Force and Laws of Motion

1. Force — Balanced and Unbalanced Forces

A force is a push or a pull. It is a vector quantity — it has both magnitude and direction. Forces can change the speed, direction, or shape of an object. The SI unit of force is the Newton (N).

What can a force do?

• Move a stationary object (push a box on the floor).
• Stop a moving object (braking a car).
• Speed up a moving object (engine force on a car).
• Slow down a moving object (friction on a rolling ball).
• Change the direction of motion (a cricket bat deflects a ball).
• Change the shape of an object (pressing dough, compressing a spring).

Balanced forces: When two or more forces acting on an object have a net (resultant) force of zero, the forces are called balanced. Balanced forces do NOT change the state of motion of the object. The object either stays at rest or continues moving at the same speed in the same direction.

• Example 1: A book on a table. Gravity pulls it down; the normal reaction of the table pushes it up. Net force = 0, the book does not move.
• Example 2: A tug-of-war where both teams pull equally. The rope does not move.
• Example 3: A car moving at constant speed on a level road — engine force equals friction force, net force = 0.

Unbalanced forces: When the net force on an object is not zero, the forces are called unbalanced. An unbalanced force ALWAYS produces a change in speed or direction — that is, acceleration.

• Example 1: Kicking a football — the kick force is not balanced, so the ball accelerates from rest.
• Example 2: A car accelerating — the engine provides a force greater than friction, so there is a net forward force.
• Example 3: A stone falling freely — gravity acts downward and is not balanced (air resistance is usually neglected), so the stone accelerates downward.

Key rule: No unbalanced force means no change in motion. This is the fundamental idea behind Newton's First Law.

2. Newton's First Law of Motion

Statement (NCERT exact): "An object remains in a state of rest or of uniform motion in a straight line unless compelled to change that state by an applied unbalanced force."

This law is also called the Law of Inertia. Galileo first showed through inclined-plane experiments that moving objects do not need a continuous force to keep moving — they naturally tend to maintain their state. Newton formalised this observation into his first law.

What the law tells us:

1. A body at rest will remain at rest unless acted on by an unbalanced external force.
2. A body moving at constant speed in a straight line will continue doing so unless acted on by an unbalanced external force.
3. Any change in speed OR direction requires an unbalanced force.

Types of Inertia — Three Kinds:

1. Inertia of Rest: The tendency of a body to remain at rest when it is at rest.

• When a bus starts suddenly, passengers lurch backward — their bodies tend to remain at rest while the bus moves forward.
• Dust particles fly off when a carpet is beaten — the carpet moves suddenly but the dust particles tend to stay where they are.
• A coin placed on a card resting on a glass falls straight into the glass when the card is flicked away quickly — the coin stays in place due to inertia of rest while the card moves.
• Seeds fall straight down from a shaken tree — the branch is shaken but the seeds tend to remain at rest.

2. Inertia of Motion: The tendency of a body to remain in motion when it is already moving.

• When a bus brakes suddenly, passengers jerk forward — their bodies tend to keep moving while the bus slows down.
• An athlete runs a few steps beyond the finish line even after stopping effort — the body tends to continue moving.
• A ball rolling on a frictionless surface would roll forever — no force is needed to maintain motion.

3. Inertia of Direction: The tendency of a body to maintain its direction of motion.

• Water flies off tangentially from a spinning wet grindstone when it stops suddenly.
• A stone tied to a string and whirled in a circle flies off in a straight line tangent to the circle if the string breaks — it tends to keep moving in the direction it was travelling at that instant.
• When a car turns a corner, passengers feel pushed to the outside — their bodies tend to maintain the original direction of motion.

3. Inertia and Mass

Inertia is the natural resistance of an object to any change in its state of rest or of uniform motion. It is NOT a force — it is an inherent property of matter.

Different objects have different amounts of inertia. The physical quantity that measures inertia is mass. This is one of the most important statements in NCERT:

The more mass an object has, the more inertia it has, and the harder it is to change its state of motion.

