A thin rod of length L and mass M is held vertically with one end on the floor and is allowed to fall. The velocity of the other end when it hits the fioor, assuming that the end which is on the floor does not slip, will be :

CONCEPT:

• Potential Energy: Potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
• Potential energy formula: The formula for potential energy depends on the force acting on the two objects. For the gravitational force the formula is:

\(W = m \times g \times h = mgh \)

where m is the mass in kilograms, g is the acceleration due to gravity, h is the height in meters.

• Rotational Kinetic Energy: The kinetic energy of a rotating body can be compared to the linear kinetic energy and described in terms of angular velocity. 
  • Rotational energy occurs due to the object's rotation and is a part of its total kinetic energy. If the rotational energy is considered separately across an object's axis of rotation, the moment of inertia observed.
    • Rotational energy is also known as angular kinetic energy defined as: "The kinetic energy due to the rotation of an object and is part of its total kinetic energy".
  • Rotational kinetic energy is directly proportional to the rotational inertia and the square of the magnitude of the angular velocity. A rolling object has both translational and rotational kinetic energy.

\(K_{R} = \frac{1}{2}Iω^{2}\)

K_R is the Rotational Kinetic energy, I is the Moment of Inertia, ω is the angular velocity

​CALCULATION:

Using conservation of Energy:

Potential energy = Rotational kinetic energy

\(mgh = (0.5)Iω^2\)

\(mg (\frac{L}{2}) = (0.5)(\frac{mL^{2}}{3})(\frac{v}{L})^2\)

\(\therefore v = \sqrt{3gL}\)

• Hence option 2 is the correct answer.