What you'll learn
Statics and forces deals with objects that are not accelerating — either stationary or moving at constant velocity — and with the conditions that keep them that way. It is the topic that explains why a beam balances, why a crane does not topple, why a bus with a low floor is harder to overturn, and how a lever multiplies force. The mathematics is straightforward, but the reasoning must be exact: an object in equilibrium requires two separate conditions to be satisfied simultaneously, and questions routinely test whether you know both. By the end of this guide you should be able to identify types of force, find resultants including at right angles, calculate moments, apply the principle of moments to balance problems, state the two conditions for equilibrium, explain centre of gravity and stability, and analyse levers and machines.
Key terms and definitions
Force — a push or pull that can change an object's shape, speed or direction, measured in newtons
Resultant force — the single force with the same effect as all the forces acting
Equilibrium — the state in which an object has zero resultant force and zero resultant moment
Moment — the turning effect of a force about a pivot, equal to force multiplied by perpendicular distance
Pivot or fulcrum — the fixed point about which an object turns
Principle of moments — for a body in equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments
Centre of gravity — the single point at which the entire weight of a body appears to act
Stability — the ability of an object to return to its original position after being tilted
Lever — a rigid bar turning about a pivot, used to multiply force or distance
Mechanical advantage — the ratio of load to effort
Couple — two equal and opposite parallel forces that produce rotation but no linear motion
Normal reaction — the force exerted by a surface perpendicular to it on an object resting on it
Core concepts
Types of force and their representation
Contact forces act between touching objects and include friction, air resistance, tension in a string, compression and the normal reaction from a surface.
Non-contact forces act at a distance and include gravitational force, magnetic force and electrostatic force.
Forces are vectors and are drawn as arrows whose length represents magnitude and whose direction shows the line of action. A free body diagram shows all the forces acting on a single object, and drawing one is often the first step to solving a problem.
Resultant forces
Forces acting along the same line are combined by adding those in one direction and subtracting those in the opposite direction.
Forces at right angles are combined using Pythagoras' theorem: the resultant is the square root of the sum of the squares of the two forces. Its direction is found using trigonometry, taking the tangent of the angle as the opposite force divided by the adjacent force.
A force can also be resolved into two perpendicular components, which is the reverse process. The component along a direction is the force multiplied by the cosine of the angle between them, and the perpendicular component uses the sine.
If the resultant force on an object is zero, the object is in translational equilibrium: it remains at rest or continues at constant velocity.
Moments
A moment is the turning effect of a force. It equals the force multiplied by the perpendicular distance from the pivot to the line of action of the force, and is measured in newton metres.
The word perpendicular is critical. If a force acts at an angle to a bar, the distance used is not the length along the bar but the perpendicular distance from the pivot to the force's line of action. Where a force acts along a line passing through the pivot, the perpendicular distance is zero and the moment is zero, which is why pushing directly towards a door's hinge does not open it.
Increasing either the force or the perpendicular distance increases the moment, which is why a long spanner loosens a tight nut more easily than a short one.
The principle of moments
For a body in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about that point.
This is the basis of every balance problem. The method is systematic: choose a pivot, identify every force and its perpendicular distance from that pivot, classify each moment as clockwise or anticlockwise, then set the two totals equal and solve.
Choosing the pivot wisely simplifies the work enormously. Taking moments about the point where an unknown force acts eliminates that force from the equation, because its perpendicular distance is zero. Where a question involves two unknown supports, taking moments about one support gives an equation containing only the other.
For a uniform beam, the weight acts at its centre, and this weight must be included unless the beam is described as light or of negligible mass.
The two conditions for equilibrium
An object is in equilibrium only when both conditions hold.
First, the resultant force in any direction must be zero, so the forces up equal the forces down and the forces left equal the forces right.
Second, the resultant moment about any point must be zero, so clockwise moments equal anticlockwise moments.
Questions involving a beam on two supports require both conditions: the moments equation finds one unknown support force, and the forces equation then finds the other. Attempting such a question with only one condition is the commonest reason for an incomplete answer.
Centre of gravity
The centre of gravity is the single point at which the whole weight of a body appears to act. For a uniform object of regular shape it lies at the geometric centre — at the midpoint of a uniform rod, at the intersection of the diagonals of a rectangle, and at the intersection of the medians of a triangular lamina.
The centre of gravity of an irregular lamina is found experimentally by suspending it freely from a point and hanging a plumb line from the same point. The centre of gravity lies somewhere on that vertical line, so repeating from a second and third point gives lines that intersect at the centre of gravity. A freely suspended object always comes to rest with its centre of gravity directly below the point of suspension.
Stability
An object topples when a vertical line drawn downwards from its centre of gravity falls outside its base.
Stability is therefore improved in two ways: by lowering the centre of gravity, and by widening the base. A racing car is stable because it is low and wide; a double-decker bus carries heavy components low in the chassis for the same reason.
Three states of equilibrium are distinguished. In stable equilibrium, a small displacement raises the centre of gravity and the object returns to its original position. In unstable equilibrium, a small displacement lowers the centre of gravity and the object moves further away. In neutral equilibrium, such as a ball on a level surface, the centre of gravity stays at the same height and the object stays where it is placed.
