1-5 Equilibrium of Forces

Chapter 1 · Mechanics

❓ Lesson question

A book remains at rest on a table and a shop sign is suspended by two chains — what is true of the resultant force acting on each object?

You will learn to

1State the condition of equilibrium for a body at rest or moving with constant velocity.
2Draw a free-body diagram, isolating the body and showing all external forces acting on it, identifying the source of each force.
3Determine an unknown force by resolving the forces into components along two perpendicular axes and applying the equilibrium conditions to each axis.

A heavy book lies at rest on a table. Its weight acts downward due to gravity, yet the book stays exactly where it is.

Nearby, a heavy shop sign hangs motionless from two chains inclined at different angles: one steep, the other closer to the horizontal.

✍️ Predict before you continue

Write your answer before reading on. A prediction you got wrong and then examined teaches you more than an answer you were handed.

(a) What force, if any, does the table exert on the book?

(b) The sign is suspended by two chains at different angles. Which chain has the greater tension — the steeper chain, or the chain that is closer to the horizontal?

Write a line for each before this opens.

🔬 Explore — two rubber bands and one bottle

What you’ll do: measure the extension of two identical rubber bands as their angle with the vertical changes, while the load stays constant.

⏱️ 12 min🧰 a filled water bottle of known weight · two identical rubber bands · a ruler · a protractor · two hooks that can be moved apart

1Suspend the bottle from two identical rubber bands, with the hooks close together so each band makes a small angle with the vertical.
2Measure the extension of each band and record the angle it makes with the vertical.
3Keep the same bottle and move the attachment points farther apart, so the bands form a larger angle with the vertical.
4Measure both extensions again, and repeat for several larger angles — without exceeding the elastic limit of the rubber.
5The extension is your measure of tension: a rubber band stretches further when it is pulled harder.

🎛️ Simulation — a sign on two cables

Only the vertical parts of the two tensions hold the weight up. The horizontal parts do nothing but cancel each other.

Mass of the sign6 kg

Left cable angle to the horizontal30 °

Right cable angle to the horizontal30 °

Left cable T₁

58.8 N

vertical part 29.4 N

Right cable T₂

58.8 N

vertical part 29.4 N

ΣF_y check

58.8 N

the two vertical parts add to exactly the weight

ΣF_x check

0.00 N

horizontal parts cancel: 50.9 − 50.9

Equal angles, so equal tensions — each cable carries exactly half the weight vertically. Now make one angle different and watch which cable takes more.

Analyse what you collected

(a) As the angle from the vertical increased, what happened to the extension of each band — and therefore to the tension in it?

(b) The bottle’s weight stayed constant, so why did the tension increase? What happened to the horizontal components of the two tensions as the bands moved farther apart?

✍️ Write your explanation in two lines before you read on.

Measurement note

Measure each band’s extension twice and take the mean. Use the same bands throughout — two rubber bands are never quite identical — and compare the average extension across angles rather than against any fixed value.

In the Explore, the bottle’s weight never changed — but widening the angle increased the extension of both bands, and so the tension in both. You might have expected the tension to stay put, since the load did not move. The evidence says the tension depends on the weight and on the angle.

Isolate the body — the free-body diagram

A force is a push or a pull exerted by one object on another. To analyse a body, first isolate it from its surroundings and show only the external forces acting on it — never the forces it exerts on other things. Label each arrow by its source.

Force	Symbol	Source and direction
Normal force	$F_{N}$	A contact surface; perpendicular to that surface. For vertical equilibrium on a horizontal surface with no other vertical forces, $F_{N} = m g$ .
Tension	$T$	A taut rope or cable; acts along the rope, directed away from the body.
Elastic force	$F_{e} = - k x$	A stretched or compressed spring; opposite to the displacement from equilibrium, within the limit of proportionality.
Weight	$F_{g} = m g$	The Earth’s gravitational field; vertically downward through the centre of gravity.

