Cosmos Coaching Centre
Resources
Show All Resources
Year 7
Maths Science
Year 8
Maths Science
Year 9
Maths Science
Year 10
Maths Science
Year 11
Maths Standard Maths Advanced Maths Extension Physics Chemistry
Year 12
Maths Standard Maths Advanced Maths Extension Physics Chemistry
Selective
Apps
ATAR calculator Worksheet Forge Uni StarMap
Pricing Contact
2A

Forces and free-body diagrams

Forces as interactions between two objects, contact and non-contact forces, Newton's three laws of motion, third-law pairs, weight and the normal force, apparent weight in a lift, and how to draw a free-body diagram.

Stand on bathroom scales in a lift. As the lift starts going up, the reading jumps by a few kilograms. As it slows at the top, you seem to lose weight. Your mass hasn't changed and neither has the Earth's pull on you. What changed is how hard the floor pushes back, and that is what scales actually measure.

This page sets up the language for the whole module. A force is always an interaction between two objects, and the most useful skill in dynamics is isolating one object and listing every force on it. Get the free-body diagram right and the maths usually follows.

Forces are interactions

A force is a push or a pull that one object exerts on another. It is a vector, measured in newtons (N). Every force has a source and a receiver, so it helps to name forces in the form "force on the book by the table".

Forces come in two broad kinds:

  • Contact forces need the objects to touch: pushes, the normal force, friction, tension in a rope or string, air resistance, buoyancy.
  • Non-contact forces act through a field across a gap: gravitational, electrostatic and magnetic forces.
Contact forces act where surfaces touch. Field forces act across a gap: gravity, magnetism and electrostatic attraction or repulsion.

At the atomic level, every contact force is actually electromagnetic. When a book sits on a table, the electron clouds of the surface atoms repel each other. The table's surface squashes very slightly, like a stiff spring, until the push balances the book's weight. That push is the normal force, so called because it acts normal (perpendicular) to the surface.

Newton's three laws

LawStatementWhat it lets you do
First (inertia)An object stays at rest or keeps moving at constant velocity unless a non-zero net force acts on it.Recognise equilibrium: constant velocity means $\Sigma \vec F = 0$.
SecondThe acceleration of an object is proportional to the net force on it and inversely proportional to its mass: $\vec F_{net} = m\vec a$.Calculate acceleration or force. The direction of $\vec a$ is the direction of $\vec F_{net}$.
ThirdWhen object A exerts a force on object B, B exerts a force on A that is equal in size and opposite in direction: $\vec F_{AB} = -\vec F_{BA}$.Find the force on the other object in an interaction.

Two points cause most of the trouble. First, the first law says nothing about speed: an object can be moving fast with zero net force. A car cruising at a steady 100 km/h on a straight road has balanced forces. Second, $F_{net}$ in the second law is the vector sum of all the forces on one object, not any single force.

Third-law pairs

The two forces in a third-law pair:

  • are equal in size and opposite in direction;
  • act on different objects;
  • are the same type of force (both gravitational, both normal, both friction);
  • exist at the same time, for the same interaction.

Because they act on different objects, a third-law pair can never cancel. A horse pulling a cart can still accelerate: the cart pulls back on the horse, but what matters for the horse is the total force on the horse, including the ground pushing its hooves forward.

Two skaters push apart with equal and opposite forces. A rocket pushes exhaust gas down, and the gas pushes the rocket up.

The classic trap is the book on a table. Its weight (Earth pulls book down) and the normal force (table pushes book up) are equal and opposite, but they are not a third-law pair: they act on the same object and are different types of force. The third-law partner of the book's weight is the book pulling the Earth up with 14.7 N. The partner of the normal force is the book pushing down on the table.

A space shuttle rising from the launch pad on a column of bright exhaust from its boosters and engines
A rocket needs nothing to push against. The engines push hot gas backwards at high speed, and by Newton's third law the gas pushes the rocket forwards with the same size of force.Photo: NASA, Public domain

Drawing free-body diagrams

A free-body diagram shows one object, usually as a dot or a box, with every force acting on it drawn as an arrow from that point. Forces the object exerts on other things do not belong on its diagram.

  1. Choose one object and draw it on its own.
  2. Add weight $mg$, straight down.
  3. Find everything the object touches. Each contact gives a force: a normal force perpendicular to each surface, friction along it, tension along each string.
  4. Add any field forces (electric, magnetic) if they act.
  5. Draw arrow lengths roughly to scale and label each force.
  6. Choose a positive direction and write $\Sigma F = ma$ along each axis.
A crate dragged by a rope at 30° above the horizontal and its free-body diagram: four forces, all acting on the crate.

