Chapter 9: Force and Laws of Motion

 

Force and Laws of Motion

1. Understanding Motion

• Previous study involved motion along a straight line described by position, velocity, and acceleration.

• Motion can be uniform or non-uniform.

• The cause of motion was not discussed earlier.

2. Key Questions

• Why does an object's speed change with time?

• Do all motions require a cause?

• What is the nature of this cause?

3. Historical Perspective

• Ancient belief: Rest is the "natural state" of an object (based on observations like a ball stopping after rolling).

• This belief persisted until Galileo Galilei and Isaac Newton introduced a new approach to understand motion.

4. Everyday Observations and Force

• To move a stationary object or stop a moving object, effort is required (e.g., push, pull, or hit).

• This effort leads to the concept of force.

5. Concept of Force

• Force is not something we can see, taste, or feel directly.

• We observe effects of force, such as:

  • Changing motion: Initiating, stopping, or altering speed/direction.

  • Deforming objects: Changing shape or size.

6. Applications of Force

• Pushing, hitting, pulling: Methods to apply force and bring objects into motion.

• A force can:

  • Change an object’s velocity (make it faster or slower).

  • Change the direction of motion.

  • Change the shape and size of objects.

Illustrations (Figures in the Text)

• Fig. 9.1: Demonstrates ways of applying force (push, pull, hit).

• Fig. 9.2: Shows force altering shape and size.

This section introduces the foundational concept of force as a cause of motion and deformation, bridging previous knowledge with deeper exploration of motion's principles.

 Balanced and Unbalanced Forces

1. Balanced Forces

• Definition: Forces acting on an object are equal in magnitude and opposite in direction.

• Effect:

  • The object remains in its state of rest or uniform motion.

  • Example: A block pulled equally from both sides remains stationary (Fig. 9.3).

2. Unbalanced Forces

• Definition: Forces acting on an object are unequal in magnitude or not opposite in direction.

• Effect:

  • The object moves in the direction of the greater force.

  • Example: A block pulled with different magnitudes of force moves towards the side with greater force.

3. Friction and Its Role

• Friction: A force that opposes motion between two surfaces in contact.

• 

• Examples:

  • Small Force: Children pushing a box lightly—box does not move because friction balances the applied force (Fig. 9.4a).

  • Increased Force: Children push harder—box still does not move as friction balances the force (Fig. 9.4b).

  • Unbalanced Force: Children push with enough force to overcome friction—box starts moving (Fig. 9.4c).

4. Motion and Unbalanced Forces

• Bicycle Example:

  • When pedaling stops, the bicycle slows down due to friction.

  • To keep the bicycle moving, continuous pedaling is required to counteract friction.

5. Key Observations

• An object in uniform motion does not require continuous unbalanced force.

• Balanced Forces: Result in no change in motion.

• Unbalanced Forces: Cause acceleration (change in speed or direction).

• Stopping Force Removal: If an unbalanced force is removed, the object continues to move with the velocity it has acquired.

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Key Takeaways

1. Balanced forces maintain an object's state of rest or uniform motion.

2. Unbalanced forces cause changes in speed or direction.

3. Friction opposes motion and must be overcome for motion to occur.

4. Motion continues with constant velocity if forces become balanced after acceleration.

 First Law of Motion (Law of Inertia)

1. Galileo's Observations

• Marble on an Inclined Plane: Galileo observed that when a marble rolls down an inclined plane, its velocity increases.

• Frictionless Ideal Plane: If the marble were on a frictionless plane, it would travel forever and would not stop unless acted upon by an unbalanced force.

• Role of Friction: In real situations, friction causes the marble to stop after some distance.

2. Newton’s First Law of Motion

• Statement: An object remains in a state of rest or uniform motion in a straight line unless compelled to change that state by an applied force.

• Inertia: This law explains the tendency of objects to resist a change in their state of motion. Inertia is the property of matter that causes objects to stay at rest or keep moving unless an unbalanced force acts on them.

• Alternate Name: The first law is also known as the Law of Inertia.