Levers and machines
A lever is a rigid bar that turns about a pivot, and it works by the principle of moments. Because moment equals force times distance, a small effort applied far from the pivot balances a large load close to it.
Mechanical advantage is the load divided by the effort. A lever with a mechanical advantage of 4 allows a load four times the effort to be lifted, at the cost of moving the effort four times as far.
Levers are classified by the relative positions of pivot, load and effort. In a first-class lever the pivot lies between the effort and the load, as in a seesaw, a crowbar or a pair of scissors. In a second-class lever the load lies between the pivot and the effort, as in a wheelbarrow or a bottle opener. In a third-class lever the effort lies between the pivot and the load, as in tongs, a fishing rod or the human forearm, and the mechanical advantage is less than one, so force is sacrificed for greater speed and range of movement.
A couple consists of two equal and opposite parallel forces whose lines of action do not coincide. It produces rotation with no resultant linear force, as when both hands turn a steering wheel.
Worked examples
Example 1: A simple balance problem (4 marks)
A uniform metre rule is pivoted at its centre. A weight of 4.0 newtons is hung 30 centimetres to the left of the pivot. At what distance to the right must a 6.0 newton weight be hung to balance the rule?
Since the rule is uniform and pivoted at its centre, its own weight acts at the pivot and produces no moment.
The anticlockwise moment is 4.0 × 0.30 = 1.2 newton metres.
Let the required distance be d. The clockwise moment is 6.0 × d.
By the principle of moments, 6.0d = 1.2, so d = 0.20 metres, which is 20 centimetres to the right of the pivot.
Example 2: A beam on two supports (5 marks)
A uniform beam of weight 200 newtons and length 4.0 metres rests on two supports, one at each end. A load of 600 newtons is placed 1.0 metre from the left support. Calculate the force exerted by each support.
Take moments about the left support, which eliminates the unknown force there since its distance is zero.
The beam is uniform, so its weight of 200 newtons acts at its centre, 2.0 metres from the left support, giving a clockwise moment of 200 × 2.0 = 400 newton metres.
The 600 newton load gives a clockwise moment of 600 × 1.0 = 600 newton metres.
The right support force R acts upwards at 4.0 metres, giving an anticlockwise moment of 4.0R.
Equating: 4.0R = 400 + 600 = 1000, so R = 250 newtons.
Now apply the condition that total upward force equals total downward force. The downward forces total 200 + 600 = 800 newtons, so the left support force is 800 − 250 = 550 newtons.
Example 3: Explaining stability (3 marks)
Explain why a bus carrying passengers on the upper deck only is more likely to overturn than one with passengers seated downstairs.
Placing passengers on the upper deck raises the centre of gravity of the bus, because the distribution of mass is shifted upwards.
When the bus tilts, for example on a bend or a camber, a vertical line from the higher centre of gravity falls outside the base at a smaller angle of tilt than it would if the centre of gravity were lower.
The bus is therefore less stable and will topple at a smaller angle, which is why lower deck seating is filled first.
Common mistakes and how to avoid them
The most frequent error is using the distance along a bar rather than the perpendicular distance from the pivot to the line of action of the force. Where a force acts at an angle, the perpendicular distance is needed.
Students often omit the weight of a uniform beam. Unless the beam is described as light or of negligible mass, its weight acts at the centre and must be included.
Another common slip is using only the principle of moments in a two-support problem. Both equilibrium conditions are required, and the forces equation is the second half of the answer.
Many candidates fail to convert centimetres to metres before calculating a moment, which gives an answer one hundred times too large.
Finally, in stability questions, answers often mention the centre of gravity without referring to the base. Toppling occurs when the vertical line through the centre of gravity falls outside the base, and both elements are needed.
Exam technique for "Statics and Forces"
Draw a diagram with every force marked as an arrow, including the weight of the beam where relevant. The diagram organises the problem and earns marks in its own right.
Choose the pivot deliberately. Taking moments about the point where an unknown force acts removes it from the equation and usually turns a difficult problem into a one-line calculation.
Convert all distances to metres before calculating, so that moments come out in newton metres.
State which moments are clockwise and which anticlockwise before equating them. Labelling prevents the sign errors that otherwise appear.
When a question asks about equilibrium, check both conditions and say so explicitly. Examiners award marks for recognising that both must be satisfied.
Quick revision summary
Forces are vectors, either contact such as friction, tension and normal reaction, or non-contact such as gravitational, magnetic and electrostatic. Forces along a line add or subtract; forces at right angles combine by Pythagoras with direction from trigonometry, and a single force can be resolved into perpendicular components. A moment is force multiplied by the perpendicular distance from the pivot, measured in newton metres, and is zero when the line of action passes through the pivot. The principle of moments states that clockwise moments equal anticlockwise moments for a body in equilibrium, and choosing the pivot at an unknown force eliminates it. Equilibrium requires both zero resultant force and zero resultant moment, and two-support problems need both. The centre of gravity is where the whole weight acts, found for a lamina by suspending it from several points with a plumb line. An object topples when a vertical line through its centre of gravity falls outside its base, so stability improves with a lower centre of gravity and a wider base. Levers multiply force through moments, with mechanical advantage equal to load divided by effort, and are classified as first, second or third class by the positions of pivot, load and effort.