A $52$-kg ice skater spins about a vertical axis through her body with her arms horizontally outstretched; she makes $2.0$ turns each second. The distance from one hand to the other is $1.50$ m. Biometric measurements indicate that each hand typically makes up about $1.25\%$ of body weight. What horizontal force must her wrist exert on her hand?

Verified step by step guidance

1. Determine the mass of one hand by calculating 1.25% of the skater's total body mass. Use the formula: \( m_{hand} = 0.0125 \times m_{body} \), where \( m_{body} = 52 \, \text{kg} \).
2. Calculate the radius of the circular motion for one hand. Since the distance between the two hands is 1.50 m, the radius for one hand is half of this distance: \( r = \frac{1.50}{2} \).
3. Determine the angular velocity \( \omega \) of the skater. The skater makes 2.0 turns per second, so \( \omega = 2 \times 2\pi \) radians per second (since one turn equals \( 2\pi \) radians).
4. Calculate the centripetal force required to keep one hand in circular motion using the formula: \( F_c = m_{hand} \cdot r \cdot \omega^2 \). Substitute the values for \( m_{hand} \), \( r \), and \( \omega \) into this equation.
5. The horizontal force exerted by the wrist on the hand is equal to the centripetal force calculated in the previous step. This force is directed toward the center of the circular motion.

Key Concepts

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. In the case of the ice skater, this force is necessary to maintain her circular motion as she spins. It can be calculated using the formula F_c = m * v^2 / r, where m is mass, v is tangential velocity, and r is the radius of the circular path.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second. For the skater, making 2.0 turns per second translates to an angular velocity of 4π radians per second. This concept is crucial for determining the tangential speed of her hands and the forces acting on them during the spin.

Torque

Torque is the rotational equivalent of linear force, representing the tendency of a force to cause rotation about an axis. In this scenario, the wrist exerts a torque to counteract the centrifugal effects experienced by the outstretched hands during the spin. The torque can be calculated by multiplying the force exerted by the wrist by the distance from the wrist to the hand.