Angular Velocity of the Rod in Vertical Position

Given:
Length of the rod (L)
Acceleration due to gravity (g)

To find:
The angular velocity of the rod when it is in the vertical position.

Solution:

1. Initial Position:
The rod is hinged from one end and brought to a horizontal position. Let's consider this as the initial position.

2. Conservation of Energy:
When the rod is released, it starts swinging due to the force of gravity. As it swings, the potential energy is converted into kinetic energy and vice versa. The total mechanical energy of the system remains constant throughout the motion.

3. Potential Energy:
At the initial horizontal position, the potential energy of the rod is maximum. When the rod reaches the vertical position, the potential energy becomes zero.

4. Kinetic Energy:
At the initial horizontal position, the kinetic energy of the rod is zero. When the rod reaches the vertical position, the kinetic energy becomes maximum.

5. Conservation of Energy Equation:
Using the conservation of energy equation:
Initial Potential Energy + Initial Kinetic Energy = Final Potential Energy + Final Kinetic Energy

6. Initial Potential Energy:
At the initial horizontal position, the potential energy is maximum and can be given as mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.

7. Initial Kinetic Energy:
At the initial horizontal position, the kinetic energy is zero.

8. Final Potential Energy:
At the vertical position, the potential energy is zero.

9. Final Kinetic Energy:
At the vertical position, the kinetic energy is maximum and can be given as (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.

10. Conservation of Energy Equation (Simplified):
Since potential energy at both initial and final positions is zero, and kinetic energy at the initial position is zero, the equation becomes:
0 + 0 = 0 + (1/2)Iω^2

11. Moment of Inertia:
The moment of inertia of a rod rotating about one end is given as (1/3)mL^2, where m is the mass of the rod and L is the length of the rod.

12. Substituting the values:
(1/2)(1/3)mL^2ω^2 = 0

13. Solving for ω:
(1/6)mL^2ω^2 = 0

14. Simplifying:
ω^2 = 0

15. Angular Velocity:
Taking the square root on both sides, we get:
ω = 0

Therefore, the angular velocity of the rod when it reaches the vertical position is zero.

Answer: The angular velocity of the rod when it is in the vertical position is zero (ω = 0).