Explanation of a rod hinged at O

Overview:
- A rod OA with uniform linear mass density and length L is hinged at point O.

Details:

1. Uniform linear mass density:
- This means that the mass per unit length of the rod remains constant along its entire length.
- The linear mass density can be denoted by λ (lambda) and is given by the mass of the rod divided by its length.

2. Hinged at point O:
- By hinging the rod at point O, it is free to rotate about that point.
- This allows for the rod to have rotational motion rather than just translational motion.

3. Equilibrium conditions:
- When the rod is in equilibrium, the sum of the torques acting on it must be zero.
- This means that the clockwise torques must balance the counterclockwise torques around the hinge point O.

4. Torques on the rod:
- The torque due to the weight of the rod acts at its center of mass.
- The torque due to any external forces or moments acting on the rod must also be considered.

5. Equations of motion:
- The equations of motion for the rod can be derived by considering the forces and torques acting on it.
- These equations can be used to analyze the rotational motion of the rod about the hinge point O.