Comparing inertia:

• A cricket ball has more inertia than a table-tennis ball — it is harder to set in motion and harder to stop with the same force.
• A loaded truck has far more inertia than an empty one — much larger force (braking force, engine force) is needed to change its state of motion.
• A heavy stone cannot be kicked as easily as a light pebble — greater mass, greater inertia.
• It is easier to push an empty shopping trolley than a full one — the full one has greater mass and thus greater inertia.

Mass vs Weight: Mass (measured in kg) is a measure of the amount of matter and is the measure of inertia. Weight (measured in N) is the gravitational force on the object. Inertia depends on mass, not on weight. An astronaut in space (weightless) still has mass and still has inertia — a floating wrench is still hard to set spinning quickly.

4. Newton's Second Law of Motion

Before stating the second law, we define momentum:

Momentum (p): The momentum of an object is defined as the product of its mass (m) and its velocity (v).

Momentum is a vector quantity in the direction of velocity. SI unit of momentum is kg m/s, which is the same as N s.

Momentum captures "quantity of motion". A heavy truck moving slowly can have the same momentum as a light car moving fast. Momentum tells us how hard it is to bring the object to rest.

Statement of Newton's Second Law (NCERT): "The rate of change of momentum of an object is directly proportional to the applied unbalanced force in the direction of force."

Mathematical derivation (NCERT method):

Let an object of mass m have initial velocity u and final velocity v after time t under the action of constant net force F.

• Initial momentum: p_i = mu
• Final momentum: p_f = mv
• Change in momentum: Delta p = mv minus mu = m(v minus u)
• Rate of change of momentum = Delta p / t = m(v minus u) / t
• Acceleration a = (v minus u) / t
• So rate of change of momentum = m x a

By Newton's Second Law, F is proportional to rate of change of momentum:

More generally, when mass can also change:

Definition of 1 Newton (N): One Newton is that force which produces an acceleration of 1 m/s^2 in a body of mass 1 kg. So 1 N = 1 kg x 1 m/s^2 = 1 kg m/s^2.

Key points about F = ma:

• F here is the net unbalanced force on the object.
• The direction of a is the same as the direction of F.
• If F = 0, then a = 0 — this reproduces Newton's First Law, so the First Law is a special case of the Second Law.
• For a given force, larger mass gives smaller acceleration (harder to accelerate heavy things).
• For a given mass, larger force gives larger acceleration.
• F = ma is valid for constant mass; otherwise use F = dp/dt.

5. Impulse

Sometimes a large force acts for a very short time — such as a bat hitting a ball, a hammer driving a nail, or two cars colliding. In such cases, we define Impulse:

Impulse is the product of force and time. It equals the change in momentum. SI unit: N s (same as kg m/s).

Impulse is useful because:

• When force is not constant during a collision, we can still compute the change in momentum = average force x time.
• The change in momentum (impulse) is fixed by the initial and final speeds — only the force and time can vary: large force for short time, OR small force for long time.

Everyday applications of Impulse:

• Catching a cricket ball: A fielder draws his hands back while catching. The ball's momentum must become zero (fixed change). By increasing time of contact, the force on hands decreases. This is why the catch hurts less.
• Car crumple zones and air bags: In a collision, the car must stop (change in momentum is fixed). Crumple zones and air bags increase the time of deceleration, so the force on passengers is much smaller, reducing injury.
• High-jump landing on foam: The foam increases contact time, reducing the impact force on the athlete's body.
• Karate chop: The expert strikes with a very short contact time. This means even a moderate impulse (change in momentum) produces a very large peak force — enough to break bricks.
• Jumping into a haystack vs a hard floor: The haystack gives during impact, increasing the time for momentum to reduce to zero, so the force is much smaller and injury is avoided.

