This chapter explores the causes of motion, introducing the concept of force and Newton’s three laws of motion. It covers balanced and unbalanced forces, inertia and mass, momentum, the mathematical formulation of laws of motion, and their applications in everyday life. Understanding these laws helps us explain why objects move or stay at rest and how forces affect their motion.
Introduction
In the previous chapter, we described the motion of an object along a straight line in terms of its position, velocity, and acceleration. We saw that such motion can be uniform or non-uniform. However, we have not yet discovered what causes the motion. Why does the speed of an object change with time? Do all motions require a cause? If so, what is the nature of this cause?
For many centuries, the problem of motion and its causes had puzzled scientists and philosophers. A ball on the ground, when given a small hit, does not move forever. Such observations suggest that rest is the “natural state” of an object. This remained the belief until Galileo Galilei and Isaac Newton developed an entirely different approach to understand motion.
In our everyday life, we observe that some effort is required to put a stationary object into motion or to stop a moving object. We ordinarily experience this as a muscular effort and say that we must push, hit, or pull an object to change its state of motion. The concept of force is based on this push, hit, or pull.
What is Force?
Force is difficult to define directly because no one has seen, tasted, or felt a force. However, we always see or feel the effect of a force. It can only be explained by describing what happens when a force is applied to an object.
Ways of applying force:
- Pushing objects
- Hitting objects
- Pulling objects
All these are ways of bringing objects in motion. They move because we make a force act on them.
Effects of Force
A force can be used to:
1. Change the magnitude of velocity (make the object move faster or slower)
- Example: Pushing a trolley makes it move faster
- Example: Applying brakes slows down a bicycle
2. Change the direction of motion
- Example: Hitting a ball with a hockey stick changes its direction
3. Change the shape and size of objects
- Example: A spring expands when force is applied
- Example: A spherical rubber ball becomes oblong when we apply force
8.1 Balanced and Unbalanced Forces
Consider a wooden block on a horizontal table. Two strings X and Y are tied to the two opposite faces of the block.
Situation 1: If we apply a force by pulling string X, the block begins to move to the right.
Situation 2: If we pull string Y, the block moves to the left.
Situation 3: If the block is pulled from both sides with equal forces, the block will not move.
Balanced Forces
Definition: When two or more forces acting on an object are equal in magnitude and opposite in direction, they are called balanced forces.
Important: Balanced forces do not change the state of rest or of motion of an object.
Example: A block pulled equally from both sides remains stationary.
Unbalanced Forces
Definition: When two opposite forces of different magnitudes pull the block, the block begins to move in the direction of the greater force. These forces are called unbalanced forces.
Important: An unbalanced force acting on an object brings it in motion.
Example: If children push a box harder on one side than friction opposes it, the box starts moving.
Understanding Through Examples
Example 1: Pushing a Box
When children try to push a box on a rough floor:
- Small push: The box does not move because friction acts in a direction opposite to the push, balancing the pushing force.
- Moderate push: The box still does not move because friction still balances the pushing force.
- Hard push: The pushing force becomes bigger than the friction force, creating an unbalanced force. The box starts moving.
Example 2: Riding a Bicycle
When we stop pedalling, the bicycle begins to slow down. This is because of friction forces acting opposite to the direction of motion. To keep the bicycle moving, we have to start pedalling again.
Important Concept:
- An object moves with uniform velocity when the forces (pushing force and frictional force) acting on it are balanced and there is no net external force.
- If an unbalanced force is applied, there will be a change in its speed or direction of motion.
- To accelerate the motion of an object, an unbalanced force is required.
- If this force is removed completely, the object would continue to move with the velocity it has acquired till then.
8.2 First Law of Motion
Galileo’s Experiment with Inclined Planes
Galileo observed the motion of objects on an inclined plane and deduced important conclusions:
Observation 1: When a marble rolls down an inclined plane, its velocity increases.
Observation 2: When it climbs up, its velocity decreases.
Observation 3: Consider a marble resting on an ideal frictionless plane inclined on both sides.
Galileo’s Argument:
- When the marble is released from the left side, it would roll down the slope and go up on the opposite side to the same height from which it was released.
- If the inclinations of the planes on both sides are equal, the marble will climb the same distance that it covered while rolling down.
- If the angle of inclination of the right-side plane were gradually decreased, the marble would travel further distances till it reaches the original height.
- If the right-side plane were ultimately made horizontal (slope reduced to zero), the marble would continue to travel forever trying to reach the same height.
Conclusion:
- The unbalanced forces on the marble in this case are zero.
- An unbalanced (external) force is required to change the motion of the marble.
