Chapter Introduction
Laws of Motion is arguably the most important chapter in Class 11 Physics. It introduces Newton's three laws, the concept of force, friction, and free body diagrams — tools you will use in every subsequent mechanics chapter. The board exam allocates 8–10 marks to this chapter, including numericals on F = ma, friction, and circular motion. Competitive exams test free body diagrams and banking of roads heavily. This chapter requires both conceptual clarity and rigorous numerical practice.
What this guide covers: Galileo and Aristotle, Newton's three laws, conservation of momentum, equilibrium, common forces in mechanics, friction (static, kinetic, rolling), circular motion (banking, centripetal force), and the method of solving problems using free body diagrams — all with worked numericals.
Key Definitions
- Inertia: The inherent property of a body to resist any change in its state of rest or uniform motion. Mass is the measure of inertia — larger mass means more inertia.
- Force: An external push or pull that can change the state of rest or motion of a body. It is a vector quantity; SI unit is newton (N = kg m s<sup>-2</sup>).
- Momentum (p): p = mv. Vector quantity; SI unit kg m s<sup>-1</sup>. Product of mass and velocity. It quantifies the "quantity of motion."
- Impulse: Change in momentum = F × Δt. It equals the area under the force-time graph. Vector; SI unit N s.
- Friction: The force that opposes relative motion between surfaces in contact. Arises from electromagnetic interactions at the atomic level.
- Normal reaction (N): The force exerted by a surface on a body, perpendicular to the surface. It adjusts to prevent interpenetration.
- Free body diagram (FBD): A diagram showing a body isolated from its surroundings with all external forces acting on it drawn as arrows.
- Equilibrium: When the net force on a body is zero (ΣF = 0), the body is in translational equilibrium. If net torque is also zero (Στ = 0), it is in complete equilibrium.
- Pseudo force: A fictitious force introduced in a non-inertial (accelerating) frame to make Newton's laws work. F<sub>pseudo</sub> = −ma<sub>frame</sub>.
- Conservative force: A force for which work done is path-independent and zero over a closed loop (e.g., gravity, spring force).
- Non-conservative force: A force for which work done depends on the path (e.g., friction).
Detailed Concepts
1. Galileo and the Aristotelian Fallacy
Aristotle believed that a force is needed to maintain motion — this is wrong. He observed that a ball stops rolling and concluded that without force, motion ceases. He missed that friction was the stopping force.
Galileo's inclined plane experiments showed that in the absence of friction, a body in motion continues indefinitely. His key insight: friction is what stops objects, not the absence of a driving force.
Galileo's thought experiment: 1. Ball released on one inclined plane rolls up the other to the same height. 2. If the second plane is less steep, the ball travels further to reach the same height. 3. If the second plane is horizontal, the ball never reaches the original height and keeps moving forever (if no friction).
This led directly to Newton's first law — the law of inertia.
2. Newton's Three Laws of Motion
| Law | Statement | Mathematical Form | Key Implication |
|---|---|---|---|
| First law (Law of Inertia) | A body continues in its state of rest or uniform motion unless acted upon by an external force. | If F<sub>net</sub> = 0, then v = constant | Defines inertia and force qualitatively |
| Second law | The rate of change of momentum is proportional to the applied force and is in the direction of the force. | F = ma or F = dp/dt | Quantitative definition of force |
| Third law | For every action, there is an equal and opposite reaction. | F<sub>AB</sub> = −F<sub>BA</sub> | Forces always occur in pairs; no isolated force |
Important notes on Newton's laws:
First law:
- It defines an inertial frame — a frame where a body with no net force has no acceleration.
- It is not a special case of the second law (F = 0 implies a = 0). The first law defines what a force-free state looks like.
Second law:
- F = ma is valid only in an inertial frame.
- F is the net external force, not just any force.
- It defines the unit of force: 1 N = force that gives 1 kg mass an acceleration of 1 m s<sup>-2</sup>.
- The second law in momentum form: F = dp/dt is more general (works even when mass changes, e.g., rocket).
Third law:
- Action and reaction act on different bodies, never on the same body. So they cannot cancel each other.