3. Practical Examples of Inertia

• Motorcar Braking: When the car stops suddenly, the body continues to move forward due to inertia, which may cause injury. Safety belts help to slow down the body.

• Bus Starting: When the bus suddenly starts, the body tends to remain at rest, causing us to feel a backward motion.

• Sharp Turns in a Car: When a car turns sharply at high speed, the body tends to continue in a straight line, causing us to be thrown to one side due to inertia.

4. Illustrative Activities of Inertia

• Carom Coins Activity:

  • When a sharp horizontal hit is applied to the bottom of a pile of carom coins, the bottom coin moves, and the other coins fall vertically due to their inertia.

• Coin on Card Activity:

  • A sharp flick of the card makes it shoot away, allowing the coin to fall vertically due to inertia.

• Water in Tumbler on Tray Activity:

  • When the tray is turned suddenly, the water spills due to inertia, as the water tends to remain at rest while the tray moves.

5. Everyday Application of Inertia

• Tea Cup Groove: The groove in a saucer prevents the tea cup from toppling over during sudden jerks, as it helps counteract the effects of inertia.

Key Takeaways

• First Law of Motion: An object will not change its state of motion (rest or uniform motion) unless an unbalanced force acts on it.

• Inertia: The tendency of objects to resist changes in their motion is called inertia.

• Practical Applications: The effects of inertia can be observed in daily life, such as when a car stops suddenly or when a bus starts.

Inertia and Mass Notes

• Inertia:

  • It is the property of an object to resist a change in its state of rest or motion.

  • If an object is at rest, it tends to remain at rest.

  • If an object is in motion, it tends to continue moving with the same velocity in a straight line unless acted upon by an unbalanced force.

• Examples of Inertia:

  • Pushing an empty box is easier than pushing a box full of books because the full box has more inertia.

  • Kicking a football causes it to fly away, but kicking a stone with the same force results in little movement, demonstrating that objects with greater mass have more inertia.

• Relation Between Inertia and Mass:

  • The more massive an object, the greater its inertia.

  • For example, it is harder to move a train than a cart because the train has more mass and thus more inertia.

• Mass as a Measure of Inertia:

  • The mass of an object quantifies its inertia.

  • Heavier objects resist changes in motion more than lighter objects, requiring more force to change their state of motion.

Key Concept:

• Inertia and Mass are related; the greater the mass, the greater the inertia of an object.

Second Law of Motion 

Introduction:

• The First Law of Motion highlights that an object will continue in its state of rest or uniform motion unless acted upon by an external force.
• The Second Law of Motion builds on this and provides a mathematical relationship between the force applied, the mass of the object, and its acceleration.

Momentum:

• Momentum (p) is the product of an object's mass (m) and its velocity (v):
  $p = mv$
  It has both magnitude and direction, and its unit is kg m/s.

Change in Momentum:

• The change in momentum ($\Delta p$) of an object is:
  $\Delta p = mv - mu$
  where m is the mass, u is the initial velocity, and v is the final velocity.

Rate of Change of Momentum:

• The rate at which momentum changes is directly proportional to the force applied:
  $\text{Rate of change of momentum} \propto \frac{(mv - mu)}{t}$
  where t is the time during which the force acts.
• 
  

Second Law of Motion (Mathematical Formulation):

• The second law mathematically states that:
  $F = \frac{m(v - u)}{t}$
  where F is the force, m is the mass, v is the final velocity, u is the initial velocity, and t is the time taken for the change in velocity.

  • This can also be written as:
    $F = ma$
    where a is the acceleration, which is the rate of change of velocity ($a = \frac{v - u}{t}$).