- But no net force is needed to sustain the uniform motion of the marble.
In practical situations, it is difficult to achieve zero unbalanced force because of friction. The effect of friction may be minimised by using a smooth marble, smooth plane, and providing lubricant.
Newton’s First Law of Motion
Newton further studied Galileo’s ideas on force and motion and presented three fundamental laws that govern the motion of objects. These are known as Newton’s Laws of Motion.
Newton’s First Law of Motion:
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 force.
In other words, all objects resist a change in their state of motion. The tendency of undisturbed objects to stay at rest or to keep moving with the same velocity is called inertia. This is why the first law of motion is also known as the law of inertia.
Applications of First Law of Motion
1. Travelling in a Motorcar:
When brakes are applied:
- We tend to remain at rest with respect to the seat.
- The car slows down but our body tends to continue in the same state of motion because of its inertia.
- We tend to fall forward.
- A sudden application of brakes may cause injury by impact or collision.
- Safety belts are worn to prevent such accidents. They exert a force on our body to make the forward motion slower.
When bus starts suddenly:
- We tend to fall backwards.
- The sudden start of the bus brings motion to the bus and our feet in contact with the floor.
- But the rest of our body opposes this motion because of its inertia.
When motorcar makes a sharp turn:
- We tend to get thrown to one side.
- We tend to continue in our straight-line motion.
- When an unbalanced force is applied by the engine to change direction, we slip to one side due to the inertia of our body.
2. Carom Coins Activity:
- Make a pile of similar carom coins on a table.
- Give a sharp horizontal hit at the bottom of the pile using another coin.
- If the hit is strong enough, the bottom coin moves out quickly.
- Once the lowest coin is removed, the inertia of the other coins makes them fall vertically on the table.
3. Coin and Card Activity:
- Set a five-rupee coin on a stiff card covering an empty glass tumbler.
- Give the card a sharp horizontal flick with a finger.
- If done fast, the card shoots away, allowing the coin to fall vertically into the glass tumbler due to its inertia.
- The inertia of the coin tries to maintain its state of rest even when the card flows off.
4. Water-Filled Tumbler Activity:
- Place a water-filled tumbler on a tray.
- Hold the tray and turn around as fast as you can.
- The water spills.
Reason: The water tends to continue in its state of rest or motion due to inertia while the tray (and tumbler) changes direction suddenly.
Note: A groove is provided in a saucer for placing a tea cup. It prevents the cup from toppling over in case of sudden jerks.
Galileo Galilei (1564-1642)
Galileo Galilei was born on 15 February 1564 in Pisa, Italy. From childhood, he had interest in mathematics and natural philosophy. His father wanted him to become a medical doctor. Galileo enrolled for a medical degree at the University of Pisa in 1581 which he never completed because of his real interest in mathematics.
Important Contributions:
1. ‘The Little Balance’ (La Balancitta) – 1586:
- Described Archimedes’ method of finding relative densities using a balance.
2. ‘De Motu’ – 1589:
- Presented theories about falling objects using an inclined plane to slow down the rate of descent.
3. Professor at University of Padua – 1592:
- Continued observations on the theory of motion.
- Through study of inclined planes and pendulum, formulated the correct law for uniformly accelerated objects: distance moved is proportional to the square of time taken.
4. Telescopes and Astronomical Discoveries:
- Developed telescopes with better optical performance.
- Discovered mountains on the moon, milky way made of tiny stars, and four small bodies orbiting Jupiter.
- Observed sunspots.
- Argued that all planets must orbit the Sun (not the earth).
5. Pendulum Clock – 1640:
- Designed the first pendulum clock.
8.3 Inertia and Mass
All the examples and activities illustrate that there is a resistance offered by an object to change its state of motion. If it is at rest, it tends to remain at rest; if it is moving, it tends to keep moving. This property of an object is called its inertia.
Question: Do all bodies have the same inertia?
No. Different objects have different amounts of inertia.
Examples of Different Inertia
1. Empty box vs. Box full of books:
- It is easier to push an empty box than a box full of books.
- The box full of books has more inertia.
2. Football vs. Stone:
- If we kick a football, it flies away.
- If we kick a stone of the same size with equal force, it hardly moves.
- We may get an injury in our foot!
- The stone has more inertia than the football.
3. Five-rupee coin vs. One-rupee coin:
- In the coin and card activity, if we use a one-rupee coin instead of a five-rupee coin, a lesser force is required.
- The five-rupee coin has more inertia.
4. Cart vs. Train:
- A force just enough to cause a small cart to pick up a large velocity will produce negligible change in the motion of a train.