- Action and reaction are simultaneous — there is no time delay.
- The third law is valid in both inertial and non-inertial frames.
- The third law fails in certain relativistic situations involving magnetic forces between moving charges (not in Class 11 syllabus).
3. Impulse and Impulse-Momentum Theorem
Impulse J = F × Δt = change in momentum = m(v − u)
The impulse-momentum theorem: The impulse of a force equals the change in momentum.
Applications:
- Catching a cricket ball: A fielder pulls his hands back — increasing Δt reduces F, so the ball hurts less.
- Seat belts and airbags: They increase the time of impact (Δt), reducing the force on the body.
- Hammering a nail: A sharp blow has a very short Δt, producing a very large F.
Graphical interpretation: Impulse = area under the force-time graph.
4. Conservation of Linear Momentum
From Newton's second law: F = dp/dt. If F<sub>net</sub> = 0, then dp/dt = 0, so p = constant.
Principle: In the absence of an external force, the total linear momentum of an isolated system is conserved.
Mathematical form: m<sub>1</sub>v<sub>1</sub> + m<sub>2</sub>v<sub>2</sub> = m<sub>1</sub>u<sub>1</sub> + m<sub>2</sub>u<sub>2</sub> (before and after interaction)
Applications:
- Recoil of a gun: m<sub>g</sub>v<sub>g</sub> + m<sub>b</sub>v<sub>b</sub> = 0 (momentum before = 0)
- Rocket propulsion: continuous ejection of exhaust gases propels the rocket forward
- Explosion problems: total momentum before = total momentum after (if no external force)
- Two-body collision: momentum is conserved in all collisions (even when KE is not)
Proof from Newton's third law: Internal forces between two bodies are equal and opposite (3rd law). So they produce equal and opposite changes in momentum, which cancel. Only external forces can change the total momentum.
5. Equilibrium of a Particle
A particle is in equilibrium when the net force on it is zero.
Conditions:
- ΣF<sub>x</sub> = 0 and ΣF<sub>y</sub> = 0 (two dimensions)
- For a rigid body: ΣF = 0 (translational equilibrium) AND Στ = 0 (rotational equilibrium)
Lami's theorem (for three forces in equilibrium): F<sub>1</sub>/sin α<sub>1</sub> = F<sub>2</sub>/sin α<sub>2</sub> = F<sub>3</sub>/sin α<sub>3</sub> where α<sub>i</sub> is the angle opposite to force F<sub>i</sub>.
6. Types of Friction
| Property | Static Friction | Kinetic Friction | Rolling Friction |
|---|---|---|---|
| When it acts | No relative motion | Surfaces sliding | Body rolling on surface |
| Value | f ≤ μ<sub>s</sub>N (adjusts up to max) | f = μ<sub>k</sub>N (constant) | f = μ<sub>r</sub>N (very small) |
| Relative magnitude | μ<sub>s</sub>N (largest) | μ<sub>k</sub>N (smaller) | μ<sub>r</sub>N (smallest) |
| Direction | Opposes tendency of motion | Opposes actual motion | Opposes rolling motion |
| Depends on? | Normal reaction + nature of surfaces | Normal reaction + nature of surfaces | Normal reaction + nature of surfaces |
| Independent of | Contact area | Contact area | Contact area |
| Self-adjusting? | Yes (up to max) | No | No |
Key relations: μ<sub>s</sub> > μ<sub>k</sub> > μ<sub>r</sub>
Laws of limiting friction: 1. Limiting friction is independent of the area of contact (depends only on normal reaction). 2. Limiting friction is proportional to the normal reaction (f<sub>max</sub> = μ<sub>s</sub>N). 3. Kinetic friction is slightly less than limiting static friction. 4. Friction depends on the nature of surfaces in contact.
Methods of reducing friction:
- Lubrication (oil, grease, graphite)
- Polishing surfaces
- Using ball bearings (rolling friction < sliding friction)
- Streamlining (reducing air friction/drag)
- Using air cushions (hovercraft)
When friction is useful: Walking, writing with a pen, braking, gripping objects, matchstick lighting. Without friction, most daily activities would be impossible.
