Problem 29 - College Physics

Question

\(\cdot\) The spinning figure skater. The outstretched hands and arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center. (See Figure \(10.53 .\) ) When the skater's hands and arms are brought in and wrapped around his body to execute the spin, the hands and arms can be considered a thin-walled hollow cylinder. His hands and arms have a combined mass of 8.0 \(\mathrm{kg}\) . When outstretched, they span 1.8 \(\mathrm{m}\) ; when wrapped, they form a cylinder of radius 25 \(\mathrm{cm} .\) The moment of inertia about the axis of rotation of the remainder of his body is constant and equal to 0.40 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) If the skater's original angular speed is 0.40 \(\mathrm{rev} / \mathrm{s}\) what is his final angular speed?

Step-by-Step Solution

Answer

The skater's final angular speed is approximately 1.14 rev/s.

Step 1: Determine Moment of Inertia for Outstretched Arms

Calculate the moment of inertia for the outstretched arms considered as a slender rod. The moment of inertia (I) of a slender rod of length L pivoted about its center is given by \( I = \frac{1}{12} m L^2 \). Here, \( m = 8.0 \, \text{kg} \) and \( L = 1.8 \, \text{m} \). Thus, \( I = \frac{1}{12} \times 8.0 \, \text{kg} \times (1.8 \, \text{m})^2 = 2.16 \, \text{kg} \cdot \text{m}^2 \).

Step 2: Determine Moment of Inertia for Wrapped Arms

Calculate the moment of inertia for the arms considered as a thin-walled hollow cylinder. The formula for moment of inertia (I) of a hollow cylinder (ring) is \( I = m r^2 \). Here, \( m = 8.0 \, \text{kg} \) and \( r = 0.25 \, \text{m} \). Thus, \( I = 8.0 \, \text{kg} \times (0.25 \, \text{m})^2 = 0.50 \, \text{kg} \cdot \text{m}^2 \).

Step 3: Use Conservation of Angular Momentum

Apply the conservation of angular momentum since there are no external torques. The initial angular momentum \( L_i \) is equal to the final angular momentum \( L_f \). Therefore, \( I_i \omega_i = I_f \omega_f \). Initially, \( I_i = 2.16 \, \text{kg} \cdot \text{m}^2 + 0.40 \, \text{kg} \cdot \text{m}^2 = 2.56 \, \text{kg} \cdot \text{m}^2 \) and \( \omega_i = 0.40 \, \text{rev/s} = 2.51 \, \text{rad/s} \) (since \( 1 \, \text{rev} = 2\pi \, \text{rad} \)).

Step 4: Calculate Final Angular Speed

The final moment of inertia \( I_f \) is the sum of the body's moment of inertia and the wrapped arms, \( I_f = 0.50 \, \text{kg} \cdot \text{m}^2 + 0.40 \, \text{kg} \cdot \text{m}^2 = 0.90 \, \text{kg} \cdot \text{m}^2 \). Using \( I_i \omega_i = I_f \omega_f \), solve for \( \omega_f \): \[ \omega_f = \frac{I_i \omega_i}{I_f} = \frac{2.56 \, \text{kg} \cdot \text{m}^2 \times 2.51 \, \text{rad/s}}{0.90 \, \text{kg} \cdot \text{m}^2} \approx 7.14 \, \text{rad/s}. \]

Step 5: Convert Final Angular Speed to Revolutions Per Second

Convert \( \omega_f \) from rad/s to rev/s by dividing by \( 2\pi \): \[ \omega_f = \frac{7.14 \, \text{rad/s}}{2\pi} \approx 1.14 \, \text{rev/s}. \]

Key Concepts

Conservation of Angular Momentum

Conservation of angular momentum is a fundamental principle in physics that states that if no external torque acts on a system, the total angular momentum of the system remains constant. This principle is pivotal in solving problems involving rotational motion, such as the spinning figure skater exercise.
In this scenario, initially, the skater's arms are extended and then brought close to his body. Although the distribution of mass changes, the system is still isolated from external torques during the spin. This ensures that the angular momentum before and after the skater changes his arm position remains the same.
Calculating angular momentum involves the product of the body's moment of inertia and angular speed:

• Initial angular momentum: \( L_i = I_i \omega_i \)
• Final angular momentum: \( L_f = I_f \omega_f \)

The exercise solution uses this principle to balance the angular momentum before and after the skater pulls in his arms, ensuring the equality \( I_i \omega_i = I_f \omega_f \) holds, which solves for the new angular speed.

Angular Speed

Angular speed is a measure of how quickly an object rotates or spins around its axis. It is analogous to linear speed but in a rotational context. The standard unit is radians per second (rad/s), but it can also be expressed in revolutions per second (rev/s).
In the context of the figure skater, the angular speed changes as his arms move from an outstretched position to wrapped around his body. Initially, his angular speed was known as \(0.40 \text{ rev/s} \), which needed to be converted to rad/s by using the conversion \(1 \text{ rev} = 2\pi \text{ rad}\).
The conversion allows understanding the relationship between the change in moment of inertia and how tight the spin becomes seamlessly:

• Influence of arm position: Extended arms imply a higher moment of inertia, meaning slower spinning speed.
• Wrapping arms implies a lower moment of inertia, leading to faster spinning.

The calculation of new angular speed \( \omega_f = 1.14 \text{ rev/s} \) demonstrates how angular momentum conservation affects rotation rate transitions.

Physics Problem-Solving

Physics problem-solving involves logical steps, critical thinking, and mathematical calculations. This approach allows us to apply theoretical physics principles to practical problems, such as determining the speed of a spinning figure skater.
In this exercise, this systematic approach begins with understanding and calculating moments of inertia for different body configurations, before using equations to solve for changes in angular speed.
Key steps for tackling similar physics exercises include:

• Clarifying the problem and identifying what is sought (e.g., final angular speed).
• Breaking down the physical setup into simpler subsystems.
• Applying relevant physics concepts and formulas (e.g., moment of inertia, conservation of angular momentum).
• Performing calculations mindfully, including unit conversions.

This discipline aids in uncovering hidden relationships in rotational dynamics and reinforces broader understanding in various physics scenarios.

Rotational Motion

Rotational motion refers to objects moving in a circular path around an axis. This motion is not only about objects spinning but also about the theory behind how forces and torques come into play.
Understanding rotational dynamics requires familiarizing oneself with several key concepts:

• Moment of Inertia: Resistance of a body to rotate about its axis, akin to mass in linear motion.
• Torque: Rotational counterparts of force that cause changes in rotational motion.
• Angular Velocity: The rate of change of angular position of a rotating body.

In the case of the figure skater, his spinning can be analyzed by calculating the initial and final moments of inertia, and observing how body mass distribution affects the rotational speed.
This type of motion is widespread, from galaxies rotating in space to wheels turning on vehicles, making it a crucial component in understanding the universe's mechanical processes fundamentally.