Units of Force:

• The unit of force in the SI system is newton (N), defined as:
  $1 \, \text{N} = 1 \, \text{kg} \times \text{m/s}^2$
  

Here is the derivation of$F$$=$$m$$a$ in a step-by-step mathematical format:

Step 1: Definition of Momentum

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

$$p = mv$$

Step 2: Change in Momentum

Let the initial velocity of the object be $u$ and the final velocity be $v$. The momentum before the time interval is:

$$p_1 = mu$$

The momentum after the time interval is:

$$p_2 = mv$$

The change in momentum is:

$$\Delta p = p_2 - p_1 = mv - mu = m(v - u)$$

Step 3: Rate of Change of Momentum

The rate of change of momentum is defined as the change in momentum ($\Delta p$) divided by the time interval ($\Delta t$):

$$\frac{\Delta p}{\Delta t} = \frac{m(v - u)}{\Delta t}$$

Step 4: Acceleration

Acceleration ($a$) is defined as the rate of change of velocity:

$$a = \frac{v - u}{\Delta t}$$

Therefore, we can rewrite the rate of change of momentum as:

$$\frac{\Delta p}{\Delta t} = m \cdot a$$

Step 5: Applying Newton’s Second Law

Newton's second law of motion states that the rate of change of momentum is equal to the applied force $F$:

$$F = \frac{\Delta p}{\Delta t}$$

Step 6: Final Substitution

From Step 4, we know that:

$$\frac{\Delta p}{\Delta t} = m \cdot a$$

Substitute this into the expression for force:

$$F = m \cdot a$$

Final Conclusion:

Thus, we arrive at the equation:

$$F = ma$$

This is Newton's second law of motion, which relates the force applied to an object to its mass and acceleration.

Real-life Applications:

1. Catching a Cricket Ball:
   When catching a fast-moving ball, a fielder gradually pulls their hands back to increase the time during which the ball slows down. This reduces the impact force, as the force is inversely proportional to the time taken to stop the ball.

2. High Jump:
   Athletes fall on cushioned or sand beds to increase the time over which they stop, reducing the impact force and preventing injury.

3. Breaking a Slab of Ice in Karate:
   A karate player strikes with great force over a short time interval to maximize the momentum change and break the ice slab.

Relation to First Law of Motion:

• The First Law of Motion is a special case of the Second Law when F = 0. This means if no external force acts on an object, it will either stay at rest or continue moving with uniform velocity.
  • For an object at rest (u = 0), the velocity will remain zero (v = 0).
  • For an object moving with constant velocity (u = v), it will continue moving at that velocity until acted upon by an external force.

Third Law of Motion

Statement of the Third Law of Motion:

• Newton's Third Law of Motion states that: "For every action, there is an equal and opposite reaction."
• This means that when one object exerts a force on another object, the second object exerts an equal and opposite force back on the first. These forces are always equal in magnitude and opposite in direction but act on different objects.

Key Concepts:

1. Action and Reaction Forces:

   • Action and reaction forces act on two different objects.
   • These forces are equal in magnitude but opposite in direction.
   • These forces are simultaneous and occur instantly.

2. Example of Action and Reaction Forces:

   • Football Game Example: When you try to kick a football, you apply a force to the ball. Simultaneously, the ball applies an equal and opposite force to your foot. Both forces are action and reaction forces.
   • Spring Balance Example: When a force is applied to one spring balance, the second spring balance shows the same reading. This indicates that the force applied by one balance on the other is equal but opposite in direction.

2. Force on Objects of Different Masses:

   • Even though action and reaction forces are equal, the resulting accelerations of the objects may not be equal. This is because each object may have a different mass.
   • Example: When a bullet is fired from a gun, the bullet moves forward, and the gun experiences a recoil in the opposite direction. Since the bullet has a much smaller mass compared to the gun, it experiences a much larger acceleration than the gun.

3. Walking Example:

   • When you walk, you push the ground backward with your feet. The ground exerts an equal and opposite force on your feet, propelling you forward.
   • The force you exert on the ground is the action, and the ground's force on you is the reaction.

Important Observations:

• Equal Magnitude and Opposite Direction: The action and reaction forces are always equal in magnitude but act in opposite directions.
• Different Objects: The action and reaction forces act on different objects and cannot cancel each other out because they don't act on the same object.

Illustrations and Applications:

1. Gun and Bullet:

   • When a gun is fired, it applies a forward force on the bullet, and the bullet exerts an equal and opposite force on the gun, causing the gun to recoil.
   • The gun, having much greater mass than the bullet, experiences a much smaller acceleration than the bullet.