- The train has much more inertia than the cart.
Observation: Heavier or more massive objects offer larger inertia.
Relationship Between Inertia and Mass
Definition: Inertia is the natural tendency of an object to resist a change in its state of motion or rest.
Important: The mass of an object is a measure of its inertia.
Quantitatively:
- Greater mass = Greater inertia
- Smaller mass = Smaller inertia
Questions (8.3) and Solutions
Q1. Which object has more inertia:
(a) A rubber ball and a stone of the same size?
(b) A bicycle and a train?
(c) A five-rupees coin and a one-rupee coin?
Solution:
(a) Stone has more inertia than a rubber ball of the same size because the stone has greater mass.
(b) Train has more inertia than a bicycle because the train has much greater mass.
(c) Five-rupee coin has more inertia than a one-rupee coin because the five-rupee coin has greater mass.
Reason: Inertia is directly proportional to mass. Objects with greater mass have greater inertia.
Q2. In the following example, try to identify the number of times the velocity of the ball changes:
“A football player kicks a football to another player of his team who kicks the football towards the goal. The goalkeeper of the opposite team collects the football and kicks it towards a player of his own team”.
Also identify the agent supplying the force in each case.
Solution:
The velocity of the ball changes 4 times:
1. First change: When the first football player kicks the ball.
- Agent supplying force: First player (his foot)
2. Second change: When the second player of the same team kicks the ball towards the goal.
- Agent supplying force: Second player (his foot)
3. Third change: When the goalkeeper collects the football (ball stops).
- Agent supplying force: Goalkeeper (his hands)
4. Fourth change: When the goalkeeper kicks it towards a player of his team.
- Agent supplying force: Goalkeeper (his foot)
Q3. Explain why some of the leaves may get detached from a tree if we vigorously shake its branch.
Solution:
When we vigorously shake the branch of a tree:
- The branch moves due to the force applied.
- The leaves tend to remain in their state of rest due to inertia.
- The leaves resist the change in motion.
- This creates a relative motion between the branch and leaves.
- If the shaking is vigorous enough, the force exceeds the strength of attachment, and leaves get detached from the branch.
Q4. Why do you fall in the forward direction when a moving bus brakes to a stop and fall backwards when it accelerates from rest?
Solution:
When a moving bus brakes to a stop:
- Our feet (in contact with the floor) stop with the bus.
- But the upper part of our body tends to continue moving forward due to inertia of motion.
- This makes us fall forward.
When a bus accelerates from rest:
- Our feet (in contact with the floor) move forward with the bus.
- But the upper part of our body tends to remain at rest due to inertia of rest.
- This makes us fall backwards.
8.4 Second Law of Motion
The first law of motion indicates that when an unbalanced external force acts on an object, its velocity changes (the object gets an acceleration). We would now like to study:
- How the acceleration depends on the force applied
- How we measure a force
Observations from Everyday Life
1. Table tennis ball vs. Cricket ball:
- If a table tennis ball hits a player, it does not hurt him.
- A fast-moving cricket ball hitting a spectator may hurt him.
2. Truck at rest vs. Moving truck:
- A truck at rest does not require attention when parked.
- A moving truck, even at speeds as low as 5 m s⁻¹, may kill a person standing in its path.
3. Bullet from a gun:
- A small mass like a bullet may kill a person when fired from a gun.
Conclusion: The impact produced by objects depends on their mass and velocity.
Momentum
To describe the impact or the quantity of motion, Newton introduced a property called momentum.
Definition: The momentum (p) of an object is defined as the product of its mass (m) and velocity (v).
Formula:
Momentum (p) = Mass (m) × Velocity (v)
p = mv
Properties of Momentum:
- Momentum has both direction and magnitude (vector quantity).
- Its direction is the same as that of velocity.
- SI Unit: kilogram-metre per second (kg m s⁻¹)
Important: Since the application of an unbalanced force brings a change in velocity, it is clear that a force produces a change of momentum.
Understanding Force and Momentum Change
Example: A car with a dead battery is to be pushed to give it a speed of 1 m s⁻¹ (sufficient to start its engine).
Observation 1: If one or two persons give a sudden push (unbalanced force), it hardly starts.
Observation 2: A continuous push over some time results in a gradual acceleration of the car to this speed.
Conclusion: The change of momentum is not only determined by the magnitude of the force but also by the time during which the force is exerted.
Important: The force necessary to change the momentum depends on the time rate at which the momentum is changed.
Newton’s Second Law of Motion
Statement: The rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of the force.
