Newton’s three laws of motion, the forces of everyday life, and frames of reference

What this chapter is for

The chapter on kinematics described motion without asking what causes it. This chapter asks. Isaac Newton answered in three laws, published in 1687, that link the forces on a body to the way it moves.⁶ With them, and the forces that act in daily life, such as weight, the push of a floor, the pull of a rope and friction, you can work out how almost anything around you will move.

The chapter first explains force and inertia, then states the three laws one at a time with examples. It then takes the everyday forces in turn, weight, the normal force, tension, friction and drag, and gives a method for drawing every force on a body and applying the second law to it. It ends with the frames in which the laws hold and the four basic forces of nature. The past questions are worked in full at the end.

The syllabus begins its mechanics with “Newtonian laws of motion”.¹ Three Part-II questions from CE-2016 to CE-2026 come from this chapter, and so does one surviving MCQ.² The laws also run through every later chapter of mechanics, from momentum and rotation to orbits.

Before you start

The words you need

The story in plain words

1. A body keeps doing what it is doing, resting or moving steadily in a straight line, unless a net force acts. This is the first law, and the tendency is inertia.
2. A net force changes the motion. The acceleration is the net force divided by the mass. This is the second law.
3. Forces come in pairs. When one body pushes another, the second pushes back equally on the first. This is the third law.
4. Everyday forces, weight, normal force, tension, friction and drag, are found by drawing every force on one body at a time.
5. The laws hold as they stand only in an inertial frame, one that is not accelerating.

1. Force and inertia

A force is a push or a pull. It has a size and a direction, so it is a vector, and it is measured in newtons, N. Forces on a body add as vectors, and their sum is the net force. Two people pushing a stalled car from behind add their pushes. Two teams in a tug of war pull against each other, and if their pulls are equal, the net force on the rope is zero.⁸

It seems natural to think that a body needs a force to keep moving, since a cart stops when the bullock stops pulling, and for centuries people believed it. Galileo saw that the cart stops because of friction. On smoother and smoother ground a rolling ball goes further and further, and with no friction at all it would roll on for ever. A body does not need a force to keep moving. It needs a force to change how it moves.⁷

This tendency of a body to keep its motion, to stay at rest if it is at rest and to keep moving steadily in a straight line if it is moving, is called inertia. You feel it every day. When a bus brakes suddenly, the passengers lurch forwards, because their bodies keep moving while the bus slows under them. When it starts suddenly, they fall backwards. A cloth pulled quickly from under a glass leaves the glass behind. The more massive a body, the more inertia it has, so the harder it is to start, stop or turn. A loaded truck needs far more force to stop than a bicycle at the same speed. The mass of a body is the measure of its inertia.

2. The first law, and the frames it holds in

Newton’s first law states the idea of inertia as a law.

A body stays at rest, or moves in a straight line at a constant speed, unless a net force acts on it.

It does not say that no force acts. A book lying on a table has two forces on it, its weight pulling down and the table pushing up. They cancel, the net force is zero, and the book stays at rest. A car cruising at a steady 100 km/h on a straight motorway has the engine’s driving force forwards and the drag of the air and road backwards, and they too cancel. A body with zero net force on it is in equilibrium, whether it is at rest or moving steadily.

The first law also says where the laws of motion hold. Stand in a bus that brakes, and a ball on the floor rolls forwards with no force pushing it. In the frame of the braking bus, the first law fails. A frame in which the first law holds is called an inertial frame. The ground is very nearly one, and any frame moving at a steady velocity relative to it is another. Frames that accelerate or turn, like the braking bus or the turning car of the last chapter, are not inertial, and there the laws hold only after fictitious forces are added.⁹

3. The second law, force and acceleration

A net force changes a body’s velocity. Newton’s second law says by how much.¹⁰

The net force on a body equals its mass times its acceleration, ${𝐅}_{\text{net}}=m𝐚$.

The acceleration points the same way as the net force, and for a given force it is smaller for a larger mass. This is what makes mass the measure of inertia. Push a shopping trolley and a loaded car with the same force, and the trolley picks up speed far faster.

The law defines the unit of force. One newton is the net force that gives a mass of 1 kg an acceleration of 1 m/s², so 1 N = 1 kg m/s². A 100 g apple weighs about one newton.

