Laws of Motion

1. What this chapter covers

Chapters 2 and 3 described how things move without once asking why. This chapter asks why.

Not in the 2026-27 chapter or syllabus

Both appear in most coaching handouts for this chapter. The relationship behind the angle of repose is still worth a look once — it is in the FAQs below — but do not spend revision time on it.

What CBSE does list: force and inertia, all three laws, momentum, impulse, conservation of linear momentum, equilibrium of concurrent forces, static and kinetic friction, laws of friction, rolling friction, lubrication, and the dynamics of circular motion for a vehicle on a level road and on a banked road.

2. Aristotle's fallacy — the question that took two thousand years

Does a body need a force to keep moving?

The chapter opens with that question and immediately says it "took ages to answer". That is not a throwaway line — it is the point of the section.

Aristotle (384 BC – 322 BC) said yes. If a body is moving, something external is required to keep it moving. An arrow keeps flying, he reasoned, because the air behind it keeps pushing it along.

And this is not a stupid view. The textbook is careful to say so: it is the natural conclusion from ordinary experience. A child dragging a toy car on a string knows perfectly well that letting go means the car stops. Every observation an ordinary person makes supports Aristotle.

Galileo, in the seventeenth century, saw what was wrong with it — and the chapter calls his answer the foundation of Newtonian mechanics and the birth of modern science.

The flaw is that the everyday observation is contaminated. The toy car stops because of friction, not because motion needs feeding. Remove friction and the need for a force disappears with it.

Why this matters for your answers: when a question asks why a body in motion eventually stops, the answer is never "because the force ran out". It is that some external force — usually friction — acted on it.

3. The law of inertia, and Newton's first law

Galileo's argument was a thought experiment about inclines.

A ball rolling down one incline rolls up an opposite incline to nearly the same height. Flatten the second incline, and the ball travels further to reach that same height. Keep flattening it, and the distance keeps growing.

Take the limit. On a perfectly flat, frictionless surface the ball would never reach that height, so it would roll on forever.

Newton's first law states it directly:

A body continues in its state of rest, or of uniform motion in a straight line, unless compelled by an external force to change that state.

Inertia is the name for that tendency to resist a change of state, and mass is its measure — more mass, more inertia.

The first law is a definition, not just a claim. It defines what a force is — the thing that changes a state of motion. Without it, "force" would be circular.

4. The second law and momentum

Linear momentum is mass times velocity, and it is a vector pointing along the velocity:

$\mathbf{p}=m\mathbf{v}$

Newton's second law is stated in terms of momentum, and this matters:

$\mathbf{F}=\frac{d\mathbf{p}}{dt}$

When the mass is constant this reduces to the familiar form:

$\mathbf{F}=\frac{d(m\mathbf{v})}{dt}=m\frac{d\mathbf{v}}{dt}=m\mathbf{a}$

Write the momentum form when a question says "state Newton's second law". $F=ma$ is the special case; the momentum form is the law. It is also the only form that survives when mass changes, as it does for a rocket burning fuel.

The SI unit of force is the newton: $1\text{N}=1{\text{kg m s}}^{-2}$.

Three consequences worth being explicit about:

• Force causes acceleration, not velocity. A body can be moving fast with no force on it at all.
• Zero net force means zero acceleration — not zero velocity.
• The same force gives a heavier body less acceleration.

The law is a vector equation, so it holds component by component. That is what lets you resolve forces along two axes and solve each direction separately, exactly as in Chapter 3.

5. Impulse

Sometimes a large force acts for a very short time and you cannot easily measure either one. A ball bouncing off a wall is the chapter's example: the contact lasts a moment, but the force is big enough to reverse the ball's momentum.

Impulse sidesteps the problem by combining them:

$\mathbf{J}=\mathbf{F}\Delta t=\Delta \mathbf{p}=m\mathbf{v}-m\mathbf{u}$

It is a vector, measured in N s, which is the same thing as kg m s⁻¹.

The useful reading is backwards. For a fixed change in momentum, the force and the contact time trade off against each other. Stretch the time, and the force must fall.