Apply the equilibrium condition

A body is in translational equilibrium when it stays at rest or moves at constant velocity — that is, when the resultant force on it is zero:

Σ F = 0

A force at an angle is resolved into components along two perpendicular axes, and the condition is applied independently along each:

Σ F_{x} = 0 Σ F_{y} = 0

For the hanging bottle: the horizontal components of the two tensions are equal and opposite, so they cancel; the vertical components both point up and together balance the weight. The resultant is zero, and the bottle stays put.

Point

If a rigid body remains at rest, or moves with constant velocity, under several forces, it is in equilibrium. To analyse it, isolate the body from its surroundings and draw a free-body diagram showing only the external forces on it. The weight is a single downward force through the centre of gravity.
$F_{1} + F_{2} + F_{3} + \dots = 0$ . The equilibrium equations are set up by resolving the forces into components along two perpendicular axes: $Σ F_{x} = 0$ and $Σ F_{y} = 0$ .
Constant velocity counts as equilibrium too. A lift descending at a steady speed is in equilibrium exactly as much as one standing still — a common exam trap.

📐 Equation box

Condition for translational equilibrium

Σ F = 0

Resolved along two perpendicular axes

Σ F_{x} = 0, Σ F_{y} = 0

Two cables at α and β to the horizontal

T_{1} = \frac{W c o s β}{s i n ( α + β )}, T_{2} = \frac{W c o s α}{s i n ( α + β )}

Elastic (spring) force

F_{e} = k x

Key terms

Translational equilibrium: — The resultant force on the body is zero: at rest, or moving at constant velocity.
Free-body diagram: — The body drawn alone, with every external force on it shown and labelled by its source.
Normal force: — The push a surface exerts, perpendicular to that surface.
Centre of gravity: — The single point through which the whole weight can be taken to act.

🛠️ Worked example

Situation

A lamp of mass 6.0 kg is suspended in equilibrium by two identical cables, each making an angle of 30° with the horizontal. Calculate the tension in each cable.

Step 1 of 6 — try it yourself before you open the next one.

The same 6.0 kg lamp is suspended by two identical cables, each making an angle of 50° with the horizontal. Determine the tension in each cable, and compare with the previous example.

A streetlamp above a public square hangs from two cables that make angles of 30° and 60° with the horizontal. The lamp has a mass of 20 kg. Determine the tension in each cable, and state which cable is under the greater tension.

A sign of mass 5.0 kg over a street hangs from two cables that each make 40° with the horizontal. Determine the tension in each cable.

A metal block hangs at rest from a spring with a spring constant of 200 N/m, stretching it by 0.15 m. Determine the weight of the block.

In a New Context

Travelling along a country road you may notice that power lines sag between poles rather than forming a straight horizontal line. The same principle of equilibrium applies as to a load supported by two cables.

At the two supports, the vertical components of the tension together balance the total weight of the span, while the horizontal components are equal and opposite. As the sag decreases, the wire becomes more nearly horizontal at the supports, so the vertical component becomes a smaller fraction of the tension — and a greater tension is needed to support the same weight.

In the idealised limit of a perfectly horizontal wire supporting a non-zero weight, the required tension would tend to infinity. Power lines are therefore installed with a controlled amount of sag.

➡️ Your turn: Using vertical components, explain why reducing the sag in a power line increases the tension in the line.

Try it in the simulation

Set both cable angles to 8° and read the tension; then to 4°. Halving the angle roughly doubles the tension. Keep going and the numbers become ridiculous — which is the answer to the question above, produced by your own hand rather than asserted.

🧪 Nature of Science — an early law of equilibrium, before Newton

Before Newton formulated the laws of motion, Simon Stevin (1586) studied equilibrium on inclined planes with a thought experiment known as the “wreath of spheres”. He imagined a closed chain of equally spaced spheres draped over two inclined planes. If one side produced a greater turning effect than the other, the chain would move continuously while returning to the same arrangement — producing perpetual motion.