Newton himself wrote the law in terms of momentum, the product of mass and velocity, $𝐩=m𝐯$. The net force is the rate of change of momentum.

$${𝐅}_{\text{net}}=\frac{d𝐩}{dt}$$

When the mass stays the same, $d𝐩/dt=md𝐯/dt=m𝐚$, and the two forms agree. The momentum form is the more general. It also covers a body whose mass changes, such as a rocket burning its fuel, and the chapter on momentum uses it. The 2026 Paper I asked in an MCQ what the rate of change of momentum equals, and the answer is force.⁵

The law is a vector equation, so it holds separately for each direction, $F_x=m{a}_x$, $F_y=m{a}_y$ and $F_z=m{a}_z$. The forces across the motion and the forces along it can be handled one direction at a time, in the same way as the projectile of the chapter on kinematics.

The 2019 Paper I gave a quick example.³ To give a 1,000 kg car an acceleration of half of $g$, 4.905 m/s², needs a net force of $1000\times 4.905=\mathrm{4,905}$ N. To give a 200 g apple the same acceleration needs $0.200\times 4.905=0.981$ N. The car needs 5,000 times more force, because it has 5,000 times the mass.

4. The third law, forces come in pairs

Push against a wall and the wall pushes back on your hands. Newton’s third law says this always happens.¹²

When body A exerts a force on body B, body B exerts a force on body A that is equal in size and opposite in direction.

The two forces of a pair act on different bodies. That is the key point, and missing it causes most of the confusion about this law. If they acted on the same body they would cancel and nothing could ever move. They do not cancel, because each acts on a different body.

Think of walking. Your foot pushes backwards on the ground. The ground pushes forwards on your foot, with an equal force, and that forward push is what moves you. On ice, your foot cannot push backwards, so the ground cannot push you forwards, and you cannot walk. A rocket pushes its hot gases backwards out of its nozzle, and the gases push the rocket forwards. A rocket needs no air to push against, which is why it works in space.

A book resting on a table shows the difference between a third-law pair and two forces that merely balance, as Figure 1 draws. The Earth pulls the book down with its weight, and by the third law the book pulls the Earth up with an equal force. The table pushes the book up, and the book pushes the table down. The book’s weight and the table’s push both act on the book and cancel, so the book stays at rest. They are equal by the first law, not by the third, and they are not a pair. They are different kinds of force, and if the table were in a lift that accelerated, they would no longer be equal.

5. Mass and weight

Mass and weight are different things, though everyday speech mixes them. Mass is the amount of inertia, in kilograms, and it is the same everywhere. Weight is the pull of gravity, a force, in newtons. Near the Earth’s surface a body falls with acceleration $g$, so by the second law the force pulling it, its weight, is¹¹

$$W=mg$$

A 60 kg student weighs $60\times 9.81=588.6$ N on Earth. On the Moon, where the surface gravity is 1.62 m/s², the same student weighs $60\times 1.62=97.2$ N, about a sixth as much, but still has a mass of 60 kg.¹⁶ Throwing a cricket ball on the Moon would be just as hard as on Earth, since that depends on mass, but lifting it would be easy.

A bathroom scale does not measure weight. It measures how hard you push down on it, which equals how hard it pushes up on you. Stand on one in a lift. When the lift is at rest or moving steadily, the scale pushes up with your weight and reads 60 kg. When the lift accelerates upwards at 2 m/s², the scale must push up with more than your weight, enough to give you that acceleration too. Taking upwards as positive, $N-mg=ma$, so $N=m(g+a)=60\times 11.81=708.6$ N, and the scale reads 72.2 kg. Accelerating downwards at 2 m/s², it reads $60\times 7.81=468.6$ N, or 47.8 kg. If the cable broke and the lift fell freely, the scale would read zero. This reading is your apparent weight, and Figure 2 draws it. Astronauts in orbit feel weightless for the same reason. They and their station fall freely together round the Earth.

6. The normal force

When a body presses on a surface, the surface pushes back, at right angles to itself. This push is the normal force, $N$, where “normal” means at right angles. A floor, a table, a wall and the seat of a chair all push this way. The surface is slightly squashed, too little to see, and pushes back like a very stiff spring. It pushes only as hard as it has to, which is why the normal force is not always equal to the weight.