Weight, normal force and apparent weight

Weight is the gravitational force on an object, $W = mg$, where $g = 9.8$ N kg⁻¹ near Earth's surface. It is a force in newtons. Mass is the amount of matter (and the object's resistance to acceleration), in kilograms. A 60 kg person has a weight of 588 N on Earth and about 97 N on the Moon, but the same mass in both places.

The normal force is not always equal to weight. It is whatever the surface needs to supply so the object doesn't sink into it. Scales read the normal force, and in an accelerating lift that changes. Taking up as positive:

$$N - mg = ma \quad\Rightarrow\quad N = m(g + a)$$
Worked example: scales in a lift

A 60.0 kg student stands on scales in a lift. Find the normal force (a) when the lift accelerates upward at 2.0 m s⁻², and (b) when it accelerates downward at 1.5 m s⁻². What mass does the scale display in each case?

Forces on the student: weight $mg = 588$ N down, normal force $N$ up. $\Sigma F = N - mg = ma$.
up is positive
$N = 60.0(9.8 + 2.0) = 708$ N. The scale divides by $g$: $708/9.8 = 72.2$ kg.
(a) a = +2.0
$N = 60.0(9.8 - 1.5) = 498$ N, displayed as $498/9.8 = 50.8$ kg.
(b) a = −1.5
Accelerating up makes you feel heavier; accelerating down makes you feel lighter. A lift moving at constant velocity (either direction) reads 60.0 kg.
check

If the cable snapped and the lift fell freely, $a = -9.8$ m s⁻², so $N = 0$. The student would float relative to the floor: apparent weightlessness. Astronauts on the International Space Station feel weightless for the same reason. Gravity there is still about 90% of its surface value, but the station and everyone in it are falling around the Earth together.

Pushing and pulling more than one object

When objects move together, two views help. Treat them as one system to find the acceleration, then isolate one object to find the force between them.

Worked example: two blocks pushed together

Block A (2.0 kg) and block B (3.0 kg) sit side by side on a frictionless floor. A hand pushes A with a horizontal 10 N force, so A pushes B. Find the acceleration and the force between the blocks.

External horizontal force = 10 N on 5.0 kg: $a = 10/5.0 = 2.0$ m s⁻².
whole system
Only A pushes B horizontally: $F_{AB} = m_B a = 3.0 \times 2.0 = 6.0$ N.
isolate B
On A: $10 - F_{BA} = 2.0 \times 2.0$, so $F_{BA} = 6.0$ N. Equal and opposite to $F_{AB}$, as the third law requires.
check on A
InteractiveFree-body diagrams and third-law pairs
Try this
  • In Book on a table, compare "Forces on the book" with "Third-law pairs". The 14.7 N weight and 14.7 N normal force appear on the book, but each has its partner on a different object.
  • In Person in a lift, set the acceleration to +2.0 m/s². The normal force rises to 708 N and the scales read 72.2 kg. Drag it to −9.8 m/s²: the normal force disappears.
  • In Car towing a trailer at 1.5 m/s², the tow bar tension is 900 N and the road pushes the car forward with 2,700 N. The car and trailer pull on each other equally, yet both accelerate.
  • In Pushing two blocks with 10 N, the contact force is 6.0 N. Press "Push from the right": now it is 4.0 N, because the hand now has to accelerate B directly and A only needs 2.0 × 2.0 N.

Test yourself

  1. A book rests on a table. Which force is the third-law partner of the book's weight?

  2. A 75 kg person stands on scales in a lift that is slowing down while moving upward at 1.2 m s⁻². The scales read

  3. A hockey puck slides across ice at constant velocity. Which statement is correct?

  4. A truck collides with a small car. During the collision, the force of the truck on the car is

  5. Which of these is a non-contact force?

Label the free-body diagram for the crate.

The situationThe free-body diagram12 kgA crate dragged by a rope at 30°weight mgnormal force Ntension Tfriction30°One dot, every force ON the crate, tails on the dot

Is each pair of forces a Newton's third-law pair?

A force mediated by a field, such as gravity, acts without the objects . The two forces in a third-law pair act on objects, so they never cancel. The force a surface exerts perpendicular to itself is the force. Bathroom scales measure this force, so in a lift accelerating upward they read than your weight. Weight is measured in .

5 marks

A student says: "When a horse pulls a cart, the cart pulls back on the horse with an equal and opposite force. These forces cancel, so the horse and cart can never accelerate." Explain why the student is wrong, using free-body diagrams to support your answer.