8.4.1 MATHEMATICAL FORMULATION OF SECOND LAW OF MOTION
Suppose an object of mass m is moving along a straight line with:
- Initial velocity = u
- Final velocity = v (after time t)
- Constant force F is applied throughout time t
- Let a be the uniform acceleration
Initial momentum: p₁ = mu
Final momentum: p₂ = mv
Change in momentum:
Δp = p₂ - p₁
= mv - mu
= m(v - u)
Rate of change of momentum:
Rate of change = m(v - u) / t
According to the second law:
F ∝ m(v - u) / t
F = k × m(v - u) / t
where k is a constant of proportionality.
We know that:
a = (v - u) / t
Therefore:
F = k × m × a
Defining the Unit of Force
The SI units of mass and acceleration are kg and m s⁻² respectively. The unit of force is so chosen that the value of constant k becomes one.
Definition of 1 Unit of Force:
One unit of force produces an acceleration of 1 m s⁻² in an object of 1 kg mass.
1 unit of force = k × (1 kg) × (1 m s⁻²)
Thus, k = 1.
Formula for Force:
F = ma
Unit of Force: kg m s⁻² or newton (N)
Definition of 1 Newton:
1 newton is the force that produces an acceleration of 1 m s⁻² in an object of mass 1 kg.
1 N = 1 kg × 1 m s⁻²
Alternative Formula for Force
From F = ma and a = (v – u)/t:
F = m(v - u) / t
Multiplying both sides by t:
Ft = mv - mu
Ft = Change in momentum
Important: When F = 0, then v = u for whatever time t. This means the object will continue moving with uniform velocity u. If u is zero, then v will also be zero (object remains at rest). This mathematically proves the first law of motion.
Applications of Second Law
1. Catching a Cricket Ball:
When a fielder catches a fast-moving cricket ball, he gradually pulls his hands backwards with the moving ball.
Reason:
- This increases the time during which the high velocity decreases to zero.
- Acceleration decreases (a = Δv/t, larger t means smaller a).
- Force decreases (F = ma).
- This reduces the impact on the hands.
If the ball is stopped suddenly:
- Time is very small
- Acceleration is very large
- Force is very large → May hurt the palm
2. High Jump Event:
Athletes are made to fall on a cushioned bed or sand bed.
Reason:
- This increases the time of the athlete’s fall to stop after the jump.
- Rate of change of momentum decreases.
- Force decreases.
- Prevents injury.
3. Karate Player Breaking Ice Slab:
A karate player breaks a slab of ice with a single blow.
Reason:
- The blow is delivered in a very short time.
- Rate of change of momentum is very large.
- Large force is exerted on the ice slab.
- The slab breaks.
Example 8.1
A constant force acts on an object of mass 5 kg for a duration of 2 s. It increases the object’s velocity from 3 m s⁻¹ to 7 m s⁻¹. Find the magnitude of the applied force. Now, if the force was applied for a duration of 5 s, what would be the final velocity of the object?
Solution:
Given:
- m = 5 kg
- u = 3 m s⁻¹
- v = 7 m s⁻¹
- t = 2 s
Part 1: Finding force
Using formula:
F = m(v - u) / t
F = 5 kg × (7 - 3) m s⁻¹ / 2 s
F = 5 × 4 / 2
F = 10 N
Part 2: Finding final velocity if t = 5 s
From Ft = m(v – u):
v = u + Ft/m
v = 3 + (10 × 5) / 5
v = 3 + 10
v = 13 m s⁻¹
Answer:
- Force = 10 N
- Final velocity (if t = 5 s) = 13 m s⁻¹
Example 8.2
Which would require a greater force — accelerating a 2 kg mass at 5 m s⁻² or a 4 kg mass at 2 m s⁻²?
Solution:
For 2 kg mass:
- m₁ = 2 kg
- a₁ = 5 m s⁻²
- F₁ = m₁a₁ = 2 kg × 5 m s⁻² = 10 N
For 4 kg mass:
- m₂ = 4 kg
- a₂ = 2 m s⁻²
- F₂ = m₂a₂ = 4 kg × 2 m s⁻² = 8 N
Since F₁ > F₂:
Answer: Accelerating a 2 kg mass at 5 m s⁻² would require a greater force.
Example 8.3
A motorcar is moving with a velocity of 108 km/h and it takes 4 s to stop after the brakes are applied. Calculate the force exerted by the brakes on the motorcar if its mass along with passengers is 1000 kg.