The 2019 Paper I asked what happens to the normal force on a block lying on a floor as a vertical force $F$ is applied to it, pushing down or pulling up.³ The block stays at rest, so the forces on it balance. Pushing down, the floor must hold up both the weight and the push, so $N=mg+F$, and the normal force grows steadily as $F$ grows. Pulling up, the pull takes some of the weight, so $N=mg-F$, and the normal force falls as $F$ grows. It reaches zero when $F=mg$, and the block then leaves the floor. The normal force cannot go below zero, since a floor can push but cannot pull. Figure 3 draws both cases for a block of 2 kg, whose weight is 19.62 N.

7. Tension, and bodies joined by a rope

A rope, a string or a chain pulls on whatever it is tied to, along its own length. That pull is the tension, $T$. A light rope that does not stretch has the same tension all along it, and a light pulley with no friction changes the rope’s direction without changing its tension.

Two blocks of 5 kg and 3 kg hang from the two ends of a rope over a light pulley, as in Figure 4. This is called an Atwood machine. The heavier block goes down and the lighter goes up, with the same acceleration $a$, since the rope joins them. Apply the second law to each block on its own, taking the direction of its motion as positive.

$$m_{1}g-T=m_{1}a,T-m_{2}g=m_{2}a$$

Adding the two removes $T$ and gives the acceleration. Putting it back gives the tension.

$$a=\frac{(m_1-m_2)g}{m_1+m_2}=\frac{2\times 9.81}{8}=2.4525{\text{m/s}}^2,T=\frac{2m_1{m}_{2}g}{m_1+m_2}=36.79\text{N}$$

The tension lies between the two weights, 29.43 N and 49.05 N. It is less than the heavy block’s weight, so that block accelerates down, and more than the light block’s weight, so that one accelerates up.

8. Friction

Push a heavy cupboard across a floor. At first it does not move, though you push. Push harder and it suddenly gives way and slides, and then it is easier to keep it moving than it was to start it. That is friction, the force a surface exerts along itself to resist sliding.¹⁴

While the cupboard does not move, the friction on it is static friction. It matches your push exactly, however hard you push, up to a limit. The limit is proportional to the normal force, $f_s\le {\mu }_{s}N$, where ${\mu }_s$ is the coefficient of static friction, a number that depends on the two surfaces. Once the cupboard slides, the friction is kinetic friction, of a fixed size $f_k={\mu }_{k}N$, where ${\mu }_k$ is the coefficient of kinetic friction. For most surfaces ${\mu }_k$ is smaller than ${\mu }_s$, which is why the cupboard lurches when it starts. Figure 5 draws the friction as the push grows. Friction does not depend on how large the area of contact is, only on the surfaces and how hard they are pressed together.

On a slope, friction holds a body against sliding down. A block resting on a slope at angle $\theta$ has its weight split into two parts, as Figure 6 shows, $mg\sin \theta$ along the slope and $mg\cos \theta$ into it. The normal force balances the part into the slope, $N=mg\cos \theta$. The block stays put as long as static friction can hold the part along the slope, $mg\sin \theta \le {\mu }_{s}mg\cos \theta$, which means $\tan \theta \le {\mu }_s$. The steepest slope on which it can rest has $\tan \theta ={\mu }_s$, and is called the angle of repose. For ${\mu }_s=0.5$, it is 26.57°. A heap of sand or gravel at a building site settles at its own angle of repose.

On a smooth slope of 30°, a block slides down with acceleration $g\sin {30}^{\circ }=4.905$ m/s². With kinetic friction of ${\mu }_k=0.2$, the friction $0.2mg\cos {30}^{\circ }$ acts up the slope, and the acceleration is $g(\sin {30}^{\circ }-0.2\cos {30}^{\circ })=3.206$ m/s².

Friction also changes a sum like the Atwood machine of section 7. A 4 kg block lies on a table, and a rope runs from it over a pulley at the table’s edge to a 2 kg block hanging below. Treat each block on its own. The hanging block has its weight down and the tension up, $m_{2}g-T=m_{2}a$. The block on the table has the tension pulling it towards the edge and kinetic friction holding it back, $T-{\mu }_k{m}_{1}g=m_{1}a$. Adding the two removes $T$.

$$a=\frac{m_{2}g-{\mu }_k{m}_{1}g}{m_1+m_2}$$

On a smooth table, ${\mu }_k=0$, and $a=2\times 9.81/6=3.27$ m/s², with a tension of $4\times 3.27=13.08$ N. With ${\mu }_k=0.2$, the friction is $0.2\times 4\times 9.81=7.848$ N, so $a=(19.62-7.848)/6=1.962$ m/s² and $T=4\times 1.962+7.848=15.70$ N. The tension is larger with friction, since the rope now has to drag the block against it. The hanging block’s own equation checks the answer, $T=2(9.81-1.962)=15.70$ N.

