Force and Laws of Motion

Force

Force is defined as the effort or influence required to bring about a change in the state of an object. This change can involve various aspects of an object's motion and properties.

Effects of Force

1. Moving a Stationary Object: A force is capable of bringing a stationary object to motion.
2. Stopping a Moving Object: Similarly, a force can also be applied to bring a moving object to a stop.
3. Changing Speed: Force can either speed up or slow down an object's motion, depending on the direction of the force and the initial velocity of the object.
4. Changing Direction: By applying a force at an angle to an object's initial motion, the direction of the object's motion can be altered.
5. Changing Shape and Size: Force can also cause changes in an object's shape and size. For instance, compression and stretching are examples of force-induced changes in an object's shape.

Overall, force is the underlying factor that enables interactions and changes in the motion and characteristics of objects in the physical world.

Balanced and Unbalanced Forces

Balanced Forces

When multiple forces act on an object and their combined effect results in no change in the object's state of motion, these forces are referred to as balanced forces. In other words, if the net force or resultant force is zero, the forces are balanced.

For example, imagine a book lying on a table. The weight of the book (gravitational force) pulls it downward, while the table exerts an upward normal force to support the book. These two forces are balanced because they cancel each other out, leading to no acceleration or change in the book's motion.

Unbalanced Forces

When a set of forces acting on an object results in a change in the object's state of motion, these forces are termed unbalanced forces. In this case, the net force or resultant force is not zero.

For instance, consider a car accelerating forward. The engine applies a force in the forward direction, while friction and air resistance provide opposing forces. If the force applied by the engine is greater than the combined resistance forces, the result is an unbalanced force that causes the car to accelerate.

Types of Forces

There are two main types of forces: contact forces and non-contact forces. Let's delve into these concepts a bit more:

Contact Forces

Contact forces are those forces that come into play when two objects are physically in contact with each other. These forces arise due to direct interaction between the surfaces of the objects. Some examples of contact forces include:

1. Muscular Force: When you push, pull, lift, or exert any effort using your muscles, you are applying a contact force. For instance, when you push a door open, your hands exert a muscular force on the door.
2. Frictional Force: Friction is a contact force that opposes the motion or attempted motion between two objects in contact. It arises due to the roughness of surfaces and plays a significant role in everyday activities like walking, driving, and gripping objects.

Non-Contact Forces

Non-contact forces, as the name suggests, are forces that can act on objects even when they are not in physical contact. These forces can exert their influence over a distance. Some examples of non-contact forces include:

1. Gravitational Force: The force of attraction between masses, such as the Earth's gravitational force pulling objects toward its centre. This force keeps objects on the ground and governs the motion of planets and celestial bodies.
2. Electrostatic Force: Also known as the Coulomb force, it's the force of attraction or repulsion between charged particles. Like charges repel, and unlike charges attract. This force is responsible for interactions between charged objects.
3. Magnetic Force: The force of attraction or repulsion between magnets or magnetic materials. Magnetic forces also affect charged particles in motion and are responsible for various phenomena, including the operation of electric motors.

Newton's First Law of Motion

Newton's First Law of Motion, also known as the Law of Inertia, is one of the fundamental principles of physics that describes how objects behave when no external forces are acting upon them. Let's break down the key concepts of this law:

Statement of the Law

"An object at rest tends to stay at rest, and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced external force."

Key Concepts:

1. Inertia: Inertia is the property of an object that resists any change in its state of motion. An object with more inertia is more resistant to changes in its velocity or direction. In simpler terms, inertia is the tendency of objects to resist changes in their motion.
2. Rest and Motion: The law emphasises that objects have a natural preference to either remain at rest or continue moving with a constant velocity in a straight line. This tendency is due to their inertia.
3. Unbalanced External Force: An unbalanced external force refers to a force that is not counteracted by an equal and opposite force. It's a force that is strong enough to overcome the object's inertia and cause a change in its state of motion.

Examples:

1. Object at Rest: Imagine a book sitting on a table. According to Newton's First Law, the book will stay at rest unless something (an external force) acts upon it. If you gently push the book, it will start moving due to the applied force.
2. Object in Motion: Consider a car moving on a straight road. If you suddenly stop the car, the passengers inside might feel pushed forward. This is because their bodies want to continue moving forward (due to inertia) even though the car has come to a stop.
3. Driving a Car: When you drive a car and take a sharp turn, your body tends to move in the direction of the turn due to inertia. This is why you may feel pushed against the door on the side of the turn.
4. Jumping off a Moving Bus: If you jump off a moving bus, you'll notice that you still have forward momentum after leaving the bus. This is because of your inertia – your body tends to keep moving at the same speed and direction as the bus.