Sailor and Boat:

1. • When a sailor jumps off a rowing boat, the force exerted by the sailor on the boat pushes the boat backward. The sailor moves forward while the boat moves backward, demonstrating action and reaction.

1. Children on Carts (Activity Example):

   • Two children standing on separate carts and playing catch with a sandbag will demonstrate the action and reaction forces. When one child throws the sandbag, the cart they are standing on will move backward, and the other cart will move forward, illustrating the action and reaction forces.
   • Different accelerations will be observed depending on the mass of the carts, showing the second law of motion.

Summary:

• The Third Law of Motion explains how forces always act in pairs, and each force has a corresponding equal and opposite force.
• These forces act on different objects and can have different effects based on the mass and acceleration of the objects involved.

Conservation of Momentum

Introduction:

• The law of conservation of momentum states that the total momentum of a system of objects remains constant if no external unbalanced forces act on it.
• Momentum is defined as the product of an object’s mass and velocity.
• Momentum of a system before a collision is equal to momentum of the system after the collision, provided no external forces act on the system.

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Explanation of Conservation of Momentum:

1. Scenario of Two Colliding Objects:
   • Suppose two objects (balls A and B) of masses $m_A$ and $m_B$ are moving along a straight line with velocities $u_A$ and $u_B$ before the collision.
   • The balls collide and exert equal and opposite forces on each other during the collision, which lasts for time $t$.
   • After the collision, the velocities of the balls change to $v_A$ and $v_B$, respectively.
2. Mathematical Formulation:
   • The momentum of ball A before the collision is $m_A u_A$ and after the collision is $m_A v_A$.
   • The momentum of ball B before the collision is $m_B u_B$ and after the collision is $m_B v_B$.
   • According to the Third Law of Motion, the force exerted by ball A on ball B is equal and opposite to the force exerted by ball B on ball A: $$F_{AB} = - F_{BA}$$
   • The rate of change of momentum for ball A is: $$\frac{m_A(v_A - u_A)}{t}$$
   • Similarly, the rate of change of momentum for ball B is: $$\frac{m_B(v_B - u_B)}{t}$$
3. Conservation of Total Momentum:
   • The forces $F_{AB}$ and $F_{BA}$ are equal and opposite, hence the total momentum before the collision is equal to the total momentum after the collision: $$m_A u_A + m_B u_B = m_A v_A + m_B v_B$$
   • This equation shows that the total momentum is conserved in the absence of external forces.
   • This is the law of conservation of momentum: The total momentum of a system remains constant if no external force acts on the system.

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To derive the Law of Conservation of Momentum, you can follow these steps:

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Derivation of the Law of Conservation of Momentum

The Law of Conservation of Momentum states that in an isolated system, where no external forces act, the total momentum of the system remains constant.

Step 1: Consider Two Colliding Objects

Let's assume two objects, object A and object B, of masses $m_A$ and $m_B$ respectively. Let these objects be moving along a straight line with initial velocities $u_A$ and $u_B$ before the collision.

• Mass of object A: $m_A$
• Mass of object B: $m_B$
• Initial velocity of object A: $u_A$
• Initial velocity of object B: $u_B$

After the collision, the velocities of the two objects change. Let:

• Final velocity of object A: $v_A$
• Final velocity of object B: $v_B$

Step 2: Apply Newton’s Second Law

The change in momentum for an object is given by the product of the force acting on it and the time during which the force acts. For object A and object B, during the collision, the forces exerted on them are equal in magnitude but opposite in direction, according to Newton's Third Law.

• The force exerted by object A on object B is $F_{AB}$
• The force exerted by object B on object A is $F_{BA}$

According to Newton's Third Law:

$$F_{AB} = -F_{BA}$$

Step 3: Change in Momentum of Each Object

• The change in momentum of object A is:

$$\Delta p_A = m_A(v_A - u_A)$$

• The change in momentum of object B is:

$$\Delta p_B = m_B(v_B - u_B)$$

Step 4: Total Change in Momentum

The total change in momentum of the system (objects A and B) is the sum of the changes in momentum of both objects:

$$\Delta p_{\text{total}} = \Delta p_A + \Delta p_B$$ $$\Delta p_{\text{total}} = m_A(v_A - u_A) + m_B(v_B - u_B)$$

Step 5: Apply the Principle of Conservation of Momentum

Now, the total momentum of the system before and after the collision must be equal if no external force is acting on the system.

