Mechanics and Motion

Constant Acceleration Equations

These equations describe the motion of objects under constant acceleration, fundamental for analyzing straight-line motion.

• Velocity as a function of time:

• Position as a function of time:

• Velocity-position relation:

• Displacement using average velocity:

• Quadratic formula (for solving kinematic equations):

Example: If a car accelerates from rest at for , its velocity is .

Newton's Laws and Forces

Newton's laws govern the relationship between forces and motion.

• Weight: (where )

• Force components: ,

• Kinetic friction:

• Static friction:

• Spring force (Hooke's Law):

• Centripetal acceleration:

Example: A block on a horizontal surface with and has .

Circular Motion & Gravitation

Circular Motion

Objects moving in circles experience centripetal acceleration and forces.

• Radial acceleration:

• Speed in a circle:

Gravitational Force

Newton's law of universal gravitation describes the force between two masses.

• Gravitational force:

• Gravitational constant:

• Orbital period:

Example: The force between two masses apart is .

Work, Energy, and Power

Work and Energy

Work is the product of force and displacement; energy is the capacity to do work.

• Work:

• Total work:

• Gravitational potential energy:

• Kinetic energy:

• Elastic potential energy:

• Conservation of energy:

Power

• Average power:

• Instantaneous power:

Momentum and Impulse

Linear Momentum

Momentum is the product of mass and velocity; impulse is the change in momentum.

• Momentum:

• Impulse:

Rotational Motion

Rotational Kinematics

Describes angular motion analogous to linear kinematics.

• Angular velocity:

• Angular displacement:

• Angular velocity squared:

• Average angular velocity:

• Arc length:

• Tangential velocity:

• Tangential acceleration:

• Radial acceleration:

Rotational Energy and Inertia

• Rotational kinetic energy:

• Moment of inertia (point mass):

• Total kinetic energy:

• Potential energy (center of mass):

Torque and Angular Momentum

• Torque:

• Sum of torques:

• Work by torque:

• Power:

• Angular momentum:

• Change in angular momentum:

• Angular momentum (point mass):

Equilibrium Conditions

For an object to be in equilibrium, both net force and net torque must be zero.

• (about any axis)

Periodic Motion

Simple Harmonic Motion (SHM)

SHM describes oscillatory motion where restoring force is proportional to displacement.

• Restoring force:

• Acceleration:

• Angular frequency:

• Frequency:

• Elastic potential energy:

• Kinetic energy:

• Position as a function of time:

• Velocity as a function of time:

• Angular frequency (spring):

• Frequency (spring):

• Period (spring):

• Angular frequency (pendulum):

• Frequency (pendulum):

• Period (pendulum):

• Total energy in SHM:

• Velocity as a function of position:

Example: A mass on a spring with has .

Summary Table: Key Equations

Additional info: This equation sheet covers the core concepts and formulas from introductory mechanics, rotational motion, and periodic motion, suitable for exam preparation in a college physics course.