Since perpetual motion is impossible, Stevin concluded that the effects on the two slopes must balance. The argument needed no equations at all: it only needed the refusal to accept an absurd consequence. This established the law of equilibrium on an inclined plane and the principle of resolving forces into components.

Your turn: Why could Stevin trust this conclusion even without modern equations of motion?

✅ Lesson question, answered

A book remains at rest on a table and a shop sign is suspended by two chains — what is true of the resultant force acting on each object?

In both cases the bodies are in translational equilibrium, so the vector sum of all forces on each body is zero: $Σ F = 0$ .

For the book, the upward normal force from the table balances its weight. For the sign, the horizontal components of the two chain tensions cancel each other, while their vertical components together balance the weight.

Because the chains make different angles with the horizontal, the tensions are generally unequal — and the steeper chain carries the greater tension.

A shop sign of mass 8.0 kg hangs in equilibrium from two light cables attached to a horizontal beam. The left cable makes 30° with the horizontal and the right cable makes 55°. Determine the tension in each cable, and state which carries the greater tension.

A lamp of mass 3.0 kg is held in equilibrium by a light cable fixed to the ceiling and a light horizontal cord. The cable makes an angle of 25° with the vertical. Determine (i) the tension

T

in the cable and (ii) the tension

P

in the horizontal cord.

A block of mass 1.5 kg hangs in equilibrium from two identical light springs. Each spring makes an angle of 40° with the vertical and has a spring constant of 60 N/m. Determine (i) the force in each spring and (ii) the extension of each spring.

📝 Exam-style question

A 6.0 kg lamp hangs at rest from two cables that make equal angles of 40° with the horizontal. Draw a free-body diagram of the lamp, then determine the tension in each cable.

How the real paper is built

The ministry sets 50% multiple choice and 50% written answer. They test different things: the MCQ half rewards eliminating fast and spotting a distractor built from a plausible wrong method; the written half asks you to show that you understand why. Practise both halves — being fluent at one and weak at the other costs you exactly half the paper.

Part 1 — multiple choice (50% of the paper)

Two of these test whether you actually believe the prediction you made at the top of the lesson.

A book of weight 20 N rests on a horizontal table. The normal force from the table is:

A sign hangs from two chains at different angles. The greater tension is in:

A lift descends at a constant 2.0 m/s. The resultant force on it is:

A 4.0 kg lamp hangs from two cables that each make 30° with the horizontal. The tension in each cable is:

As two supporting cables are moved towards horizontal while holding the same weight, the tension in them:

A spring of constant 50 N/m supports a hanging block and stretches 0.40 m. The block’s weight is:

Part 2 — written answer (the other 50%)

Same routine every time: isolate the body, draw every external force, resolve along two axes, set each sum to zero. Each question has its own notebook page — work it there, then compare with the model answer.

Level 1 — direct substitution

One step. You are given the formula; put the numbers in.

1.1

A 4.0 kg box rests on a horizontal table. Determine the normal force from the table.

1.2

A spring of constant 150 N/m is stretched by 0.20 m. Determine the force in the spring.

Level 2 — application in a new context

Two steps, or you must choose the right component before you can start.

2.1

A 12 kg sign hangs from two cables that each make 35° with the horizontal. Determine the tension in each cable.

2.2

A 2.0 kg lamp hangs from a cable making 20° with the vertical, held aside by a horizontal cord. Determine the tension in the cable and in the cord.

Level 3 — exam challenge

Multi-step, or a case the textbook does not set and examiners do.

3.1

A 15 kg sign hangs from two cables making 25° and 65° with the horizontal. Determine both tensions and state which is greater.

3.2

A power line span carries an effective weight of 800 N. The wire leaves each support at 8° to the horizontal. Determine the tension. Then determine it again if the sag is reduced so the wire leaves at 4°, and explain the pattern.

Keep reading — this lesson marks itself done automatically.