A rod of length L is hinged from one end. It is brought to a...

A rod of length $L$ is hinged from one end. It is brought to a horizontal position and released. The angular velocity of the rod when it is in vertical position is : (a) $\sqrt{\left(\frac{2 g}{L}\right)}$ (b) $\sqrt{\left(\frac{3 g}{L}\right)}$ (c) $\sqrt{\left(\frac{g}{2 L}\right)}$ (d) $\sqrt{\left(\frac{g}{L}\right)}$

A rod of length $L$ is hinged from one end. It is brought to a horizontal position and released. The angular velocity of the rod when it is in vertical position is :
(a) $\sqrt{\left(\frac{2 g}{L}\right)}$
(b) $\sqrt{\left(\frac{3 g}{L}\right)}$
(c) $\sqrt{\left(\frac{g}{2 L}\right)}$
(d) $\sqrt{\left(\frac{g}{L}\right)}$

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Key Concepts

Conservation of Energy

This principle states that in the absence of non-conservative forces, the total mechanical energy of a system remains constant. In the context of rotational motion, the loss in gravitational potential energy of a falling body is converted into rotational kinetic energy.

Rotational Kinetic Energy

Rotational kinetic energy is the energy associated with objects in rotation and is given by the expression (1/2)I?², where I is the moment of inertia and ? is the angular velocity. This concept is used to determine how a rotating body’ s speed changes as its energy state changes.

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in its rotational motion and depends on the mass distribution relative to the axis of rotation. For a rod hinged at one end, the moment of inertia is specifically I = (1/3)mL².

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. The change in the vertical position of the object’s center of mass results in a change in potential energy, which can be converted to kinetic energy such as rotational kinetic energy.

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