A slender 9-lb rod can rotate in a vertical plane about a...

A slender $9-$lb rod can rotate in a vertical plane about a pivot at $B .$ A spring of constant $k=30 \mathrm{lb} / \mathrm{ft}$ and of unstretched length 6 in. is attached to the rod as shown. Knowing that the rod is released from rest in the position shown, determine its angular velocity after it has rotated through $90^{\circ} .$

Key Concepts

Conservation of Energy

This principle states that in an isolated system the total energy remains constant. In dynamics problems like this, the transformation between different forms of energy (such as gravitational potential energy, elastic potential energy, and kinetic energy) is key to determining the resulting motion. It allows one to set up an energy balance equation to solve for quantities like angular velocity when other energies change due to work done by forces like gravity and spring force.

Rotational Motion and Kinetic Energy

Rotational motion involves the movement of a body around a fixed axis. The kinetic energy associated with such motion is given by (1/2) times the moment of inertia times the square of the angular velocity. Understanding this form of kinetic energy is crucial when relating the energy changes in a rotating system to its angular velocity.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a slender rod pivoting about one end, the moment of inertia depends on the mass distribution relative to the axis of rotation. This concept is central to analyzing rotational dynamics, as it directly influences the system's kinetic energy for a given angular velocity.

Elastic Potential Energy

This concept refers to the stored energy in a spring when it is either stretched or compressed from its natural length. It is calculated using Hooke's law, which shows that the energy stored is proportional to the square of the displacement from the equilibrium position. This stored energy can be converted to kinetic energy or other forms as the system evolves.

Gravitational Potential Energy

Gravitational potential energy arises from the position of an object within a gravitational field, usually calculated relative to a chosen reference level. As the rod rotates, the change in height of its center of mass leads to a change in gravitational potential energy, which is a pivotal aspect of the energy conversions taking place in the system.

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