In every case the momentum change is identical. Only the time is being bought.

6. The third law, and the trap

To every action there is always an equal and opposite reaction.

The trap is thinking action-reaction pairs cancel. They never do, and the reason is one line:

They act on different bodies.

Two forces can only cancel if they act on the same body. An action-reaction pair, by definition, does not.

Worked through: you push the ground backward with your foot; the ground pushes your foot forward. The forward push is on you, the backward push is on the Earth. Only the force on you decides how you accelerate — so you move.

Same structure for a rocket (gas pushed back, rocket pushed forward) and a gun (bullet forward, gun backward).

When answering, name both bodies. "The ball pushes the wall, the wall pushes the ball" earns the point; "action and reaction are equal and opposite" on its own does not show you know which body each force acts on.

7. Conservation of momentum

This is not a fourth law — it follows from the second and third together. The chapter derives it from a gun firing a bullet.

1. The gun exerts force $\mathbf{F}$ on the bullet, so by the third law the bullet exerts $-\mathbf{F}$ on the gun.
2. The two forces act for the same interval $\Delta t$.
3. By the second law, $\mathbf{F}\Delta t$ is the bullet's momentum change and $-\mathbf{F}\Delta t$ is the gun's.
4. They are equal and opposite, so the total momentum change is zero.

${\mathbf{p}}_{\text{total}}=\text{constant}\text{when the net external force is zero}$

The condition is what gets dropped in answers. Momentum is conserved when the net external force is zero. Internal forces always come in third-law pairs and cancel, which is why they can never change a system's total momentum.

This is why a gun recoils, why a rocket works without pushing against anything, and why the fragments of an exploding shell have a combined momentum equal to the shell's.

8. Equilibrium of concurrent forces

Concurrent forces all act at the same point. A particle is in equilibrium when they sum to zero:

$\sum \mathbf{F}=0⟺\sum F_x=0\text{ and }\sum F_y=0$

Equilibrium does not mean at rest. It means zero acceleration, which includes moving at constant velocity. A box sliding at steady speed across a floor is in equilibrium.

For three concurrent forces in equilibrium, each one balances the resultant of the other two — which is why such problems are usually solved by resolving along two convenient perpendicular directions and setting each sum to zero.

9. Friction

Friction is the component of the contact force parallel to the surface, and it opposes relative motion.

The laws of friction

$(f_s{)}_{max}={\mu }_{s}N{f}_k={\mu }_{k}N{\mu }_s>{\mu }_k$

Two properties the chapter states explicitly:

• The maximum static friction is independent of the area of contact.
• ${\mu }_s$ depends only on the nature of the two surfaces in contact.

The most common error in this chapter is writing $f_s={\mu }_{s}N$. Static friction is not equal to ${\mu }_{s}N$ — it is at most ${\mu }_{s}N$:

$f_s\le {\mu }_{s}N$

Push a heavy crate gently and it does not move, so friction exactly matches your push. Push harder and friction rises to match. Only at the instant it starts to slide has friction reached ${\mu }_{s}N$.

Why ${\mu }_s>{\mu }_k$ has a visible consequence: a box is harder to start moving than to keep moving. Once it breaks free, the opposing force drops, which is why a crate often lurches forward the moment it gives.

Rolling friction, and why the wheel mattered

Rolling friction is far smaller than sliding friction, which the chapter calls the reason the wheel was a major milestone in human history. Its origin is that the surfaces deform slightly during rolling, giving a finite contact area rather than a point.

Reducing friction — lubrication

CBSE lists lubrication explicitly. The chapter gives three methods:

Friction in a machine dissipates power as heat, which is the practical reason all three exist.

10. Circular motion: level roads and banked roads

Chapter 3 showed that circular motion needs an acceleration $v^2/R$ pointing at the centre. This chapter supplies the force that produces it.

$F_c=\frac{m{v}^2}{R}=m{\omega }^{2}R$

Centripetal force is not a new kind of force. It is a role. Whatever real force happens to point at the centre is playing it — tension for a stone on a string, gravity for a satellite, friction for a car on a flat road.

