NDA Physics: Mechanics Overview

Mechanics in NDA Physics

1. Kinematics

Definitions: Study of motion without considering forces.
Key Equations:
- Displacement: ( s = ut + \frac{1}{2}at^2 )
- Final Velocity: ( v = u + at )
- ( v^2 = u^2 + 2as )
Types of Motion:
- Uniform Motion: Constant speed, straight path.
- Non-Uniform Motion: Varying speed or direction.

2. Dynamics

Newton's Laws of Motion:
- First Law: An object remains at rest or in uniform motion unless acted on by a net force.
- Second Law: ( F = ma ) (Force equals mass times acceleration).
- Third Law: For every action, there is an equal and opposite reaction.
Applications:
- Friction: Static and kinetic friction coefficients.
- Circular Motion: Centripetal force ( F_c = \frac{mv^2}{r} ).

3. Work, Energy, and Power

Work Done: ( W = F \cdot d \cdot \cos(\theta) )
Kinetic Energy: ( KE = \frac{1}{2}mv^2 )
Potential Energy: ( PE = mgh ) (gravitational potential energy).
Conservation of Energy: Total energy in a closed system remains constant.
Power: ( P = \frac{W}{t} )

4. Momentum

Linear Momentum: ( p = mv )
Conservation of Momentum: Total momentum before an event equals total momentum after.
Elastic and Inelastic Collisions:
- Elastic: Kinetic energy conserved.
- Inelastic: Kinetic energy not conserved.

5. Gravitation

Newton’s Law of Gravitation: ( F = G \frac{m_1m_2}{r^2} )
Gravitational Potential Energy: ( U = -\frac{Gm_1m_2}{r} )
Satellite Motion:
- Orbital velocity: ( v = \sqrt{\frac{GM}{r}} )
- Period of orbit: ( T = 2\pi \sqrt{\frac{r^3}{GM}} )

6. Rotational Motion

Angular Displacement: ( \theta = s/r ) (where ( s ) is arc length).
Angular Velocity: ( \omega = \frac{d\theta}{dt} )
Torque: ( \tau = rF \sin(\theta) )
Moment of Inertia: ( I = \sum m_ir_i^2 ) (depends on mass distribution).
Conservation of Angular Momentum: ( L = I\omega ) remains constant if no external torque acts.

7. Simple Harmonic Motion (SHM)

Characteristics: Motion about an equilibrium position, restoring force proportional to displacement.
Key Equations:
- Displacement: ( x(t) = A \cos(\omega t + \phi) )
- Velocity: ( v(t) = -A\omega \sin(\omega t + \phi) )
- Acceleration: ( a(t) = -A\omega^2 \cos(\omega t + \phi) )
Period of SHM: ( T = 2\pi \sqrt{\frac{m}{k}} ) (where ( k ) is spring constant).

8. Fluid Mechanics

Density: ( \rho = \frac{m}{V} )
Pressure: ( P = \frac{F}{A} )
Archimedes' Principle: A body submerged in fluid experiences an upward buoyant force equal to the weight of the fluid displaced.
Bernoulli's Equation: Relates pressure, velocity, and height in a flowing fluid.

These key concepts of mechanics are essential for understanding the fundamental principles of physics, particularly in the context of the NDA syllabus.

Kinematics

Study of motion without considering forces.
Key equations include displacement (( s = ut + \frac{1}{2}at^2 )), final velocity (( v = u + at )), and the relation ( v^2 = u^2 + 2as ).
Types of motion:
- Uniform Motion: Constant speed in a straight path.
- Non-Uniform Motion: Speed or direction varies.

Dynamics

Newton's Laws of Motion:
- First Law: Objects remain at rest or in uniform motion unless acted on by a net force.
- Second Law: Force (( F )) equals mass (( m )) times acceleration (( a )), formulated as ( F = ma ).
- Third Law: Every action has an equal and opposite reaction.
Applications include:
- Friction: Characterized by static and kinetic coefficients.
- Circular Motion: Governed by centripetal force ( F_c = \frac{mv^2}{r} ).

Work, Energy, and Power

Work done quantified by ( W = F \cdot d \cdot \cos(\theta) ).
Kinetic energy represented as ( KE = \frac{1}{2}mv^2 ).
Gravitational potential energy given by ( PE = mgh ).
Conservation of Energy principle states total energy in a closed system is constant.
Power calculated with ( P = \frac{W}{t} ).

Momentum

Linear momentum defined by ( p = mv ).
Conservation of momentum asserts total momentum before an event equals total momentum after.
Collisions categorized as:
- Elastic: Kinetic energy is conserved.
- Inelastic: Kinetic energy is not conserved.

Gravitation

Newton's Law of Gravitation expressed as ( F = G \frac{m_1m_2}{r^2} ).
Gravitational potential energy is calculated using ( U = -\frac{Gm_1m_2}{r} ).
Satellite motion characterized by:
- Orbital velocity formula ( v = \sqrt{\frac{GM}{r}} ).
- Orbital period given by ( T = 2\pi \sqrt{\frac{r^3}{GM}} ).

Rotational Motion

Angular displacement determined by ( \theta = s/r ) (arc length ( s )).
Angular velocity defined as ( \omega = \frac{d\theta}{dt} ).
Torque calculated as ( \tau = rF \sin(\theta) ).
Moment of inertia (( I )) depends on mass distribution, calculated as ( I = \sum m_ir_i^2 ).
Conservation of angular momentum states ( L = I\omega ) remains constant with no external torque.

Simple Harmonic Motion (SHM)

SHM characteristics include motion around an equilibrium position with a restoring force proportional to displacement.
Key equations for SHM include:
- Displacement: ( x(t) = A \cos(\omega t + \phi) ).
- Velocity: ( v(t) = -A\omega \sin(\omega t + \phi) ).
- Acceleration: ( a(t) = -A\omega^2 \cos(\omega t + \phi) ).
Period of SHM calculated by ( T = 2\pi \sqrt{\frac{m}{k}} ) where ( k ) is the spring constant.

Fluid Mechanics

Density defined by ( \rho = \frac{m}{V} ).
Pressure calculated with ( P = \frac{F}{A} ).
Archimedes' Principle states a submerged body experiences an upward buoyant force equivalent to the weight of displaced fluid.
Bernoulli's Equation connects pressure, velocity, and height in a moving fluid.