"Laws of Motion"

About

This chapter explores the cause of motion — force. Building on the kinematics from Chapter 2, you will learn Newton's three laws of motion which govern how objects move when forces act on them. The concepts of inertia, momentum, impulse, and action-reaction pairs are essential for understanding everything from walking to rocket propulsion.

Key Concepts

3.1 Force

Force is an external agent that changes or tends to change the state of rest or uniform motion of a body, its direction, or its shape and size.

Effects of a force:

1. Change the state of rest or uniform motion (accelerate/decelerate)
2. Change the direction of motion
3. Change the shape and size of the body (deformation)
4. Cause rotation about an axis

Force is a vector quantity — it has both magnitude and direction.

3.2 Newton's First Law (Law of Inertia)

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

Inertia is the property of a body that resists any change in its state of rest or motion.

• Mass is the measure of inertia — greater the mass, greater the inertia.
• A body does NOT always move in the direction of the net external force. It does so only if it was initially at rest. If already in motion, the force changes the velocity vector, which may not be in the direction of the force.

3.3 Newton's Second Law

The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force.

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

For constant mass: $\vec{F}=m\vec{a}$

Momentum: $\vec{p}=m\vec{v}$

• Momentum is a vector quantity
• SI unit: kg⋅m/s

Impulse: $\vec{J}=\vec{F}\Delta t=\Delta \vec{p}$

• Impulse = Change in momentum
• SI unit: N⋅s (same as kg⋅m/s)

3.4 Conservation of Momentum

When no external force acts on a system, the total momentum of the system remains conserved.

${\vec{p}}_{\text{total, initial}}={\vec{p}}_{\text{total, final}}$

• Important for collision problems
• Explains recoil of a gun, rocket propulsion, etc.

3.5 Newton's Third Law

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

${\vec{F}}_{AB}=-{\vec{F}}_{BA}$

• Action and reaction act on different bodies, so they do NOT cancel each other.
• The forces are equal in magnitude and opposite in direction.
• They act simultaneously.

Examples:

• Walking: You push the ground backward; the ground pushes you forward.
• High jumper: Pushes down on the ground; ground pushes the jumper upward.
• Rocket propulsion: Rocket expels gases downward; gases push rocket upward.

3.6 Common Forces in Mechanics

INTEXT QUESTIONS 3.1

Q1. Is it correct to state that a body always moves in the direction of the net external force acting on it?

Ans: No, this statement is not always correct. A body moves in the direction of the net external force only when the body was initially at rest before the application of the force. If the body was already in motion, the net force changes the velocity but not necessarily in the direction of the force. The direction of motion depends on the resultant of the initial velocity and the change in velocity due to the applied force.

Q2. What physical quantity is a measure of the inertia of a body?

Ans: Mass is the physical quantity that measures the inertia of a body. Greater the mass, greater is the inertia, and more difficult it is to change the state of rest or motion of the body.

Q3. Can a force change only the direction of velocity of an object keeping its magnitude constant?

Ans: Yes, a force can change only the direction of velocity while keeping its magnitude constant. This happens when the force is always perpendicular to the velocity vector. The best example is uniform circular motion, where centripetal force acts perpendicular to velocity, changing only the direction but not the magnitude of velocity.

Q4. State the different types of changes which a force can bring in a body when applied on it.

Ans: A force can bring about the following changes in a body:

1. Change the state of rest or uniform motion (accelerate or decelerate)
2. Change the direction of motion
3. Change the shape and size of the body (deformation)
4. Cause rotation about an axis

INTEXT QUESTIONS 3.2

Ans: The object with smaller mass is moving faster.

Since momentum $p=mv$, if two objects have the same momentum: $m_1{v}_1=m_2{v}_2$

If $m_1<m_2$, then $v_1>v_2$. The object with smaller mass must have greater velocity to have the same momentum.

Q2. A boy throws up a ball with a velocity $v_0$. If the ball returns to the thrower with the same velocity, will there be any change in:

(a) Momentum of the ball?

Ans: Yes, there is a change in momentum.

• Initial momentum = $+m{v}_0$ (upward)
• Final momentum = $-m{v}_0$ (downward)
• Change in momentum = $m{v}_0-(-m{v}_0)=2m{v}_0$

(b) Magnitude of the momentum of the ball?

Ans: No, the magnitude of momentum remains the same ($m{v}_0$) as the speed is the same in both cases.

Q3. When a ball falls from a height, its momentum increases. What causes increase in its momentum?

Ans: The gravitational force acting on the ball causes the increase in its momentum. According to Newton's second law, $F=\frac{dp}{dt}$, the gravitational force $mg$ acts downward, continuously changing the momentum of the ball in the downward direction.

Q4. In which case will there be larger change in momentum of the object?

(a) A 150 N force acts for 0.1 s on a 2 kg object initially at rest. (b) A 150 N force acts for 0.2 s on a 2 kg object initially at rest.

Ans: Case (b) will have a larger change in momentum.

