home physics online guide physics help my info terms of use contacts

 What is Physics? Mechanics Electricity and Magnetism Electric Field Gauss' Law about below subjects Electric Potential Capacity Direct Current Magnetic Field Magnetic Field Laws Magnetic Interactions Electromagnetic Induction Maxwell's Equations Oscillations and Waves Simple Harmonic Motion Damped Harmonic Motion Driven Harmonic Motion Electric Oscillation Alternating Current Wave Motion Elastic Waves Electromagnetic Waves Optics Light Waves Geometrical Optics Interference Polarization Diffraction Fraunhofer Diffraction Dispersion, Absorption, Diffusion Doppler Effect Thermodynamics Ideal Gas Molecular Statistics Transport Phenomena First Law of Thermodynamics Second and Third Laws of Thermodynamics Imperfect Gas Liquids Solids Quantum Physics Thermal Radiation Quantum Properties of Light Wave Properties of Particles Planetary Model of Atom X-Rays Particle in Potential Well Pauli Exclusion Principle Nuclear Physics Solid State Physics Appendices

# Rotational Dynamics

SI units & Physics constants

Rotational dynamics investigates rotational motion of objects and deals with effects that forces have on motion

## Rotation of point particle

Here (all units see here):

m is mass of the particle moving in x-y plane

is force vector applied in the plane of motion

is velocity vector, tangent to trajectory

is linear momentum vector, parallel to

is radius vector of curvature of trajectory, normal to trajectory

is angular velocity vector, normal to plane of motion

is angular acceleration vector, normal to plane of motion

is angular momentum, parallel to

is torque associated with the force , normal to plane of motion

d is level arm of

#### General formulas

Moment of inertia of the particle about center of rotation

Angular momentum vector is defined by vector product

where  is linear momentum vector, perpendicular to

Relation between angular momentum and angular velocity vectors

The magnitude of angular momentum

Torque vector is defined by vector product

The magnitude of torque

where:

is angle between vectors  and , shown in the above diagram

is level arm (or moment arm) of

Newton's Second Law in angular form:

- for general case

- for constant moment of inertia

## Plane rotation of symmetric solid about its axis of symmetry

Moment of inertia of the solid about axis of rotation

where

mi is small portion of mass number i at distance Ri between its center and axis of rotation (for i = 1, 2, 3, ... , n)

dV is infinitesimal volume with density at distance R from axis of rotation

Parallel Axis Theorem

where:

I is moment of inertia of solid of mass m about axis located at distance l from its center of mass

Icm is moment of inertia of the solid about axis passing thought the ceneter of mass and parallel the the previous axis

Angular momentum

where  is angular velocity of the solid

Newton's Second Law in angular form:

- for general case

- for constant moment of inertia

where  is net torque about axis of rotation associated with net external force

## General case for rotation of system of particles

Resultant angular momentum vector of the system about arbitrary point C

where  and  are position vector and linear momentum vector for i-th particle with respect to the point C (for  i = 1, 2, 3, ..., n)

Resultant torque about point C associated with external forces

where is external force applied at point  with respect to the point C (for j = 1, 2, 3, ..., k)

Newton's Second Law in angular form

Law of conservation of angular momentum of the system

## Gyroscopic motion of spinning top

Here:

is angular velocity of the top about its axis

is vertical external force applied to the top

is radius-vector of the point where the force  is applied to the top

is precessional frequency of the top about z-axis

Equation of motion for the top

where I is moment of inertia of the top about it's axis

The value of precessional frequency