Kinematics

Definitions and Equations of Motion

Kinematics is the study of motion without considering its causes. The following equations describe the relationships between displacement, velocity, acceleration, and time for objects moving in one dimension with constant acceleration.

• Displacement (): The change in position of an object.

• Velocity (): The rate of change of displacement.

• Acceleration (): The rate of change of velocity.

Key Equations:

Example: If a car starts from rest () and accelerates at for $5v = 0 + 2 \times 5 = 10\ \mathrm{m/s}$.

Length and Integration

Calculating Distance and Area

Integration is used to find the total displacement or area under a curve, such as velocity vs. time.

Example: The area under a velocity-time graph gives the displacement.

Other Useful Mathematical Tools

Vector Addition and Differentiation

Vectors are quantities with both magnitude and direction. Addition and differentiation of vectors follow component-wise rules.

Example: If and , then .

Dynamics

Newton's Second Law and Forces

Dynamics deals with the causes of motion, primarily forces. Newton's Second Law relates force, mass, and acceleration.

Example: A block on a frictionless surface with a force applied has .

Work and Energy

Work, Kinetic Energy, and Potential Energy

Work is the energy transferred by a force acting over a distance. Kinetic energy is the energy of motion, and potential energy is stored energy due to position.

• Work-Energy Theorem:

Example: Lifting a box vertically: .

Oscillations

Simple Harmonic Motion (SHM)

Oscillatory motion occurs when an object moves back and forth about an equilibrium position. SHM is characterized by sinusoidal motion.

• Period:

• Frequency:

• Angular frequency:

Example: A mass on a spring with has .

Fundamental Constants and Physical Properties

Physical Constants

Physical constants are universal values used in physics calculations.

• Speed of light,

• Gravitational constant,

• Planck's constant,

• Elementary charge,

• Mass of electron,

• Mass of proton,

• Earth's radius,

Example: The gravitational force between two masses apart is .

Rotational Motion

Moments of Inertia

The moment of inertia quantifies an object's resistance to rotational acceleration about an axis. It depends on the mass distribution relative to the axis.

Example: A disk of radius has .

Additional info: Some equations and constants have been expanded for clarity and completeness. The table of moments of inertia is reconstructed from the image for study purposes.