysing rotational equilibrium.
The Concept of a Couple
A couple occurs when two equal and opposite forces act on an object but not along the same line of action. This arrangement produces a turning effect, known as a torque, which causes rotation without linear motion. Couples are central to the study of rotational systems and are commonly seen in devices like steering wheels, door handles, and screwdrivers.
A couple does not result in the object accelerating linearly because the resultant force is zero, but it produces angular acceleration about an axis.
Characteristics of a Couple
The two forces are equal in magnitude.
They are opposite in direction.
Their lines of action are separated by a perpendicular distance.
They create rotation only, not translation.
Defining Torque of a Couple
Torque of a Couple: The turning effect produced by a couple, equal to one of the forces multiplied by the perpendicular distance between their lines of action.
The torque of a couple is independent of the point about which it is measured because both forces create the same rotational effect in the same direction.
EQUATION
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Torque (τ) = F × d
τ = torque (N m)
F = magnitude of one of the forces (N)
d = perpendicular distance between lines of action of the forces (m)
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The torque (symbol τ) is measured in newton-metres (N m) and represents the moment of the couple. It quantifies how strongly the couple can rotate an object.
After applying this equation, it is important to note that the direction of the torque determines whether rotation is clockwise or anticlockwise. By convention, anticlockwise torques are taken as positive.
The torque of a couple (also called the couple moment) is given by M=Fd⊥M = F d_\perpM=Fd⊥, where FFF is the magnitude of either force and d⊥d_\perpd⊥ is the perpendicular separation.
A pair of equal and opposite forces forms a couple whose moment is M = Fd. The diagram labels the forces and the perpendicular separation d, making the dependence explicit. This image focuses solely on the geometry of a couple without extraneous detail. Source
Visualising and Analysing Couples
A couple can be visualised as two forces acting on opposite sides of a pivoted object. For example, when turning a spanner, one hand pushes upward while the other pushes downward. The spanner experiences a rotational effect around the nut because of the torque produced by this couple.
Rotational Equilibrium and Couples
When the sum of torques (moments of couples) acting on a body equals zero, the body is in rotational equilibrium. This means it does not start spinning faster or slower. However, it might still be rotating at a constant angular speed if torques are balanced.
Key ideas:
A single couple always produces a pure rotation.
Two couples in opposite directions may cancel each other if their torques are equal in magnitude.
Net torque = 0 implies no change in angular motion.
Moment and Torque: The Relationship
Although moment and torque both represent rotational effects, the moment of a single force is the turning effect about a point, whereas torque specifically refers to a pair of forces (a couple).
A moment is caused by one force acting at a distance, while a torque requires two forces. The two terms are sometimes used interchangeably, but for clarity, OCR expects “torque” when describing the effect of a couple.
EQUATION
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Moment (M) = F × perpendicular distance (d)
M = moment about a point (N m)
F = force applied (N)
d = perpendicular distance from pivot to line of action of force (m)
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In the special case of a couple, the moment of each force about any point is equal and adds to produce the same total torque. This makes torque a vector quantity, since it has both magnitude and direction (clockwise or anticlockwise).
Because a couple’s moment depends only on F and the perpendicular separation between the lines of action, it is independent of the point about which moments are taken (a free vector).
Two equal and opposite forces separated by distance d act on a rigid bar. The calculated moment M = Fd is unchanged by shifting the reference point (parameter x)—a defining feature of couples. The layout is intentionally minimal to highlight the variables. Source
Applications of Couples in Physics and Engineering
Everyday Applications
Turning a steering wheel: Equal and opposite forces from each hand form a couple, rotating the wheel.
Opening a jar lid: Fingers and thumb apply a couple to twist the lid without moving it linearly.
Using a wrench: One end is pushed up while the other is pulled down, generating torque to loosen or tighten bolts.
Common everyday exemplars include turning a steering wheel or loosening a wheel nut with a cross-shaped lug wrench, where the hands apply equal and opposite forces.
A cross lug wrench used to loosen wheel nuts: applying forces at opposite arms creates a couple that produces a pure turning effect on the nut. This image adds real-world context; note that it does not display labels or arrows, unlike the diagrams. Source
Engineering and Mechanics
Balancing beams and levers: Engineers calculate torque to ensure structures do not rotate under load.
