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

m_i is small portion of mass number i at distance R_i 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

I_cm 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

If  then about point C

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