9. Drag and terminal speed

A body moving through air or water feels a drag force against its motion, which grows with its speed. For a fast body in air, such as a skydiver or a cricket ball, it is close to $\frac{1}{2}C\rho A{v}^2$, where $\rho$ is the density of the air, $A$ the area the body presents to the flow and $C$ a number for its shape, called the drag coefficient.¹⁵

A skydiver leaving an aircraft speeds up at first, at nearly $g$. As the speed grows, so does the drag, and the net force falls. When the drag equals the weight, the net force is zero, and by the first law the skydiver falls on at a steady speed, the terminal speed. Setting $\frac{1}{2}C\rho A{v}^2=mg$ gives $v=\sqrt{2mg/(C\rho A)}$. For a 75 kg skydiver spread flat, with $C=1.0$, $A=0.70$ m² and $\rho =1.21$ kg/m³, the terminal speed is 41.7 m/s, about 150 km/h. Opening the parachute multiplies the area many times, and the terminal speed drops to a few metres per second, slow enough to land. A raindrop falls at its own terminal speed for the same reason, which is why rain does not hurt.

10. How to solve a problem with Newton’s laws

Every problem in this chapter is solved with the same steps.¹³

1. Choose one body, and draw it on its own, as a dot or a box.
2. Draw every force that acts on that body, and only those, each as an arrow from the body. Weight always, then one force for each thing that touches the body, such as a normal force, a tension or friction.
3. Choose axes, one along the acceleration if you know its direction.
4. Write $F_{\text{net}}=ma$ along each axis, with the forces’ components.
5. If there are several bodies, repeat for each, and use what joins them, such as a rope’s shared acceleration, to tie the equations together.
6. Solve, and check that the answer has sensible units and size.

The drawing of step 2 is the free-body diagram, and it is where most marks are won or lost. Never draw a force that the body exerts on something else, only forces on the body. Never add “the force of motion” or “the centripetal force” as extra arrows. They are not separate forces.

Take two blocks of 2 kg and 3 kg side by side on a smooth floor, pushed with 10 N on the 2 kg block. Treated as one body of 5 kg, they accelerate at $10/5=2$ m/s². Now draw the 3 kg block alone. The only horizontal force on it is the push of the 2 kg block, which must be $3\times 2=6$ N. By the third law, the 3 kg block pushes back on the 2 kg block with 6 N, so the 2 kg block has a net force of $10-6=4$ N, which is $2\times 2$, as it should be.

11. The forces of nature

All the forces of this chapter, the normal force, tension, friction, drag and the push of a hand, are at bottom electric. They are the pushes and pulls between the electrons and nuclei of the atoms in the two surfaces, which resist being squeezed together. Only weight is different. It is gravity.

Physics knows four basic forces. Gravity acts between all masses. It is by far the weakest, but it adds up over huge masses and rules the motion of planets. The electromagnetic force acts between charges and holds atoms, molecules and solids together. The strong nuclear force holds protons and neutrons together in the nucleus. The weak nuclear force drives beta decay. The electromagnetic and weak forces have been shown to be two sides of one force, now called the electroweak force. For that work the 1979 Nobel Prize in Physics went jointly to Sheldon Glashow, Abdus Salam and Steven Weinberg, “for their contributions to the theory of the unified weak and electromagnetic interaction between elementary particles”.¹⁷ The chapters on nuclear and particle physics return to the four forces.

12. Past questions worked in full

The first is the 2019 Paper I, Question 3(b), for 5 marks. “Estimate the net force needed to accelerate (i) a 1000kg car at ½ g; (ii) a 200g apple at the same rate.”³ By the second law, $F=ma$ with $a=\frac{1}{2}\times 9.81=4.905$ m/s². For the car, $F=1000\times 4.905=\mathrm{4,905}$ N, about 4.9 kN. For the apple, convert 200 g to 0.200 kg first, and $F=0.200\times 4.905=0.981$ N. Add one line on why the answers differ, the mass, and the marks are full.

