Rotational energy - Angular velocity of a beam

Problem Statement:

A homogeneous beam of mass M and length L is attached to the wall by means of a joint and a rope as indicated in the figure. The angle between the beam and the vertical axis is θ. If the rope is cut, determine the angular velocity of the beam as it reaches the horizontal.

Givens: The moment of inertia of the beam with respect to an axis passing through its center of mass is: ICM = (1/12)ML2

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Solution:

To determine the angular velocity of the beam when it reaches the horizontal we will use the principle of conservation of energy. In the following figure we have represented the two states we will use as well as the origin of the heights to calculate the potential energy.

In state A the beam is at rest, so that its energy is only the gravitational potential energy. The gravitational potential energy of a solid is calculated taking as a reference the height of its center of mass.

The mechanical energy in state A is therefore given by:

We must write the height h according to the givens of the problem statement. We have represented this height as well as the angle we will use to calculate it in the figure below:

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In state B the center of mass of the beam is at the origin of heights, so it will not have gravitational potential energy. The beam is rotating and therefore it will have rotational kinetic energy:

The moment of inertia of the beam with respect to an axis passing through point O is determined using Steiner’s theorem:

The mechanical energy of the beam is conserved since there is no non-conservative force (friction) that is acting on it. So:

From the last expression we can deduce the angular velocity of the beam:

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