EXAMPLE 2: Euler's 2nd law for 2-D motion

The uniform bar of mass m and length l is released from rest in the horizontal position. Calculate the equation of its angular velocity as a function of the angle of rotation.

Solution:

Since O is a fixed point we can use the special form of Euler’s second law for rotation about fixed points.

This is the equation of motion for the bar. We must integrate it to get the angular velocity .

The constant of integration C is evaluated using the initial condition

at to give C=0. Therefore,