7. Circular Motion
Centripetal force: The inward force required to keep a body moving in a circle. F<sub>c</sub> = mv<sup>2</sup>/r.
Source of centripetal force in common situations:
| Situation | Source of centripetal force |
|---|---|
| Car on level curve | Friction between tyres and road |
| Car on banked curve | Component of normal reaction + friction |
| Satellite in orbit | Gravitational force |
| Electron in atom | Electrostatic force |
| Stone on string | Tension in the string |
Banking of roads: At a curved road, the road is tilted inward so that a component of normal reaction provides the centripetal force.
Derivation (frictionless case):
- N cos θ = mg (vertical equilibrium)
- N sin θ = mv<sup>2</sup>/r (centripetal force)
- Dividing: tan θ = v<sup>2</sup>/(rg)
- Optimum speed: v = √(rg tan θ)
With friction (maximum and minimum safe speeds on a banked road):
- v<sub>max</sub> = √(rg(μ<sub>s</sub> + tan θ)/(1 − μ<sub>s</sub> tan θ))
- v<sub>min</sub> = √(rg(μ<sub>s</sub> − tan θ)/(1 + μ<sub>s</sub> tan θ))
Bending of a cyclist: The cyclist leans inward so that the horizontal component of the normal reaction provides the centripetal force. At the correct angle: tan θ = v<sup>2</sup>/(rg).
8. Solving Problems in Mechanics — Free Body Diagrams
Steps to draw an FBD: 1. Identify the body whose motion you want to analyse. 2. Isolate it from all surroundings. 3. Draw all external forces acting on it as arrows:
4. Choose a convenient coordinate system (often along and perpendicular to the surface). 5. Write ΣF = ma along each axis.
- Weight (mg, always downward)
- Normal reaction (N, perpendicular to surface, away from surface)
- Friction (f, along surface, opposing motion or tendency)
- Tension (T, along the string, away from the body)
- Applied forces
Common force pairs in mechanics:
| Force | Direction | Value/Magnitude |
|---|---|---|
| Weight (W) | Vertically downward | mg |
| Normal (N) | Perpendicular to surface, outward | Adjusts to prevent penetration |
| Tension (T) | Along the string, away from body | Same throughout a massless string |
| Spring force | Along spring, opposing deformation | F = kx (Hooke's law) |
| Static friction | Along surface, opposing tendency | f ≤ μ<sub>s</sub>N |
| Kinetic friction | Along surface, opposing motion | f = μ<sub>k</sub>N |
Tips for FBD:
- Never include forces on other bodies in the FBD of your chosen body.
- Internal forces of the body are not shown.
- If two bodies are connected by a massless string, tension is the same at all points.
- For a pulley problem, draw separate FBDs for each mass.
Important Numericals
Numerical 1: F = ma
A body of mass 5 kg starts from rest under a force of 10 N. Find the velocity after 4 s and the distance covered.
Numerical 2: Friction on an Inclined Plane
A body of mass 10 kg is placed on a rough incline of 30°. If μ<sub>s</sub> = 0.5, will the body slide? What is the friction force? (g = 10 m s<sup>-2</sup>)
Numerical 3: Banking of Roads
A circular road of radius 50 m is banked at 30°. Find the optimum speed. (g = 10 m s<sup>-2</sup>)
Numerical 4: Conservation of Momentum
A gun of mass 5 kg fires a bullet of mass 0.01 kg with a muzzle velocity of 400 m s<sup>-1</sup>. Find the recoil velocity of the gun.
Numerical 5: Pulley System (Atwood Machine)
Two masses of 3 kg and 5 kg are connected by a massless string over a frictionless pulley. Find the acceleration and tension. (g = 10 m s<sup>-2</sup>)
Numerical 6: Impulse
A ball of mass 0.2 kg moving at 10 m s<sup>-1</sup> is caught and brought to rest in 0.1 s. Find the average force exerted by the hands.
