A rigid body of moment of inertia I is rotating with uniform angular velocity ω about a given axis. Derive the expression for angular momentum associated with body.

Consider a rigid body consisting of particles of masses m₁, m₂........m_n at the distances r₁, r₂...........r_n respectively from the axis of rotation.
Axis of rotation XY is revolving with angular velocity .
The body is rigid, therefore the different particles revolve in a circular orbit of radii equal to their distances from the axis of rotation with their centers on XY axis, but all the particles complete one revolution in equal time.
Therefore, the linear speed of different particles is different.
Linear velocity of ith particle at a distance r_i from the axis of rotation is,

                       v_i =r_i ω

The linear momentum of i^th particle is, 

                      p_i =m_i v_i

Linear momentum is along the tangent to the circular path followed by particle.
Therefore, the linear momentum vector and radius vector are perpendicular to each other.
Angular momentum is equal to product of linear momentum and perpendicular distance from axis of rotation to direction of motion, therefore angular momentum of a particle about XY-axis is,

where,
I₁ is the moment of inertia of i^th particle about XY-axis.
Now, the total angular momentum of the body about XY is the vector sum of the angular momentum of all the constituent particles.
All the particles have angular momentum in the same direction, therefore the magnitude of the angular momentum of body is equal to the sum of the magnitude of angular momentum of all the constituent particles of the body and is directed along the XY-axis.