Propeller systems: Engines apply couples to generate controlled rotational motion.
Gyroscopes: The stability of spinning systems depends on torques acting as couples to change angular momentum.
These examples demonstrate how couples are used to initiate, control, or counteract rotation in practical systems.
Mathematical Treatment of Couples
To analyse motion in physics problems, the vector nature of torque must be considered. The torque vector is perpendicular to the plane containing the forces and the distance between them. The direction is given by the right-hand rule: curl the fingers in the direction of rotation, and the thumb points in the direction of the torque vector.
Important Properties of Torque
Independent of reference point: The total torque from a couple remains constant regardless of where it is measured.
Constant magnitude: Changing the line of action of the forces without altering their separation does not change torque.
Additivity: Multiple torques acting on a body can be summed algebraically (taking clockwise and anticlockwise signs into account).
These properties make torque a convenient quantity for analysing rotational dynamics and equilibrium conditions.
Distinguishing Between Force and Torque Effects
A force causes a linear acceleration according to Newton’s Second Law (F = ma), while a torque causes an angular acceleration according to τ = Iα, where I is the moment of inertia and α is angular acceleration. Though this rotational analogue lies beyond the current subsubtopic, understanding the conceptual link clarifies the unique role of torque in rotational motion.
Torque thus bridges forces and rotational motion, describing how balanced or unbalanced couples determine whether an object remains in rotational equilibrium or begins to spin.
FAQ
A single force causes a moment about a specific point, producing rotation only if the force acts at a distance from that point. The effect depends on where the force is applied.
A couple, however, involves two equal and opposite forces whose lines of action do not meet. It produces a pure rotation that is independent of any chosen pivot. In other words, a couple’s torque is a free vector, whereas a single force’s moment depends on the reference point.
Each of the two forces in a couple produces a moment about any point. These two moments are equal in magnitude and act in the same rotational sense, so they add to give the same total torque.
Mathematically, shifting the point of reference changes the perpendicular distance for each force equally but in opposite ways, cancelling out any change. This is why torque from a couple is independent of the pivot position.
No. A pure couple always has zero resultant force, because the two forces are equal and opposite. This means they cancel translational effects and leave only rotation.
However, if the forces are not perfectly equal or not exactly parallel, there will be a net force as well as torque, leading to both rotation and translation. This is then not a pure couple but a combination of rotational and linear motion.
A simple way is to use a metre rule or lightweight bar balanced on a pivot:
Apply equal forces at both ends in opposite directions and perpendicular to the bar.
Observe the rotation without sideways movement, showing a pure turning effect.
Repeating with different distances between forces shows that greater separation increases torque.
This practical illustrates that the turning effect depends on both force and perpendicular distance, confirming the relationship torque = F × d.
Reversing one of the forces changes the direction of rotation. The couple still exists, but the torque now acts in the opposite rotational sense.
For example:
If the original couple caused clockwise rotation, reversing one force will produce anticlockwise rotation.
The magnitude of the torque remains unchanged (since F and d are constant), but its direction vector reverses according to the right-hand rule.
Practice Questions
Question 1 (2 marks)
Define a couple and state one condition required for a pair of forces to form a couple.
Mark Scheme:
1 mark for correctly defining a couple as two equal and opposite forces whose lines of action do not coincide, producing rotation without translation.
1 mark for stating a valid condition such as forces must be equal in magnitude and opposite in direction but act along parallel lines separated by a perpendicular distance.
Question 2 (5 marks)
A uniform metal bar of length 0.80 m is pivoted at its centre. Two equal and opposite forces of 12 N are applied at opposite ends of the bar, each perpendicular to its length.
(a) Calculate the torque of the couple acting on the bar. (2 marks)
(b) Explain what effect this couple has on the bar and why the torque is independent of the position of the pivot. (3 marks)
Mark Scheme:
(a)
1 mark for using correct equation: torque = F × d.
1 mark for correct substitution and answer: torque = 12 × 0.80 = 9.6 N m.
(b)
1 mark for stating that the couple causes the bar to rotate about its centre (produces a turning effect).
1 mark for explaining that no linear acceleration occurs because the resultant force is zero.
1 mark for stating that torque is independent of the pivot because both forces produce equal and opposite moments about any point, giving the same total rotational effect.
