Physics Chapters 4 & 5

Momentum

The product of an object's mass 'm' and velocity 'v' is called momentum, denoted by 'p'.

p = mv

Momentum is a vector quantity that points in the same direction as the velocity. Its SI Unit is kilogram-meter per second (kg m/s), or newton-second (Ns).

Force and Change in Momentum

The time rate of change of linear momentum of a body is equal to the net force acting on the body. A large force is required for a sudden change in momentum.

F_avg = Δp / Δt = (mv_f - mv_i) / Δt

Impulse and Change in Momentum

Impulse is the product of the force exerted on an object and the time interval over which the force acts. It is equal to the change in momentum.

J = FΔt = Δp

Law of Conservation of Momentum

For an isolated system (where no net external force acts), the total momentum remains constant. The initial momentum equals the final momentum.

p_f = p_i

A common example is the recoil of a gun. Before firing, the total momentum is zero. After firing, the forward momentum of the bullet is equal and opposite to the backward (recoil) momentum of the gun, keeping the total momentum of the system zero.

Parallel Forces

Forces that are parallel to each other are called parallel forces. They can be categorized into two types:

• Like Parallel Forces: Forces that are parallel and act in the same direction. For example, pushing a cart with both hands.
• Unlike Parallel Forces: Forces that are parallel but act in opposite directions. For example, turning the handle of a bike.

Moment of a Force (Torque)

The turning effect produced in a body about a fixed point (axis of rotation) due to an applied force is called torque (τ). It's the rotational equivalent of force.

Torque is the product of the force 'F' and the perpendicular distance from the axis of rotation, known as the moment arm 'd'.

τ = F × d

Its SI unit is Newton-meter (Nm). By convention, anti-clockwise moments are taken as positive and clockwise moments are taken as negative.

Center of Mass & Center of Gravity

Center of Mass (CM): The point in a body where the mass is considered to be equally distributed. A force acting through the CM will not cause rotation.

Center of Gravity (CG): The point where the entire weight of the body appears to act vertically downwards. For most objects on Earth, CM and CG are at the same location.

Equilibrium

An object is in equilibrium when the net force and net torque acting on it are both zero. There are two conditions for equilibrium:

1. First Condition: The vector sum of all forces acting on the body is zero (ΣF = 0). This ensures no translational acceleration (i.e., it is in translational equilibrium).
2. Second Condition (Principle of Moments): The vector sum of all torques acting on the body is zero (Στ = 0). This ensures no rotational acceleration (i.e., it is in rotational equilibrium).

Types of Equilibrium

• Static Equilibrium: The body is at rest.
• Dynamic Equilibrium: The body is moving at a constant velocity (either translational or rotational).

Stability and Types of Equilibrium

Stability is the ability of an object to return to its original position after being slightly displaced.

• Stable Equilibrium: If displaced, the object returns to its original position. Its center of gravity (CG) is at its lowest possible point and rises when displaced.
• Unstable Equilibrium: If displaced, the object moves further away and topples over. Its CG is at its highest point and is lowered when displaced.
• Neutral Equilibrium: If displaced, the object stays in its new position. Its CG is neither raised nor lowered.

Friction

Friction is a force that resists relative motion between surfaces in contact. It always acts opposite to the direction of motion or intended motion.

Types of Friction

Includes sliding friction, rolling friction (which is much smaller than sliding), and fluid friction (also known as drag force, which opposes motion through fluids like air or water).

Advantages & Disadvantages

Advantages: Essential for walking, driving, and holding things together.
Disadvantages: Causes wear and tear, wastes energy, and generates heat in machinery.

Centripetal Force

A force that pulls an object towards the center of a circular path, causing it to continuously change direction and maintain uniform circular motion.

F_c = mv² / r

Where 'm' is mass, 'v' is velocity, and 'r' is the radius of the circular path.