Equilibrium of Forces – SS2 Physics Lesson Note (First Term)

A body is in equilibrium when the forces acting on it do not make it accelerate or rotate. Two conditions must be met: the upward forces must equal the downward forces, and the clockwise moments about a point must equal the anticlockwise moments about the same point.

Behavioural objectives

By the end of the lesson, students should be able to:

• state the two conditions for equilibrium;
• define the moment of a force and give its unit;
• define a couple, its moment and its arm, and give examples of its use;
• use moments to find the reaction forces on a beam resting on two pivots;
• define centre of gravity and centre of mass;
• describe stable, unstable and neutral equilibrium with examples.

What are the conditions for equilibrium?

A body is said to be in equilibrium if, under the action of several forces, it does not accelerate or rotate.

1. The sum of the upward forces must be equal to the sum of the downward forces.
2. The sum of the clockwise moments about a point must be equal to the sum of the anticlockwise moments about the same point.

For the forces in the diagram:

F₁ + F₂ = F₃ + F₄

(F₁ + F₂) – (F₃ + F₄) = 0

Clockwise moment = F₂X₂ + F₄X₄

Anticlockwise moment = F₁X₁ + F₃X₃

(F₁X₁ + F₃X₃) – (F₂X₂ + F₄X₄) = 0

The sum of the clockwise moments = the sum of the anticlockwise moments.

What is the moment of a force?

The moment of a force is the product of the force and the perpendicular distance.

M = F × distance

Unit = Nm

What is a couple?

A couple is a system of two parallel, equal and opposite forces whose lines of action do not coincide.

The moment of a couple is the product of one of the forces and the perpendicular distance between the lines of action of the two forces.

M = f × 2r

M = f × d

The distance between the two equal forces is called the arm of the couple. The moment of a couple is also called a torque.

Application of the effect of couples

1. It is easier to turn a tap on or off by applying a couple.
2. It is easier to turn the steering wheel of a vehicle by applying a couple with our two hands instead of a single force with one arm.

Worked example: reaction forces on a beam

A light beam AB sits on two pivots C and D. A load of 10 N hangs at O, 2 m from the support at C. Find the value of the reaction forces P and Q at C and D respectively.

P + Q = 10 N

Taking moments about C:

10 × 2 = Q (2 + 6)

20 = 8Q

Q = 20/8 = 2.5 N

Or, taking moments about D:

P × 8 = 10 × 6

P = 60/8

P = 7.5 N

Q = 10 – 7.5

Q = 2.5 N

What is the centre of gravity?

The centre of gravity of a body is the point through which the line of action of the weight of the body always passes, irrespective of the position of the body. It is also the point at which the entire weight of the body appears to be concentrated.

The centre of mass of a body is the point at which the total mass of the body appears to be concentrated. Sometimes, the centre of mass may coincide with the centre of gravity for small objects.

What are the types of equilibrium?

There are three types of equilibrium: stable, unstable and neutral.

Stable equilibrium

A body is in stable equilibrium if it tends to return to its original position when slightly displaced. A low centre of gravity and a wide base put objects in stable equilibrium. Examples: a cone resting on its base; a racing car with a low C.G. and wide base; a ball or sphere in the middle of a bowl.

Unstable equilibrium

A body is in unstable equilibrium if, when slightly displaced, it tends to move further away from its original position. A high C.G. and a narrow base usually cause unstable equilibrium. Examples: a cone or an egg resting on its apex.

Neutral equilibrium

A body is in neutral equilibrium if, when slightly displaced, it tends to come to rest in its new position. Examples: a cone, a cylinder or an egg resting on its side.

Evaluation

1. State the two conditions for a body to be in equilibrium.
2. Define the moment of a force and state its unit.
3. What is a couple? Write the formula for the moment of a couple.
4. Give the difference between the centre of gravity and the centre of mass of a body.
5. Name the three types of equilibrium and give one example of each.

Assignment

1. Two forces A and B act at a point at right angles. If their resultant is 50 N and their sum is 70 N, their magnitudes are: (a) 50 N and 20 N (b) 20 N and 40 N (c) 40 N and 30 N (d) 60 N and 10 N
2. A uniform metre rule of mass 100 g balances at the 40 cm mark when a mass X is placed at the 10 cm mark. What is the value of X? (a) 33.33 g (b) 43.33 g (c) 53.33 g (d) 63.33 g
3. The equilibrant of a system of forces is (a) equal and opposite to the resultant of the forces (b) the force which has the same effect as the system (c) equal to the resultant of the system (d) the force that makes the system unstable
4. Two forces forming a couple are separated by a distance of 25 cm. If one of the forces equals 40 N, what is the moment of the couple? (a) 1000 Nm (b) 500 Nm (c) 10 Nm (d) 5 Nm
5. Two forces each of magnitude 10 N act in opposite directions at the end of a table. If the length of the table is 50 cm, find the moment of the couple on the table. (a) 0.5 Nm (b) 5 Nm (c) 50 Nm
6. A pole AB of length 5 m and weight 300 N has its centre of gravity 2.0 m from the end A and lies on horizontal ground. Calculate the force required to begin to lift this end. (a) 60 N (b) 120 N (c) 240 N