Classification of Inertia

1. Inertia of Rest: This refers to the tendency of an object to remain at rest unless acted upon by an external force. In other words, objects resist changes from a state of rest. An example of this is when fruits fall from a tree when its branches are shaken. The fruits were initially at rest, and the shaking of the branches applies an external force that overcomes their inertia of rest, causing them to fall.
2. Inertia of Motion: Also known as "momentum" or "linear inertia," this type of inertia describes the tendency of an object in motion to continue moving with the same speed and direction unless acted upon by an external force. For example, when a passenger is standing in a moving bus and the bus suddenly stops, the passenger tends to lean forward due to their inertia of motion. Their body wants to keep moving forward even though the bus has stopped.
3. Inertia of Direction: This type of inertia pertains to the tendency of an object to resist changes in its direction of motion. When an object is moving in a particular direction, it resists any attempts to change that direction. An example of this is water particles sticking to the tires of a moving bicycle. These particles have inertia of direction, and when the cycle suddenly changes its direction, the particles tend to move off tangentially due to their resistance to change in direction.

Momentum

Momentum is a fundamental concept in physics that quantifies the motion of an object. It's defined as the product of an object's mass and its velocity. Mathematically, momentum (p) is given by the equation:

Where:
p represents the momentum of the object.
m is the mass of the object.
v is the velocity of the object.

Momentum is a vector quantity, which means it has both magnitude and direction. Its direction is the same as the direction of the velocity vector. The SI unit of momentum is a kilogram metre per second (kg ms^-1).

Momentum is particularly useful when analysing the behaviour of objects in motion.

Newton's Second Law of Motion

The second law of motion describes the relationship between the force applied to an object, its mass, and the resulting acceleration. The law can be stated as follows:

"The rate of change of momentum of an object is proportional to the net applied unbalanced force. The direction of change of momentum is the same as the direction of the net force."

Mathematical Formulation

The mathematical formulation of Newton's Second Law of Motion provides a quantitative relationship between force, mass, and acceleration. It allows us to calculate the force required to produce a certain acceleration on an object.

Suppose we have an object of mass m that is initially moving with a velocity u. If a constant force F is applied to the object for a certain time t, it will accelerate to a final velocity v. The initial and final momentum of the object are p₁ = mu and p₂ = mv respectively.

The change in momentum (Δp) can be calculated as:

The rate of change of momentum (Δp/t) is proportional to the applied force:

Rearranging the equation and introducing the acceleration (a) as a = (v−u)/t, we get:

k is the constant of proportionality. For 1 unit of force on 1 kg mass with the acceleration of 1 ms^-2, the value of k = 1.
Therefore the equation becomes:

This equation represents Newton's Second Law of Motion. It states that the force applied to an object is directly proportional to its mass and its acceleration. In other words, the greater the mass of an object, the greater the force needed to accelerate it, and the greater the acceleration desired, the greater the force required.

Key concepts of Newton's Second Law

Force and Acceleration: The greater the force applied to an object, the greater its acceleration will be. If you push or pull an object with more force, it will accelerate faster. This emphasises the direct proportionality between force and acceleration.

Mass and Acceleration: For a given force, an object with a smaller mass will experience a larger acceleration, while an object with a larger mass will experience a smaller acceleration. This demonstrates the inverse proportionality between mass and acceleration.

Direction: The direction of the net force and the resulting acceleration are the same. If the force is applied in the same direction as the object's initial motion, it will speed up. If the force is applied in the opposite direction, it will slow down or change direction.

Units: The SI (International System of Units) unit of force is the Newton, abbreviated as "N". The newton is defined as the amount of force required to accelerate a one-kilogram mass by one metre per second squared (1 kg ms^-2). In equation form:

1 N = 1 kg ms^-2

In other words, if you apply a force of 1 Newton to an object with a mass of 1 kilogram, it will experience an acceleration of 1 metre per second squared.

Impulse

Impulse is defined as the product of force and the time for which it acts. It can also be measured by the change in momentum produced in a body. This concept is derived from Newton's Second Law.

Mathematically, impulse (I) is given by:

Where:
I is the impulse
F is the force applied
t is the time for which the force acts

Impulse can also be calculated as the change in momentum (Δp) of an object:

Where:
m is the mass of the object
v is the final velocity
u is the initial velocity

SI unit of Impulse: The SI unit of impulse is kilogram metre per second (kg ms^-1) or newton second (N s). This unit is equivalent to the unit of momentum, emphasising the relationship between impulse and momentum.