NCERT Exercise Reference
| NCERT Exercise | Topic | Marks to Expect |
|---|---|---|
| 5.1–5.4 | Newton's first and second law | 1–2 marks (definitions) |
| 5.5–5.7 | Third law, conservation of momentum | 2–3 marks (proof + numerical) |
| 5.8–5.11 | Friction (types, incline) | 2–3 marks (numerical) |
| 5.12–5.14 | Circular motion, banking | 3–5 marks (derivation + numerical) |
| 5.15–5.17 | FBD, pulley systems | 3–5 marks (numerical) |
Common Mistakes
1. Adding action and reaction on the same body: They act on different bodies. Do not include both in the same FBD. The book pushes the table down (action on table); the table pushes the book up (reaction on book). 2. Confusing mass and weight: Mass (kg) is constant; weight (N = mg) changes with g (e.g., on the Moon g ≈ 1.67 m s<sup>-2</sup>, so weight is about 1/6th). 3. Taking friction as always μN: Static friction adjusts from 0 to μ<sub>s</sub>N; only the maximum (limiting) value equals μ<sub>s</sub>N. For a body at rest on a horizontal surface with no applied force, friction = 0. 4. Forgetting to resolve forces on an incline: Always resolve into components along and perpendicular to the plane (mg sin θ along the plane, mg cos θ perpendicular). 5. Applying F = ma in non-inertial frames without pseudo force: In an accelerating lift or car, you must add a pseudo force (−ma<sub>frame</sub>) to apply Newton's laws. 6. Wrong direction of centripetal force: Centripetal force is always directed towards the centre, not away from it. "Centrifugal force" exists only in rotating frames. 7. Normal reaction is not always mg: On an incline, N = mg cos θ. In a lift accelerating up, N = m(g + a). Always calculate N from the FBD.
Exam Tips
- FBD is the single most important skill — every numerical in this chapter starts with drawing one. Practise 10+ FBD problems of different types (incline, pulley, horizontal, circular).
- Newton's second law numericals (F = ma) are straightforward marks — do not lose them to sign errors. Always write the equation for each axis.
- Friction on incline problems are very common: practise resolving forces along and perpendicular to the plane. Know how to determine whether the body slides or stays.
- Banking of roads derivation is a 3–5 mark favourite — memorise the formula and be able to derive it from the FBD.
- Conservation of momentum is often asked in 2-mark questions — remember it applies to isolated systems only (no external force).
- Pulley (Atwood machine) problems are classic: draw separate FBDs for each mass, write equations, add them to eliminate tension.
- Always state the direction of forces clearly in your answer. Use "→", "↑", "along the plane", "towards the centre" etc.
- For long-answer numericals, draw the FBD as part of your solution — it carries marks.
- Impulse problems: Remember J = FΔt = Δp. If you know the change in momentum and the time, you can find the average force.
Quick Revision Notes
- Newton's 1st law: No net force → no change in motion (defines inertia). Mass measures inertia.
- Newton's 2nd law: F = ma = dp/dt (quantitative definition of force). Valid in inertial frames.
- Newton's 3rd law: Action = −Reaction, on different bodies, simultaneous.
- Impulse: J = FΔt = Δp. Area under F-t graph. Longer Δt → smaller F.
- Momentum conservation: If F<sub>ext</sub> = 0, then Σp = constant. Applies to all collisions.
- Friction: Static (adjusts, max = μ<sub>s</sub>N) > Kinetic (constant, μ<sub>k</sub>N) > Rolling (μ<sub>r</sub>N).
- Friction is independent of contact area but depends on normal reaction and surface nature.
- Banking: tan θ = v<sup>2</sup>/(rg); optimum speed v = √(rg tan θ).
- FBD steps: Isolate body → draw all external forces → write ΣF = ma per axis.
- Incline: Along plane = mg sin θ; perpendicular = mg cos θ; N = mg cos θ.
- Centripetal force: F = mv<sup>2</sup>/r, always towards centre, provided by friction/normal/tension/gravity.
- Pulley: separate FBDs for each mass; tension same in massless string; a is same for both masses.
- Non-inertial frame: add pseudo force F<sub>pseudo</sub> = −ma<sub>frame</sub>.