• The total momentum before the collision is:

$$p_{\text{initial}} = m_A u_A + m_B u_B$$

• The total momentum after the collision is:

$$p_{\text{final}} = m_A v_A + m_B v_B$$

According to the law of conservation of momentum, the total momentum before and after the collision should be the same:

$$m_A u_A + m_B u_B = m_A v_A + m_B v_B$$

This equation shows that the total momentum of the system (objects A and B) is conserved during the collision.

Step 6: Conclusion

Thus, the Law of Conservation of Momentum states that if no external forces act on a system, the total momentum of the system remains constant. This is mathematically expressed as:

$$m_A u_A + m_B u_B = m_A v_A + m_B v_B$$

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Example to Understand the Concept

Let’s consider two billiard balls colliding. The total momentum of the two balls before and after the collision will be the same, provided there are no external forces like friction or air resistance.

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In Simple Words:

The law tells us that in any closed system, the momentum before an event (like a collision) is equal to the momentum after the event. It is one of the most fundamental principles in physics, and it holds true for all types of collisions and interactions, whether elastic or inelastic.

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Illustrations and Applications:

1. Rubber Balloon Activity (Momentum in Air):

   • Inflate a balloon and fix a straw on the surface. Pass a thread through the straw and secure the ends.
   • When the thread is cut, the air escapes from the balloon, and the balloon moves in the opposite direction, demonstrating conservation of momentum. The escaping air (action) exerts an equal and opposite force on the balloon (reaction), causing it to move forward.

2. Test Tube and Cork Activity (Recoil of Test Tube):

   • A test tube with a small amount of water is heated until the water vaporizes and the cork is ejected.
   • The test tube recoils in the opposite direction of the cork, demonstrating the principle of conservation of momentum. The force exerted by the cork on the test tube causes the test tube to move in the opposite direction.

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Key Takeaways:

• Conservation of Momentum states that in an isolated system (no external forces), the total momentum before a collision is equal to the total momentum after the collision.
• Action and Reaction: In collisions, the forces that objects exert on each other are equal and opposite. The total momentum of the system does not change unless an external force acts on it.
• This principle is demonstrated in various physical activities, such as balloon propulsion and the recoil of the test tube.

Conservation Laws 

Introduction to Conservation Laws:

• Conservation laws are fundamental principles in physics that state certain properties of a system remain constant over time, provided no external influence acts on it.
• These laws include the conservation of momentum, energy, angular momentum, charge, and others.
• Conservation laws are based on experimental observations and cannot be "proved" in the traditional sense. They can only be verified through experiments, where results align with the law. However, a single experiment that contradicts the law is sufficient to disprove it.

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Law of Conservation of Momentum:

• The law of conservation of momentum has been formulated over three centuries ago and has been verified by countless experiments and observations.
• Momentum is the product of mass and velocity. The law states that the total momentum of an isolated system remains constant before and after a collision, provided no external forces act on the system.

Fundamental Principles:

1. Verification, Not Proof:

   • The law of conservation of momentum is supported by experimental evidence, but it cannot be proven. It can only be verified when results align with the law.
   • If a single experiment contradicts the law, it would disprove the law.

2. Widespread Acceptance:

   • The law of conservation of momentum is widely accepted because no experiment has yet contradicted it.
   • It forms the basis for understanding various physical phenomena, such as collisions and explosions.

3. Real-life Examples:

   • Everyday Life Applications: The law helps explain several experiences, such as how objects in motion behave during collisions (e.g., car crashes, billiard ball interactions).
   • In sports, the motion of objects like balls in football, cricket, or table tennis can be analyzed using the law of conservation of momentum.