Using the impulse-momentum theorem: $\Delta p=F\Delta t$

• Case (a): $\Delta p=150\times 0.1=15$ kg⋅m/s
• Case (b): $\Delta p=150\times 0.2=30$ kg⋅m/s

Q5. An object is moving at a constant speed in a circular path. Does the object have constant momentum? Give reason for your answer.

Ans: No, the object does not have constant momentum. Although the speed is constant, the direction of velocity changes continuously in circular motion. Since momentum is a vector quantity ($\vec{p}=m\vec{v}$), and the direction of the velocity vector changes, the momentum vector also changes continuously even though its magnitude remains constant.

INTEXT QUESTIONS 3.3

Q2. Identify the action-reaction forces in each of the following situations:

(a) A man kicks a football.

Ans:

• Action: Force exerted by man's foot on the football.
• Reaction: Force exerted by the football on man's foot.

(b) Earth pulls the moon.

Ans:

• Action: Gravitational force exerted by Earth on the moon.
• Reaction: Gravitational force exerted by the moon on Earth.

(c) A ball hits a wall.

Ans:

• Action: Force exerted by the ball on the wall.
• Reaction: Force exerted by the wall on the ball.

Q3. "A person exerts a large force on an almirah to push it forward but he is not pushed backward because the almirah exerts a small force on him." Is the argument given here correct? Explain.

Ans: No, this argument is incorrect. According to Newton's third law, the almirah exerts exactly the same magnitude of force on the person as the person exerts on the almirah.

The person doesn't get pushed backward because:

1. There is sufficient friction between the person's feet and the floor to prevent sliding
2. The person can brace themselves against the applied force

On a slippery surface, the person would indeed be pushed backward when trying to push the almirah.

Terminal Exercise

3. A force of 20 N acts on a body of mass 5 kg for 4 seconds. If the body is initially at rest, calculate: (a) the acceleration produced, (b) the final velocity, (c) the change in momentum.

4. State the law of conservation of linear momentum. Use it to explain the recoil of a gun.

5. A bullet of mass 20 g is fired horizontally with a velocity of 150 m/s from a pistol of mass 2 kg. What is the recoil velocity of the pistol?

6. Explain why: (a) A cricketer moves his hands backward while catching a fast ball. (b) It is difficult for a fireman to hold a hose ejecting large amounts of water at high velocity.

7. A body of mass 5 kg is acted upon by two perpendicular forces of 8 N and 6 N. Find the magnitude and direction of the acceleration.

8. Distinguish between mass and weight. Which one is a measure of inertia?

9. Discuss the concept of apparent weight of a person in an elevator moving: (a) upward with acceleration, (b) downward with acceleration.

10. A ball of mass 0.15 kg is dropped from a height of 10 m, strikes the ground and rebounds to the same height. Find: (a) the impulse delivered to the ball by the ground, (b) the average force exerted by the ground on the ball if the time of contact is 0.01 s. (g = 10 m/s²)

Worked Examples

Example 1: Newton's Second Law

Problem: A constant force acts on a body of mass 10 kg and changes its velocity from 5 m/s to 25 m/s in 5 seconds. Find the magnitude of the force.

Solution:

• $m=10$ kg, $u=5$ m/s, $v=25$ m/s, $t=5$ s
• $a=\frac{v-u}{t}=\frac{25-5}{5}=4$ m/s²
• $F=ma=10\times 4=40$ N

Example 2: Impulse

Problem: A cricket ball of mass 150 g moving with a speed of 12 m/s is hit straight back to the bowler with a speed of 15 m/s. Find the impulse imparted to the ball.

Solution:

• $m=0.15$ kg, $u=12$ m/s, $v=-15$ m/s (negative because direction is reversed)
• Impulse = Change in momentum = $m(v-u)=0.15(-15-12)=0.15(-27)=-4.05$ N⋅s

The magnitude of impulse = 4.05 N⋅s. The negative sign indicates the impulse direction is opposite to the initial direction.

Example 3: Conservation of Momentum

Problem: A bullet of mass 10 g moving with velocity 400 m/s gets embedded in a freely suspended wooden block of mass 1.99 kg. Find the velocity acquired by the block.

Solution: By conservation of momentum: $m_b{u}_b+m_w(0)=(m_b+m_w)v$

$0.01\times 400=(0.01+1.99)v$

$4=2v⟹v=2\text{ m/s}$

Common Mistakes

1. Thinking action and reaction cancel: They act on DIFFERENT bodies, so they never cancel each other.
2. Assuming a body always moves in the direction of net force: True only if initially at rest. If already moving, the change in velocity may not be in the force's direction.
3. Forgetting momentum is a vector: Direction matters — use proper sign conventions.
4. Confusing mass and weight: Mass is the measure of inertia (scalar, constant everywhere); weight is the gravitational force (vector, varies with g).
5. Misapplying F = ma: This is valid only when mass is constant. The general form is $\vec{F}=\frac{d\vec{p}}{dt}$.

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