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Circular motion refers to the movement of an object along the circumference of a circle. This is what happens when you swing an object around a central point, like the hands of a skater spinning around their body. In this context, each hand of the ice skater moves in a circular path around the vertical axis passing through the skater's body.

As the skater spins, each hand requires a force to keep moving in that circular path. This is known as the centripetal force, and it must always act towards the center of the circular path. The lack of any horizontal force would cause the hands to move in a straight line due to inertia, flying outward from the body like a frisbee.

This force is provided by the muscles in the skater's wrist. The muscles pull the hands inward, allowing them to stay in circular motion at the radius determined by the distance from her body.

A free-body diagram is a visual tool used to show all the forces acting on an object. It helps in analyzing the forces to solve physics problems, such as determining the net force and the resulting motion of an object.

In the skater's problem, we draw a free-body diagram for just one hand. The hand experiences two main forces:

• An inward force, which is the centripetal force provided by the wrist. This force keeps the hand moving in a circle.
• The gravitational force, otherwise known as the weight of the hand, pulls downward.

The centripetal force vector should be directed horizontally towards the center of the circle (the skater's body), while the weight vector should be pointing downward due to gravity. This diagram helps us visualize and compute the necessary forces for the skater's hand to remain in circular motion.

Rotational motion involves objects that rotate around a central point or axis. In this situation, the ice skater exhibits rotational motion as she spins around her vertical axis. This type of motion is described by the angular velocity, which in this problem, is given as the number of turns she makes per second.

The angular velocity, denoted as \(\omega\), is a crucial aspect of rotational motion. It's calculated by converting revolutions per second into radians per second, utilizing the formula \(\omega = 2\pi \times \,\text{turns per second}\). In our case, because the skater makes two turns per second, \(\omega = 4\pi\,\text{rad/s}\).

This angular speed determines how fast each hand moves along the circular path when the skater spins. Together with the radius of the hand's path, it helps calculate the centripetal force necessary to maintain the hand's circular path, thus establishing the interplay between circular and rotational motions in maintaining equilibrium.