The second is the 2019 Paper I, Question 3(c), for 5 marks. “A vertical force is applied to a block of mass m that lies on a floor. What happens to the magnitude of the normal force on the block from the floor as magnitude F is increased from zero if force is (a) downward and (b) upward?”³ Draw the block with $mg$ down, $N$ up and $F$. The block is at rest, so the forces balance. For a downward $F$, $N=mg+F$, which grows steadily from $mg$. For an upward $F$, $N=mg-F$, which falls steadily from $mg$ and reaches zero when $F=mg$. Beyond that the block lifts off, $N$ stays zero, and the block accelerates upwards at $(F-mg)/m$.

The third is the 2026 Paper I, Question 3(b), for 10 marks. “A 6 kg block initially at rest is pulled to the right along a horizontal surface by a constant horizontal force of 12N. Find the speed of the block after it has moved 3m if the surfaces in contact have a coefficient of kinetic friction of 0.15.”⁵ Draw the block with the 12 N pull to the right, kinetic friction to the left, weight down and the normal force up. Vertically, $N=mg=6\times 9.81=58.86$ N. The friction is ${\mu }_{k}N=0.15\times 58.86=8.829$ N. The net force is $12-8.829=3.171$ N, so the acceleration is $3.171/6=0.5285$ m/s². From rest, $v^2=2as=2\times 0.5285\times 3=3.171$, so $v=1.781$ m/s. The chapter on work and energy gets the same answer by setting the net work, $3.171\times 3=9.513$ J, equal to the kinetic energy gained.

The fourth is the first half of the 2023 Paper I, Question 3(b). “What do you mean by the term ‘inertia’ in physics?”⁴ Inertia is a body’s tendency to keep its state of motion, at rest or at a steady velocity in a straight line, unless a net force acts on it. Its measure is the mass. Give one example, the passengers thrown forwards when a bus brakes. The rest of that question, on the rotational inertia of cylinders, is worked in the chapter on rotation.

13. Where Newton’s laws are at work

Road safety is Newton’s laws. A car’s tyres stop it through friction, and on a dry road a kinetic friction coefficient of about 0.7 gives a deceleration of $0.7\times 9.81=6.867$ m/s², so from 20 m/s, 72 km/h, a skidding car stops in ${20}^2/(2\times 6.867)=29.12$ m.¹⁴ On a wet road the coefficient is lower and the distance longer. A seat belt exists because of inertia. In a crash the car stops in a fraction of a second, and an unbelted passenger keeps moving at the car’s old speed until something, the dashboard or the windscreen, stops them. The belt stops them instead, over a longer distance and with a smaller force.

Lifts in tall buildings are designed with the apparent weight of section 5 in mind, so that they speed up and slow down gently enough to be comfortable. A rocket is the third law, gases pushed backwards and the rocket pushed forwards. Every bridge, building and tower stands because the forces on each part of it add up to zero, the first law applied to things that must not move.

14. Where Newton’s laws stand now

Newton’s laws are not the last word, but they are the right word for nearly everything on a human scale. At speeds close to that of light they give way to Einstein’s special relativity, in which the momentum is not simply $m𝐯$. For atoms and electrons they give way to quantum mechanics. Near very large masses, gravity is better described by general relativity. In each case, though, the newer theory gives Newton’s results back when speeds are low, bodies are large and gravity is weak.

So engineers still design bridges, cars and aircraft with Newton’s laws, and space agencies still plan the paths of spacecraft with them, adding the small corrections of relativity only where precision demands it, as in satellite navigation. The laws are among the best tested ideas in science.

15. What people get wrong

Many think a moving body needs a force to keep it moving. It needs a force only to change its motion. A body moving steadily in a straight line has zero net force on it.

Many think the two forces of a third-law pair cancel. They act on different bodies, so they never cancel each other.

Many think the normal force always equals the weight. It equals the weight only on a level surface with no other vertical forces and no vertical acceleration. Push down on the body, tilt the surface or accelerate the lift, and it changes.

Many mix up mass and weight. Mass in kilograms is the same everywhere. Weight in newtons depends on $g$.

