Problem 28 - College Physics

Question

\(\cdot\) On an old-fashioned rotating piano stool, a woman sits holding a pair of dumbbells at a distance of 0.60 m from the axis of rotation of the stool. She is given an angular velocity of 3.00 rad/s, after which she pulls the dumbbells in until they are only 0.20 m distant from the axis. The woman's moment of inertia about the axis of rotation is 5.00 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) and may be considered constant. Each dumbbell has a mass of 5.00 \(\mathrm{kg}\) and may be considered a point mass. Neglect friction. (a) What is the initial angular momentum of the system? (b) What is the angular velocity of the system after the dumbbells are pulled in toward the axis? (c) Compute the kinetic energy of the system before and after the dumbbells are pulled in. Account for the difference, if any.

Step-by-Step Solution

Answer

(a) 25.8 kg·m²/s (b) 4.78 rad/s (c) Initial KE: 38.7 J, Final KE: 61.9 J; difference due to work pulling in dumbbells.

Step 1: Calculate Initial Angular Momentum

The initial angular momentum of the system can be calculated using the formula: \[ L_i = I_i \cdot \omega_i \]where \( L_i \) is the initial angular momentum, \( I_i \) is the initial moment of inertia, and \( \omega_i \) is the initial angular velocity. First, calculate the initial moment of inertia \( I_i \) which includes the woman and dumbbells: \[ I_i = I_w + 2 \cdot m_d \cdot r_i^2 \]where \( I_w = 5.00 \, \mathrm{kg} \cdot \mathrm{m}^2 \) is the woman's moment of inertia, \( m_d = 5.00 \, \mathrm{kg} \) is the mass of the dumbbell, and \( r_i = 0.60 \, \mathrm{m} \) is the initial distance from the axis.Thus, \[ I_i = 5.00 + 2 \times 5.00 \times (0.60)^2 = 5.00 + 3.60 = 8.60 \, \mathrm{kg} \cdot \mathrm{m}^2 \]Now, substitute into the angular momentum formula:\[ L_i = 8.60 \cdot 3.00 = 25.8 \, \mathrm{kg} \cdot \mathrm{m}^2/\mathrm{s} \].

Step 2: Calculate Final Angular Momentum

Assuming no external torques, angular momentum is conserved. Thus, the final angular momentum \( L_f \) equals the initial angular momentum \( L_i \):\[ L_f = 25.8 \, \mathrm{kg} \cdot \mathrm{m}^2 / \mathrm{s} \]

Step 3: Calculate Final Angular Velocity

The final moment of inertia \( I_f \) when the dumbbells are pulled in is:\[ I_f = I_w + 2 \cdot m_d \cdot r_f^2 \]where \( r_f = 0.20 \, \mathrm{m} \) is the final distance from the axis.Thus,\[ I_f = 5.00 + 2 \times 5.00 \times (0.20)^2 = 5.00 + 0.40 = 5.40 \, \mathrm{kg} \cdot \mathrm{m}^2 \]With \( L_f = I_f \cdot \omega_f \), solve for \( \omega_f \):\[ \omega_f = \frac{L_f}{I_f} = \frac{25.8}{5.40}= 4.78 \, \mathrm{rad/s} \]

Step 4: Compute Initial Kinetic Energy

Initial kinetic energy \( KE_i \) is given by:\[ KE_i = \frac{1}{2} I_i \omega_i^2 \]Substituting the known values:\[ KE_i = \frac{1}{2} \cdot 8.60 \cdot (3.00)^2 = \frac{1}{2} \cdot 8.60 \cdot 9 = 38.7 \, \mathrm{J} \]

Step 5: Compute Final Kinetic Energy

Final kinetic energy \( KE_f \) is:\[ KE_f = \frac{1}{2} I_f \omega_f^2 \]Substituting the values:\[ KE_f = \frac{1}{2} \cdot 5.40 \cdot (4.78)^2 = \frac{1}{2} \cdot 5.40 \cdot 22.8484 = 61.9 \, \mathrm{J} \]

Step 6: Account for Difference in Kinetic Energy

The increase in kinetic energy from 38.7 J to 61.9 J is due to the work done by the woman as she pulls the dumbbells closer to the axis, reducing their radius and increasing their speed without external torque.

Key Concepts

Moment of Inertia

The moment of inertia is often compared to mass in linear motion. It indicates how much resistance an object has to change in its rotational motion. Just as heavier objects require more force to accelerate, an object with a higher moment of inertia requires more torque to change its angular velocity.

For a point mass, the moment of inertia depends on how far the mass is from the axis of rotation. Specifically, it's given by the formula: \( I = m imes r^2 \), where \( m \) is the mass and \( r \) is the distance from the rotation axis.

In our exercise, the woman and her dumbbells together form a rotating system with their own moment of inertia. As she pulls the dumbbells closer, their contribution to the system's moment of inertia decreases because the distance \( r \) becomes smaller.

This change directly affects rotational motion since less inertia means it is easier to increase rotation speed. Understanding this is key in many real-world applications, from figure skating spins to the balance you maintain while riding a bicycle.

Rotational Kinematics

Rotational kinematics describes the motion of objects rotating around a fixed axis. It’s similar to linear kinematics but involves angular quantities like angular displacement, velocity, and acceleration.

Angular velocity, represented by \( \omega \), is the rate at which an object rotates. It's akin to linear velocity and is measured in radians per second (rad/s).

In the scenario of the rotating piano stool, the initial angular velocity is given as \( 3.00 \) rad/s. This means every second, the woman rotates through a 3 radian angle. When she draws the dumbbells in, her angular velocity increases as a result of conserving angular momentum—more on that shortly.

Just like in linear motion, when you change an object's mass distribution, its motion changes. Here, pulling the dumbbells changes the effective moment of inertia, leading to a change in angular velocity once the system undergoes a shift in mass distribution.

Understanding rotational kinematics is crucial when analyzing anything from the wheels of a car to the blades of a fan, where rotational motion is evident.

Conservation of Angular Momentum

In closed systems where no external torques act, angular momentum remains constant. This is the principle of the conservation of angular momentum.

In our particular exercise, the initial and final angular momentum are the same because no external torques intervene. This principle is captured by the equation \( L_i = L_f \), where \( L \) represents angular momentum.

Angular momentum \( L \) is the product of moment of inertia \( I \) and angular velocity \( \omega \): \( L = I \times \omega \).

Initially, the system had a certain amount of angular momentum due to the woman's initial rotation with outstretched dumbbells. When she pulls them closer, her moment of inertia decreases—but, because \( L \) is conserved, her rotational speed must increase to maintain the same total angular momentum.

This concept can be observed in everyday life, especially in activities involving spins or turns, like when an ice skater pulls their arms in to spin faster. Understanding this conservation law is fundamental for predicting the results of rotational motion in isolated systems.