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 xy 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 ith 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 radiusvector of the point where the force is applied to the top
is precessional frequency of the top about zaxis
Equation of motion for the top
where I is moment of inertia of the top about it's axis
The value of precessional frequency