Many draw the weight and the normal force on a book as a third-law pair. They act on the same body and are different kinds of force. The pair to the book’s weight is the book’s pull on the Earth.

Some take static friction as always ${\mu }_{s}N$. That is only its greatest value. Below it, static friction is just as large as it needs to be.

Some write $F=ma$ with a single force instead of the net force. Add every force on the body first.

16. What FPSC has asked

Three Part-II questions from CE-2016 to CE-2026 come from this chapter, and one surviving MCQ.² The 2026 question on whether a normal force can do work belongs to the chapter on work and energy.

Each is a direct use of the laws with a free-body diagram. Learn to draw one quickly, and the marks follow.

17. Facts and formulas to carry into the hall

• First law. With zero net force, a body stays at rest or moves at a steady velocity. The tendency is inertia, and its measure is mass.
• Second law. ${𝐅}_{\text{net}}=m𝐚=d𝐩/dt$, with 1 N = 1 kg m/s².
• Third law. The forces between two bodies are equal and opposite, and act on different bodies.
• Weight is $W=mg$. Surface gravity is 9.81 m/s² on Earth and 1.62 m/s² on the Moon.
• Apparent weight in a lift accelerating upwards at $a$ is $m(g+a)$, and zero in free fall.
• The normal force acts at right angles to a surface and is only as large as it needs to be.
• Friction obeys $f_s\le {\mu }_{s}N$ and $f_k={\mu }_{k}N$, with ${\mu }_k<{\mu }_s$. The angle of repose has $\tan \theta ={\mu }_s$.
• On a slope, the weight splits into $mg\sin \theta$ along it and $mg\cos \theta$ into it.
• In an Atwood machine, $a=(m_1-m_2)g/(m_1+m_2)$ and $T=2m_1{m}_{2}g/(m_1+m_2)$.
• Terminal speed comes where the drag, about $\frac{1}{2}C\rho A{v}^2$, equals the weight.
• The four basic forces are gravity, electromagnetism, and the strong and weak nuclear forces. Glashow, Salam and Weinberg shared the 1979 Nobel Prize for unifying the weak and electromagnetic forces.

18. What loses marks on Newton’s laws

A free-body diagram that shows forces the body exerts on others, or leaves out the weight, or adds a “force of motion”. Only forces on the body belong in it.

Using the weight where the normal force is needed in a friction sum. On a slope or with an extra vertical push, $N$ is not $mg$.

Forgetting to convert grams to kilograms, as with the 200 g apple, or km/h to m/s.

Stating the third law as “action and reaction cancel”. They act on different bodies.

Giving a force without units, or a mass in newtons.

19. A worked answer plan

Here is the plan for the 2019 Paper I, Question 3, made in the first five minutes.³ Its parts carry 10, 5 and 5 marks, so with 28.5 minutes of writing they get about 14, 7 and 7.

Parts (b) and (c) are short and sure, so do them carefully in about seven minutes each, and give part (a) the time it needs.

What to remember from this chapter

• A body keeps its state of motion unless a net force acts. That tendency is inertia, and mass measures it.
• The net force equals the rate of change of momentum, which is $m𝐚$ for a fixed mass.
• The forces between two bodies are equal and opposite, and act on different bodies, so they never cancel.
• Weight is $mg$, a force. Mass is the same everywhere, but weight depends on $g$.
• The normal force and static friction are only as large as they need to be, up to their limits.
• Solve every problem with a free-body diagram for each body and $F_{\text{net}}=ma$ along each axis.
• The laws hold as they stand only in inertial frames.

Check yourself

Answer each question in your own words before you open the answer.

1. A car moves at a steady 80 km/h along a straight road. What is the net force on it?

2. A horse pulls a cart. By the third law the cart pulls back on the horse equally. How can they move?

3. What does a scale read for a 60 kg person in a lift accelerating upwards at 2 m/s²?

4. A block rests on a slope. As the slope is tilted further, at what angle does it begin to slide if ${\mu }_s=0.5$?

5. Blocks of 5 kg and 3 kg hang over a light pulley. What is their acceleration?

6. From the chapter on kinematics. A 6 kg block has a net force of 3.171 N on it from rest. How fast is it going after 3 m?

7. From the chapter on circular motion. What supplies the centripetal force on a car on a flat roundabout, and which law says it must be there?