Impulsive Force: An impulsive force is a large force applied for a short duration of time. When a force is applied to an object for a brief moment, it can lead to a significant change in the object's momentum. For example, if you hit a baseball with a bat, the force exerted by the bat over a short time causes the baseball to change its velocity rapidly, resulting in a change in momentum.

The impact of an impulsive force can be reduced by increasing the time over which the force acts. This is why safety mechanisms often involve increasing the time of impact, such as using airbags in vehicles to extend the duration of collision forces.

Force due to Gravity

The force due to gravity, often referred to as the weight of an object, is the force exerted on an object by the gravitational attraction of another massive object, such as the Earth. It is one of the fundamental forces of nature and is responsible for the phenomenon of weight and the motion of objects in a gravitational field.

Key Concepts:

1. Gravitational Force: Every mass in the universe attracts every other mass with a force known as gravitational force. This force is directly proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between their centres.
2. Acceleration Due to Gravity: The acceleration due to gravity (g) is a measure of how fast an object falls under the influence of gravity. Near the surface of the Earth, the standard value of g is approximately 9.8 metres per second squared (ms^-2).
3. Equation: The force due to gravity (F) can be calculated using the equation:

   

   where m is the mass of the object and g is the acceleration due to gravity. This force acts vertically downward toward the centre of the massive object.
4. Gravitational Field: The presence of a massive object creates a gravitational field around it. Any other object with mass placed within this field will experience a force due to gravity.
5. Universal Law of Gravitation: The force due to gravity is described by Newton's law of universal gravitation, which states that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

Newton's Third Law of Motion

The third law of motion states that for every action, there is an equal and opposite reaction. When one object exerts a force on another object, the second object instantaneously exerts a force back on the first object. These forces are always equal in magnitude but opposite in direction, and they act on different objects simultaneously.

Implications of the Third Law

1. Equal Magnitude, Opposite Direction: The forces involved in an interaction between two objects are always equal in magnitude and opposite in direction.
2. Different Objects: The action and reaction forces occur between two different objects. If object A exerts a force on object B, object B simultaneously exerts an equal and opposite force on object A.
3. Pairs of Forces: The two opposing forces are often referred to as an action-reaction pair. While one force is the action, the other is the reaction.
4. Examples:

• Football Collision: In the game of football, when two players collide while attempting to kick the ball, they both experience a force. Each player exerts a force on the other, leading to a pair of equal and opposite forces.
• Spring Balances: When two spring balances are connected, applying a force on one balance results in both balances showing the same reading. The forces exerted by the two balances are equal in magnitude and opposite in direction.
• Walking on the Road: When you start walking on a road, your feet exert a force on the road backwards. In response, the road exerts an equal and opposite force on your feet, propelling you forward.
• Gun Recoil: When a gun is fired, the bullet is propelled forward due to the force exerted by the gun. Simultaneously, the gun experiences an equal and opposite force (recoil) that moves it backwards. However, since the gun has a much greater mass, its acceleration is much less compared to the bullet's acceleration.
• Rowing Boat: When a sailor jumps out of a rowing boat, the force exerted by the sailor on the boat pushes it backwards in the water, allowing the sailor to move forward.

Conservation of Momentum

Conservation of momentum is a fundamental principle in physics that states that the total momentum of an isolated system remains constant if no external forces are acting on it. In other words, in the absence of external influences, the total amount of momentum before an event or interaction is the same as the total momentum after the event.
The conservation of momentum is expressed as:

Total Initial Momentum = Total Final Momentum

This equation reflects the idea that the sum of the momenta of all objects in the system before an event is equal to the sum of their momenta after the event.

Key Points:

1. Total Momentum: Momentum is a vector quantity that accounts for both the mass and velocity of an object. The total momentum of a system is the sum of the individual momentum of all the objects within the system.
2. Isolated System: An isolated system is one that doesn't interact with its surroundings through external forces. This means that the system's total momentum is unaffected by external influences like friction, gravity, or other forces.
3. Conservation: The conservation of momentum implies that the total momentum of the system remains unchanged over time. This principle applies to various interactions, such as collisions, explosions, and other events involving multiple objects.

Mathematical Expression

Consider two objects, A and B, with masses m₁ and m₂ respectively and initial velocities u₁ and u₂ before a collision. When these objects collide, they exert forces on each other, resulting in changes in their velocities. Let v₁ and v₂ be their velocities after the collision.

According to the law of conservation of momentum, the total initial momentum of the system (m₁u₁ + m₂u₂) is equal to the total final momentum of the system (m₁v₁ + m₂v₂)

Mathematically, this can be expressed as:

This equation signifies that the sum of the momenta of the two objects before the collision is equal to the sum of the momenta after the collision. This conservation principle holds true for any interactions where no external unbalanced forces are acting.