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Key Takeaways:

• Conservation laws like momentum, energy, and charge are fundamental to understanding physical processes.
• They are based on experimental evidence and can be verified but not proved.
• The law of conservation of momentum has stood the test of time and aligns with all observed experiments and real-life scenarios.
• If external forces (like friction or air resistance) are absent or negligible, the total momentum in a system remains constant before and after an interaction (e.g., collision).

Galileo Galilei: Life and Contributions

Early Life and Education:

• Born: 15 February 1564 in Pisa, Italy.
• Father: Vincenzo Galilei, who wanted him to become a medical doctor.
• Education: Galileo enrolled in the University of Pisa in 1581 to study medicine but did not complete his degree due to his deep interest in mathematics and natural philosophy.

Scientific Career and Achievements:

• First Scientific Book (1586): Galileo's first major work was "The Little Balance" (La Balancitta), where he described Archimedes’ method for determining the relative densities (specific gravities) of substances using a balance.
• Theories on Motion (1589): In his work De Motu, Galileo presented theories about falling objects. He famously used an inclined plane to slow down the fall and measure the motion, establishing the idea that the distance moved by an object is proportional to the square of the time taken.
• Professor at University of Padua (1592): Galileo was appointed Professor of Mathematics at the University of Padua in the Republic of Venice. Here, he made significant contributions to the study of motion and developed his theories on the acceleration of falling bodies.

Inventions and Innovations:

• Telescopes: Galileo designed and improved telescopes, significantly enhancing their optical performance compared to others of his time. This enabled him to make revolutionary astronomical discoveries.
• Pendulum Clock: Around 1640, Galileo designed the first pendulum clock, laying the foundation for precise timekeeping.

Astronomical Discoveries:

• Starry Messenger (1610): In his famous book "Starry Messenger", Galileo described his astronomical discoveries. Among them:

  • Mountains on the Moon.
  • The Milky Way composed of tiny stars.
  • Four moons orbiting Jupiter, later named the Galilean moons.

• Sunspots: In "Discourse on Floating Bodies" and "Letters on the Sunspots", Galileo shared his observations of sunspots, challenging the belief that the Sun was perfect and unchanging.

• Heliocentric Theory: Galileo used his telescopic observations of Saturn and Venus to argue that the planets, including Earth, orbit the Sun, contradicting the geocentric model that placed Earth at the center of the universe.

Legacy:

• Galileo’s work laid the groundwork for modern physics and astronomy.
• He is often referred to as the father of modern science for his use of experimentation and observation to validate scientific theories.
• His advocacy of the heliocentric model eventually led to conflict with the Catholic Church, but his contributions revolutionized our understanding of the universe.

Table of Contents

Sr. No.	Topic	Subtopics
1	Introduction to Motion and Laws	Definition and Basic Concepts, Historical Context
2	First Law of Motion (Law of Inertia)	Statement of the Law, Examples of Inertia in Everyday Life, Applications of the Law, Numerical Problems on Inertia
3	Second Law of Motion	Statement of the Law, Mathematical Formulation of Second Law, Explanation of Force and Acceleration, Unit of Force (Newton), Applications in Daily Life (e.g., Catching a Cricket Ball), Numerical Problems on Second Law
4	Third Law of Motion	Statement of the Law, Action and Reaction Forces, Examples in Everyday Life (Football, Gun Recoil), Applications (Rowing Boat, Jumping Sailor), Mathematical Formulation
5	Conservation of Momentum	Definition and Explanation of Momentum, Principle of Conservation of Momentum, Example: Collision of Two Balls, Derivation of Momentum Conservation Formula, Activity on Conservation of Momentum
6	Conservation Laws	Importance of Conservation Laws in Physics, Law of Conservation of Momentum, Verification of Conservation Laws, Everyday Applications of Momentum Conservation
7	Galileo Galilei: Contributions to Physics	Early Life and Education, Scientific Career and Major Works, Developments in Motion Theory, Telescopes and Astronomical Discoveries, Law of Falling Bodies and Pendulum Clock, Heliocentric Theory and Conflict with the Church, Legacy in Modern Science
8	Conclusion	Summary of Laws of Motion, Real-life Applications and Impact on Modern Physics