6. Newton's Third Law of Motion

Statement (NCERT exact): "To every action, there is an equal and opposite reaction and they act on two different objects."

Four key ideas packed in this law:

1. Forces always occur in pairs. You cannot have a lone force; every force has a reaction.
2. Action and reaction are equal in magnitude.
3. Action and reaction are opposite in direction.
4. Action and reaction act on two different objects — NEVER on the same object. This is why they do NOT cancel each other.

Examples from daily life:

• Walking: You push the ground backward with your foot (action). The ground pushes you forward (reaction). This forward reaction force propels you.
• Swimming: A swimmer pushes water backward with hands and feet (action). Water pushes the swimmer forward (reaction).
• Rowing a boat: The oar pushes water backward (action). Water pushes the oar — and thus the boat — forward (reaction).
• Rocket propulsion: The rocket engine burns fuel and pushes hot exhaust gases backward at very high speed (action). The gases push the rocket forward (reaction). Crucially, this works in outer space where there is no air to "push against" — the rocket only needs to push its own exhaust gases.
• Gun recoil: When a gun is fired, it exerts a force on the bullet, sending it forward (action). The bullet exerts an equal and opposite force on the gun, causing the gun to recoil backward (reaction). Because the gun has much greater mass than the bullet, its recoil velocity is much smaller.
• Jumping off a boat: You push the boat backward (action). The boat pushes you forward (reaction). Since the boat has less friction with water, it slides backward as you jump forward.
• Inflated balloon released: Air rushes out of the balloon backward (action). The balloon moves forward (reaction). This is the same principle as rocket propulsion.

Common misconception — action and reaction DO NOT cancel: They are equal and opposite, but they act on two different bodies. The ground's push on you (forward) and your push on the ground (backward) do not cancel because they act on different objects. If they acted on the same object they would cancel, but they do not.

7. Conservation of Linear Momentum

Statement: "The total momentum of an isolated system of objects remains constant (is conserved) provided no external unbalanced force acts on the system."

Complete NCERT derivation from Newton's Third Law:

Consider two objects A (mass m_A, initial velocity u_A) and B (mass m_B, initial velocity u_B) that interact (collide) for a time interval t. Let their velocities after the collision be v_A and v_B.

During the collision:

• Force exerted by A on B = F_AB (action)
• Force exerted by B on A = F_BA (reaction)
• By Newton's Third Law: F_AB = minus F_BA

By Newton's Second Law applied to A and B separately (same time t for both):

• F_BA = m_A(v_A minus u_A) / t
• F_AB = m_B(v_B minus u_B) / t

Since F_AB = minus F_BA:

Multiplying both sides by t and rearranging:

Total momentum before collision = Total momentum after collision. This is the Law of Conservation of Linear Momentum.

Condition for validity: No external unbalanced force must act on the system. For example, in a gun-bullet system just at the instant of firing, gravity and the floor's normal force cancel (balanced external forces), so total momentum is conserved. Before firing, both gun and bullet are at rest, so total momentum = 0. After firing:

This means: m_bullet x v_bullet = minus M_gun x V_gun. The bullet goes forward, the gun recoils backward with equal and opposite momentum.

Applications of Conservation of Momentum:

• Gun recoil velocity calculation
• Rocket and jet propulsion
• Two objects colliding and sticking together (perfectly inelastic collision)
• Two objects bouncing off each other (elastic collision)
• An exploding bomb — fragments fly off so that total momentum remains zero (if bomb was at rest)

8. Key Formulae Card

Important relationships:

• Inertia is measured by mass — not by weight, not by velocity.
• Momentum has both magnitude and direction.
• Impulse = change in momentum — useful when force x time product is given or asked.
• Newton's First Law is a special case of the Second Law (when F = 0, a = 0).
• Conservation of Momentum is derived from Newton's Third Law.
• In an isolated system, if one object gains momentum, another loses the same amount in the opposite direction.

9. NCERT Worked Examples