The slender 6-kg bar A B is horizontal and at rest and the...

The slender $6-\mathrm{kg}$ bar $A B$ is horizontal and at rest and the spring is unstretched. Determine the angular velocity of the bar when it has rotated clockwise $45^{\circ}$ after being released. The spring has a stiffness of $k=12 \mathrm{~N} / \mathrm{m}$.

Key Concepts

Elastic Potential Energy

Elastic potential energy is the energy stored in deformable objects such as springs when they are compressed or stretched. It is typically expressed as ½ k x², where k is the stiffness of the spring and x is the displacement from the natural length. This concept is fundamental when analyzing systems where the recovery force of a spring is involved, as it facilitates understanding how stored energy converts into kinetic energy.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. Understanding the moment of inertia is crucial for problems involving rotational dynamics and energy conversion, as it directly influences the angular acceleration and kinetic energy of the system.

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to the rotation of an object and is given by ½ I ?², where I is the moment of inertia and ? is the angular velocity. This concept is essential for analyzing systems in which objects rotate about an axis, as it provides a way to quantify how much energy is associated with the motion of a rotating object.

Conservation of Energy

This principle states that within an isolated system, the total energy remains constant over time, with energy merely transforming from one form to another. In mechanics problems, it is used to equate the loss in potential energy (such as elastic or gravitational potential energy) to the gain in kinetic energy (often rotational or translational), enabling the calculation of variables like velocity or angular velocity without directly computing forces or accelerations.

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