Physics A Level
Chapter 4: Forces 4.5 The torque of a couple
Physics A Level
Chapter 4: Forces 4.5 The torque of a couple
Figure 4.25 shows the forces needed to turn a car’s steering wheel. The two forces balance up and down ($15 N$ up and $15 N$ down), so the wheel will not move up, down or sideways. However, the wheel is not in equilibrium. The pair of forces will cause it to rotate.
A pair of forces like that in Figure 4.25 is known as a couple.
A couple has a turning effect, but does not cause an object to accelerate. To form a couple, the two forces must be:
- equal in magnitude
- parallel, but opposite in direction
- separated by a distance d.
The turning effect or moment of a couple is known as its torque.
We can calculate the torque of the couple in Figure 4.25 by adding the moments of each force about the centre of the wheel:
$\begin{array}{l}
torque\,of\,couple\, = \,(15 \times 0.20) + (15 \times 0.20)\\
= 6.0\,N\,m
\end{array}$
We could have found the same result by multiplying one of the forces by the perpendicular distance
between them:
$torque\,of\,a\,couple = 15 \times 0.4 = 6.0\,N\,m$
The torque of a couple is defined as follows:
torque of a couple = one of the forces $ \times $ perpendicular distance between the forces
Question
10) The driving wheel of a car travelling at a constant velocity has a torque of $137 N m$ applied to it by the axle that drives the car (Figure 4.26). The radius of the tyre is $0.18 m$. Calculate the driving force provided by this wheel.
Pure turning effect
When we calculate the moment of a single force, the result depends on the point or pivot about which the moment acts. The further the force is from the pivot, the greater the moment. A couple is different; the moment of a couple does not depend on the point about which it acts, only on the perpendicular distance between the two forces. A single force acting on an object will tend to make the object accelerate (unless there is another force to balance it). A couple, however, is a pair of equal and opposite forces, so it will not make the object accelerate. This means we can think of a couple as a pure ‘turning effect’, the size of which is given by its torque.
For an object to be in equilibrium, two conditions must be met at the same time:
- The resultant force acting on the object is zero.
- The resultant moment is zero.
EXAM-STYLE QUESTIONS
1) A force F is applied at a distance d from the hinge H and an angle x to the door.
What is the moment of the force F about the point H? [1]
A: $Fd\,\cos \,x$
B: $\frac{{Fd\,}}{{\cos \,x}}$
C: $Fd\,\sin \,x$
D: $\frac{{Fd\,}}{{\sin \,x}}$
2) The angle between two forces, each of magnitude $5.0 N$, is ${120^ \circ }$.
What is the magnitude of the resultant of these two forces? [1]
A: $1.7 N$
B: $5.0 N$
C: $8.5 N$
D: $10 N$
3) A ship is pulled at a constant speed by two small boats, A and B, as shown. The engine of the ship does not produce any force.
The tension in each cable between A and B and the ship is $4000 N$.
a: Draw a free-body diagram showing the three horizontal forces acting on the ship. [2]
b: Draw a vector diagram to scale showing these three forces and use your diagram to find the value of the drag force on the ship. [2]
[Total: 4]
4) A block of mass $1.5 kg$ is at rest on a rough surface which is inclined at ${20^ \circ }$ to the horizontal as shown.
a: Draw a free-body diagram showing the three forces acting on the block. [2]
b: Calculate the component of the weight that acts down the slope. [2]
c: Use your answer to part b to determine the force of friction that acts on the block. [2]
d: If the angle of the surface is actually measured as ${19^ \circ }$ and ${21^ \circ }$ determine the absolute uncertainty in this angle and the uncertainty this produces in the value for part b. [3]
e: Determine the normal contact force between the block and the surface. [3]
[Total: 12]
5) This free-body diagram shows three forces that act on a stone hanging at rest from two strings.
a: Calculate the horizontal component of the tension in each string. State why these two components are equal in magnitude? [5]
b: Calculate the vertical component of the tension in each string. [4]
c: Use your answer to part b to calculate the weight of the stone. [2]
d: Draw a vector diagram of the forces on the stone. This should be a triangle of forces. [1]
e: Use your diagram in part d to calculate the weight of the stone. [2]
[Total: 14]
6) The force F shown here has a moment of $40 N m$ about the pivot. Calculate the magnitude of the force F. [4]
7) The asymmetric bar shown has a weight of $7.6 N$ and a centre of gravity that is $0.040 m$ from the wider end, on which there is a load of $3.3 N$. It is pivoted a distance of $0.060 m$ from its centre of gravity. Calculate the force P that is needed at the far end of the bar in order to maintain equilibrium. [4]
8) a: State what is meant by:
i- a couple [1]
ii- torque. [2]
b: The engine of a car produces a torque of $200 N m$ on the axle of the wheel in contact with the road. The car travels at a constant velocity towards the right:
Copy the diagram of the wheel and show the direction of rotation of the wheel, and the horizontal component of the force that the road exerts
i- on the wheel. [2]
ii- State the resultant torque on the wheel. Explain your answer. [2]
iii- The diameter of the car wheel is $0.58 m$. Determine the value of the horizontal component of the force of the road on the wheel. [1]
[Total: 8]
9) a: Explain what is meant by the centre of gravity of an object. [2]
b: A flagpole of mass $25 kg$ is held in a horizontal position by a cable as shown. The centre of gravity of the flagpole is at a distance of $1.5 m$ from the fixed end.
i- Write an equation to represent taking moments about the left-hand end of the flagpole. Use your equation to find the tension T in the cable. [4]
ii- Determine the vertical component of the force at the left-hand end of the flagpole. [2]
[Total: 8]
10) a: State the two conditions necessary for an object to be in equilibrium. [2]
b: A metal rod of length $90 cm$ has a disc of radius $24 cm$ fixed rigidly at its centre, as shown. The assembly is pivoted at its centre.
Two forces, each of magnitude $30 N$, are applied normal to the rod at each end so as to produce a turning effect on the rod. A rope is attached to the edge of the disc to prevent rotation.
Calculate:
i- the torque of the couple produced by the $30 N$ forces [1]
ii- the tension T in the rope. [3]
[Total: 6]
11) a: State what is meant by the torque of a couple. [2]
b: Three strings, A, B and C, are attached to a circular ring, as shown in Figure 4.35.
The strings and the ring all lie on a smooth horizontal surface and are at rest. The tension in string A is $8.0 N$. Calculate the tension in strings B and C. [4]
[Total: 6]
12) This diagram shows a picture hanging symmetrically by two cords from a nail fixed to a wall. The picture is in equilibrium.
a: Explain what is meant by equilibrium. [2]
b: Draw a vector diagram to represent the three forces acting on the picture in the vertical plane. Label each force clearly with its name and show the direction of each force with an arrow. [2]
c: The tension in the cord is $45 N$ and the angle that each end of the cord makes with the horizontal is ${50^ \circ }$. Calculate:
i- the vertical component of the tension in the cord [1]
ii- the weight of the picture. [1]
[Total: 6]
SELF-EVALUATION CHECKLIST
After studying the chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| add forces using a vector triangle | 4.1 | |||
| resolve forces into perpendicular components | 4.2 | |||
| represent the weight of a body at a single point known as its centre of gravity | 4.3 | |||
| define and apply the moment of a force and the torque of a couple | 4.4, 4.5 | |||
| state and apply the principle of moment | 4.4 | |||
| use the idea that, when there is no resultant force and no resultant torque, a system is in equilibrium. | 4.5 |