Classical Mechanics: Kinematics and Dynamics

One- and Two-Dimensional Kinematics

Kinematics describes the motion of objects without considering the forces that cause the motion. It includes both one-dimensional and two-dimensional motion.

• Displacement, Velocity, and Acceleration: Displacement is the change in position, velocity is the rate of change of displacement, and acceleration is the rate of change of velocity.

• Equations of Motion (Constant Acceleration):

• Projectile Motion: Involves two-dimensional motion under gravity, with horizontal and vertical components analyzed separately.

• Example: Calculating the trajectory of a disk leaving a track (see Problem 5).

Forces and Newton's Laws

Newton's Laws of Motion describe the relationship between forces and the motion of objects.

• Newton's First Law: An object remains at rest or in uniform motion unless acted upon by a net force.

• Newton's Second Law: The net force on an object is equal to its mass times its acceleration.

• Newton's Third Law: For every action, there is an equal and opposite reaction.

• Applications: Calculating net torque on a satellite (Problem 1), forces in a torsion balance (Problem 2).

Circular Motion and Gravitation

Circular Motion

Objects moving in a circle experience a centripetal acceleration directed toward the center of the circle.

• Centripetal Acceleration:

• Centripetal Force:

• Example: Satellite in elliptical orbit around Earth (Problem 1).

Universal Gravitation

Newton's Law of Universal Gravitation describes the attractive force between two masses.

• Gravitational Force:

• Gravitational Constant: is determined experimentally, e.g., using a torsion balance (Problem 2).

Work, Energy, and Power

Work and Kinetic Energy

Work is done when a force causes displacement. The kinetic energy theorem relates work to changes in kinetic energy.

• Work:

• Kinetic Energy:

• Work-Energy Theorem:

Potential Energy and Conservation of Energy

• Gravitational Potential Energy:

• Elastic Potential Energy:

• Conservation of Mechanical Energy: (remains constant in absence of non-conservative forces)

• Example: Disk rolling down a track and projectile motion (Problem 5).

Momentum and Collisions

Linear Momentum

Momentum is the product of mass and velocity. In a closed system, total momentum is conserved.

• Momentum:

• Impulse:

• Conservation of Momentum:

• Example: Collisions in oscillators (Problem 4).

Rotational Dynamics and Angular Momentum

Rotational Kinematics and Dynamics

Rotational motion involves angular displacement, velocity, and acceleration. Rotational inertia depends on mass distribution.

• Angular Displacement: (radians)

• Angular Velocity:

• Angular Acceleration:

• Rotational Inertia (Moment of Inertia): (varies by shape; see table below)

• Rotational Kinetic Energy:

• Torque:

• Angular Momentum:

• Conservation of Angular Momentum:

• Example: Torsion balance and disk rolling (Problems 2 and 5).

Table: Rotational Inertia for Common Shapes

Fluid Dynamics

Flow and Pressure

Fluid dynamics studies the motion of liquids and gases. Key principles include the continuity equation and Bernoulli's equation.

• Continuity Equation: (for incompressible fluids)

• Bernoulli's Equation:

• Pressure Difference in Manometer:

• Example: Flow velocity and pressure in a tube (Problem 3).

Simple Harmonic Motion

Oscillators and Springs

Simple harmonic motion (SHM) describes systems where the restoring force is proportional to displacement.

• Equation of Motion:

• Angular Frequency:

• Period:

• Mechanical Energy:

• Example: Block-spring oscillator (Problem 4).

Waves and Sound

Standing Waves in Tubes

Standing waves are formed by the interference of two waves traveling in opposite directions. The boundary conditions depend on whether the tube is open or closed at its ends.

• Open-Open Tube: Antinodes at both ends; fundamental frequency

• Open-Closed Tube: Node at closed end, antinode at open end; fundamental frequency

• Relationship of Harmonics: Frequencies of higher harmonics are integer multiples of the fundamental for open-open tubes, odd multiples for open-closed tubes.

• Example: Standing wave analysis in a tube (Problem 1, Part 2).

Additional Mathematical Tools

Useful Integrals and Trigonometry

• Integrals: Used for calculating areas, work, and energy in continuous systems.

• Trigonometric Identities: Useful for resolving vector components and analyzing oscillatory motion.

Summary Table: Key Equations

Additional info: These study notes are based on a final exam covering topics from classical mechanics, rotational dynamics, fluid dynamics, oscillations, and waves, as indicated by the problems and equation sheet provided in the file.