Rotational Dynamics
Rotational dynamics investigates rotational motion of objects and deals with effects that forces have on motion
Rotation of point particle
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
where 
is linear momentum vector, perpendicular to Relation between angular momentum and angular velocity vectors
The magnitude of angular momentum
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
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-axisEquation of motion for the top
where I is moment of inertia of the top about it's axis
The value of precessional frequency
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