Solution:
Given:
- u = 108 km/h = 108 × (1000/3600) = 108 × (5/18) = 30 m s⁻¹
- v = 0 m s⁻¹ (car stops)
- t = 4 s
- m = 1000 kg
Using formula:
F = m(v - u) / t
F = 1000 kg × (0 - 30) m s⁻¹ / 4 s
F = 1000 × (-30) / 4
F = -7500 N
Answer: Force = -7500 N
The negative sign indicates that the force exerted by the brakes is opposite to the direction of motion of the motorcar.
Magnitude of force = 7500 N
Example 8.4
A force of 5 N gives a mass m₁ an acceleration of 10 m s⁻² and a mass m₂ an acceleration of 20 m s⁻². What acceleration would it give if both the masses were tied together?
Solution:
From F = ma:
m = F/a
For mass m₁:
m₁ = F/a₁ = 5 N / 10 m s⁻² = 0.50 kg
For mass m₂:
textm₂ = F/a₂ = 5 N / 20 m s⁻² = 0.25 kg
If both masses are tied together:
textTotal mass (m) = m₁ + m₂ = 0.50 + 0.25 = 0.75 kg
Acceleration of combined mass:
a = F/m = 5 N / 0.75 kg = 6.67 m s⁻²
Answer: Acceleration = 6.67 m s⁻²
Example 8.5
The velocity-time graph of a ball of mass 20 g moving along a straight line on a long table is given below. The velocity decreases from 20 cm s⁻¹ to 0 in 10 seconds. How much force does the table exert on the ball to bring it to rest?
Solution:
Given:
- m = 20 g = 20/1000 kg = 0.02 kg
- u = 20 cm s⁻¹ = 0.20 m s⁻¹
- v = 0 cm s⁻¹
- t = 10 s
Finding acceleration:
a = (v - u) / t
a = (0 - 0.20) / 10
a = -0.02 m s⁻²
Finding force:
F = ma
F = 0.02 kg × (-0.02 m s⁻²)
F = -0.0004 N
Answer: Force = -0.0004 N or -4 × 10⁻⁴ N
The negative sign indicates that the frictional force exerted by the table is opposite to the direction of motion of the ball.
8.5 Third Law of Motion
The first two laws tell us how a force changes motion and how to measure force. The third law tells us about the nature of forces.
Newton’s Third Law of Motion:
When one object exerts a force on another object, the second object instantaneously exerts a force back on the first. These two forces are always equal in magnitude but opposite in direction.
Important Points:
- These forces act on different objects, never on the same object.
- The two opposing forces are called action and reaction forces.
Alternative Statement:
To every action, there is an equal and opposite reaction.
Experiment with Spring Balances
Two spring balances A and B are connected together. The fixed end of balance B is attached to a wall. When force is applied through the free end of spring balance A:
Observation: Both spring balances show the same readings on their scales.
Conclusion:
- The force exerted by balance A on balance B is equal but opposite in direction to the force exerted by balance B on balance A.
- Any of these two forces can be called action and the other as reaction.
Important: Action and reaction always act on two different objects, simultaneously.
Examples of Third Law
1. Walking on a Road:
When you intend to start walking:
- You push the road backwards with your feet.
- The road exerts an equal and opposite force on your feet to make you move forward.
Important: The force we exert is not in the direction we intend to move. We push backward, and the road pushes us forward.
2. Firing a Gun:
When a gun is fired:
- The gun exerts a forward force on the bullet.
- The bullet exerts an equal and opposite force on the gun.
- This results in the recoil of the gun (gun moves backward).
Why does the gun move less than the bullet?
- The gun has a much greater mass than the bullet.
- From F = ma, if F is same, larger mass means smaller acceleration.
- The acceleration of the gun is much less than the acceleration of the bullet.
3. Sailor Jumping from Rowing Boat:
As the sailor jumps forward:
- The sailor exerts a backward force on the boat.
- The boat exerts an equal and opposite forward force on the sailor (making him jump).
- The force on the boat moves it backwards.
4. Playing Catch with a Bag:
Activity: Request two children to stand on two separate carts. Give them a bag full of sand. Ask them to play catch with the bag.
Observation:
- Each child experiences an instantaneous force as a result of throwing the sand bag.
- When one throws the bag forward, their cart moves backward.
- When one catches the bag, their cart moves in the direction of the bag’s motion.
Extension: Place two children on one cart and one on another cart. The cart with two children will show different acceleration for the same force (demonstrating second law).
Important Note About Action-Reaction Forces
Even though action and reaction forces are always equal in magnitude, they may not produce accelerations of equal magnitudes.
Reason: Each force acts on a different object that may have different mass.
Example:
- Bullet and gun experience equal and opposite forces.
- Bullet has small mass → large acceleration
- Gun has large mass → small acceleration
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