Problem 3 - College Physics

Question

Force on a skater's wrist. A 52 \(\mathrm{kg}\) ice skater spins about a vertical axis through her body with her arms horizontally out-stretched, making 2.0 turns each second. The distance from one hand to the other is 1.50 \(\mathrm{m} .\) Biometric measurements indicate that each hand typically makes up about 1.25\(\%\) of body weight. (a) Draw a free-body diagram of one of her hands. (b) What horizontal force must her wrist exert on her hand? (c) Express the force in part (b) as a multiple of the weight of her hand.

Step-by-Step Solution

Answer

The force her wrist exerts on her hand is approximately 77.3 N, about 12.11 times the weight of her hand.

Step 1: Calculate Skater Hand Mass

First, determine the mass of one of the skater's hands. The total body weight of the skater is 52 kg, and each hand makes up 1.25% of the body weight.Calculate the mass of one hand as follows:\[ m_{hand} = 0.0125 \times 52 \text{ kg} = 0.65 \text{ kg} \]

Step 2: Determine Tangential Velocity

The skater makes 2.0 complete turns each second. Calculate the tangential velocity of one hand. The radius \( r \) from the axis of rotation to the hand is half the total distance between the hands.\[ r = \frac{1.50}{2} = 0.75 \text{ m} \]The angular velocity \( \omega \) is:\[ \omega = 2 \times 2\pi \text{ rad/s} = 4\pi \text{ rad/s} \]The tangential velocity \( v \) is:\[ v = r \times \omega = 0.75 \times 4\pi \text{ m/s} = 3\pi \text{ m/s} \]

Step 3: Calculate Centripetal Force

The centripetal force \( F_{c} \) needed to keep the hand moving in a circle is given by:\[ F_{c} = \frac{m_{hand} \times v^2}{r} \]Substituting the values:\[ F_{c} = \frac{0.65 \times (3\pi)^2}{0.75} \text{ N} \]This results in:\[ F_{c} = \frac{0.65 \times 9\pi^2}{0.75} \text{ N} \approx 77.3 \text{ N} \]

Step 4: Express Force as a Multiple of Hand's Weight

Finally, express this force as a multiple of the weight of the hand. The weight \( W_{hand} \) is:\[ W_{hand} = m_{hand} \times g = 0.65 \times 9.81 \text{ N} \approx 6.38 \text{ N} \]The multiple is:\[ \frac{F_{c}}{W_{hand}} = \frac{77.3}{6.38} \approx 12.11 \]Thus, the horizontal force exerted by her wrist on her hand is approximately 12.11 times the weight of her hand.

Key Concepts

Free-body Diagram

A free-body diagram is a simple and helpful tool to visualize the forces acting on an object. In our scenario with the ice skater, we focus on one of her hands as it spins with her body. This diagram helps us understand how the forces keep the hand moving in a circle.

In the case of the skater's hand, the following forces are generally considered:

• Centripetal force: This is the force that acts towards the center of the circle, allowing the hand to maintain its circular motion.
• Weight: The gravitational force acting downwards on the hand.

Drawing these forces on a diagram helps clarify how the centripetal force is much larger than the weight of the hand, an important insight when analyzing circular motion.

Tangential Velocity

Tangential velocity is the speed at which an object moves along the edge of a circle. It points in the direction perpendicular to the radius at any point along its path. For the skater, her hand's tangential velocity is how fast the hand travels in a circle given the skater's spin rate.

To calculate this, we use the formula:\[ v = r \times \omega \]where:

• \( r \): The radius of the circular path, which is half the distance between her hands.
• \( \omega \): Angular velocity, the rate of rotation in radians per second.

By knowing these values, we can fully determine the tangential velocity, allowing us to understand the dynamics at play in circular motion.

Angular Velocity

Angular velocity (\(\omega\)) measures how fast an object rotates around a circle. For the skater, it tells us how quickly she completes turns around her axis.

Calculated in radians per second, angular velocity can be found using the skater's frequency of rotation multiplied by \(2\pi\):\[ \omega = 2 \times 2\pi \text{ rad/s} \]Angular velocity is different from linear velocity. Instead of focusing on the linear distance covered, it focuses on the angle turned in a specific time.

Understanding this helps explain why, even when rotating at a constant speed, different points on the arm can have different velocities. It is fundamental in analyzing rotational motion.

Centripetal Acceleration

Centripetal acceleration is the rate of change of tangential velocity as an object moves along a circular path. It always points towards the center of the circle, perpendicular to the tangential velocity.

In our exercise, the skater's hand experiences centripetal acceleration as it spins. We can calculate it using the formula:\[ a_{c} = \frac{v^2}{r} \]where:

• \( v \): The tangential velocity of her hand.
• \( r \): The radius of the circular path.

Since an object in circular motion is constantly changing direction, even at constant speed, it experiences this acceleration. This explains why a stronger force is needed to keep the hand spinning, resulting in the wrist exerting a force much greater than the weight of the hand.