A car on a level road

Three forces act: weight $mg$, normal reaction $N$, and friction $f$.

Vertically there is no acceleration, so $N=mg$. Horizontally, friction is the only thing available to supply the centripetal force:

$\frac{m{v}^2}{R}\le {\mu }_{s}N={\mu }_{s}mg⟹v_{max}=\sqrt{{\mu }_{s}Rg}$

Notice the mass cancels. The maximum safe cornering speed does not depend on how heavy the car is.

A car on a banked road

Raise the outer edge and the normal reaction tilts inward, so it now has a horizontal component pointing at the centre. Part of the job is taken off friction.

At one particular speed the banking does the whole job and no friction is needed at all — the optimum speed:

$v_0=\sqrt{Rg\tan \theta }$

At this speed the tyres suffer no sideways wear, which is exactly why roads and racetracks are banked.

Allowing friction to help as well gives the maximum permissible speed before slipping:

$v_{max}=\sqrt{Rg\left(\frac{{\mu }_s+\tan \theta }{1-{\mu }_s\tan \theta }\right)}$

Check the formula against the level road. Set $\theta =0$, so $\tan \theta =0$, and it collapses to $v_{max}=\sqrt{{\mu }_{s}Rg}$ — the level-road result. That is a fast way to confirm you have written it correctly.

11. Solving problems in mechanics — the method

The chapter closes with a procedure, and it is worth following exactly rather than improvising.

1. Pick the body you are analysing, and draw it alone.
2. Draw every external force acting on that body — and nothing else. This is the free-body diagram.
3. Choose axes, usually along and perpendicular to the acceleration.
4. Resolve every force onto those axes.
5. Apply $F=ma$ separately in each direction.

Step 2 is where marks are lost. Only forces acting on the chosen body belong in its diagram. Forces that body exerts on other things belong in their diagrams. Mixing the two is what makes connected-body problems collapse.

For bodies connected by a string, draw a separate diagram for each body, note that the tension is the same throughout a light inextensible string, and that both bodies share the same magnitude of acceleration.

Summary

• The chapter's opening question — does a moving body need a force to keep moving? — took two thousand years to answer, and the natural answer is wrong.
• Aristotle said motion needs a sustaining force. Galileo saw that friction was disguising the truth.
• Newton's first law defines force as whatever changes a state of rest or uniform motion. Inertia is the resistance to that change, and mass measures it.
• The second law is $\mathbf{F}=d\mathbf{p}/dt$; $F=ma$ is only its constant-mass special case.
• Force causes acceleration, not velocity. Zero net force means zero acceleration, not zero speed.
• Impulse $\mathbf{J}=\mathbf{F}\Delta t=\Delta \mathbf{p}$ lets you trade force against contact time — the physics behind airbags and pulling your hands back to catch.
• Action-reaction pairs never cancel, because they act on different bodies. Name both bodies in your answer.
• Conservation of momentum follows from the second and third laws together, and holds when the net external force is zero.
• Equilibrium means zero net force, which includes moving at constant velocity — not only being at rest.
• $f_s\le {\mu }_{s}N$ and $f_k={\mu }_{k}N$, with ${\mu }_s>{\mu }_k$. Static friction adjusts; it is not fixed at ${\mu }_{s}N$.
• Maximum static friction is independent of the contact area, and ${\mu }_s$ depends only on the surfaces.
• Rolling friction is much smaller than sliding friction — the reason the wheel mattered. Lubricants, ball bearings and air cushions all reduce friction.
• Centripetal force is a role, not a new force. On a level road friction plays it, giving $v_{max}=\sqrt{{\mu }_{s}Rg}$, independent of mass.
• Banking tilts the normal reaction inward. At $v_0=\sqrt{Rg\tan \theta }$ no friction is needed at all.
• Draw a free-body diagram showing only the forces acting on the chosen body, then apply $F=ma$ along each axis.
• Angle of repose and the conical pendulum are in neither the 2026-27 chapter nor the CBSE syllabus.