A gymnast of rotational inertia \(62 \mathrm{kg} \cdot \mathrm{m}^{2}\) is tumbling head over heels with angular momentum \(470 \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s} .\) What's her angular speed?

When delving into the realm of rotational motion, rotational inertia, also known as the moment of inertia, plays an integral role. This concept signifies an object's resistance to change in its rotation. In other words, it measures the degree to which an object resists being spun about an axis. The higher the rotational inertia, the harder it is to change the object's rotational velocity. It is akin to mass in linear motion and heavily depends on how the mass is distributed relative to the axis of rotation: the farther the mass is from the axis, the greater the rotational inertia.

For example, consider two objects of identical mass - one shaped like a rod and the other shaped like a disc. If both objects are rotated about axes through their centers, the rod, with its mass distributed further from the axis, will have a higher rotational inertia compared to the disc. In the context of our exercise, the gymnast's rotational inertia of \(62 \text{ kg} \text{ m}^2\) indicates how her mass distribution affects her tumbling motion.

The concept of angular momentum is highly relevant in this discussion. Angular momentum is a measure of the quantity of rotation of a body, which is the product of its moment of inertia and its angular velocity. In essence, it represents the rotational equivalent of linear momentum. One key principle to remember is that, in the absence of external torques, the angular momentum of a system remains conserved.

This conservation can be observed in figure skaters pulling their arms in to spin faster or extending them to slow down, without any external push or pull. The gymnast in our problem would spin at a rate determined by her angular momentum of \(470 \text{ kg m}^2 / \text{s}\), which is conserved during her tumbling unless acted on by external forces. By understanding this conservation, we can predict the effects of changes in the system on the motion of the objects involved.

Approaching physics problem solving effectively requires a strong grasp of the relevant concepts and a systematic methodology. It often involves identifying the correct principles, formulating the necessary equations, and then substituting the known values to find the unknown variables.

Strategies for Problem Solving

• Identify and understand the relevant physics principles involved.
• Create a plan by deciding what formulae and concepts are applicable.
• Execute the plan by manipulating the equations and inputting the known quantities.
• Verify your results to ensure they are physically feasible and consistent with the problem's context.

By following these strategies, students can tackle a variety of physics problems more efficiently. In regards to our exercise, by recognizing that we deal with angular momentum and forming a strategy around it, students can solve for the gymnast's angular speed with confidence.