Problem 42 - University Physics with Modern Physics

Question

A diver comes off a board with arms straight up and legs straight down, giving her a moment of inertia about her rotation axis of 18 kg \(\cdot\) m\(^2\). She then tucks into a small ball, decreasing this moment of inertia to 3.6 kg \(\cdot\) m\(^2\). While tucked, she makes two complete revolutions in 1.0 s. If she hadn't tucked at all, how many revolutions would she have made in the 1.5 s from board to water?

Step-by-Step Solution

Answer

0.6 revolutions.

Step 1: Understanding the Problem

The problem involves the conservation of angular momentum. Initially, the diver has a large moment of inertia and a certain angular velocity. She then tucks, reducing her moment of inertia, which increases her angular velocity while her angular momentum stays constant.

Step 2: Define Angular Momentum Conservation

Angular momentum, \( L \), is conserved in the absence of external torques. This means:\[ I_1 \cdot \omega_1 = I_2 \cdot \omega_2 \]where \( I_1 \) and \( \omega_1 \) are the initial moment of inertia and angular velocity, and \( I_2 \) and \( \omega_2 \) are the values when she is tucked.

Step 3: Calculate Initial Angular Velocity

We know \( I_2 = 3.6 \text{ kg} \cdot \text{m}^2 \) and she makes 2 revolutions in 1 second, thus:\[ \omega_2 = 2 \times 2\pi \text{ rad/s} = 4\pi \text{ rad/s} \]Substitute into the conservation equation:\[ 18 \cdot \omega_1 = 3.6 \cdot 4\pi \]Solve for \( \omega_1 \):\[ \omega_1 = \frac{3.6 \cdot 4\pi}{18} = \frac{4\pi}{5} \text{ rad/s} \]

Step 4: Determine Number of Revolutions Without Tucking

If she hadn't tucked, she would have kept her initial angular velocity, \( \omega_1 = \frac{4\pi}{5} \text{ rad/s} \), for the entire 1.5 seconds.Calculate the number of revolutions:\[ \text{Total angle in radians} = \omega_1 \cdot 1.5 = \frac{4\pi}{5} \cdot 1.5 = \frac{6\pi}{5} \]Convert this angle from radians to revolutions:\[ \text{Number of revolutions} = \frac{\frac{6\pi}{5}}{2\pi} = \frac{3}{5} \approx 0.6 \text{ revolutions} \]

Step 5: Conclusion

Without tucking, the diver makes approximately 0.6 revolutions in 1.5 seconds.

Key Concepts

Moment of Inertia

Moment of inertia is a critical concept in rotational dynamics as it represents how difficult it is to change an object's state of rotation. Think of it as the rotational equivalent of mass in linear motion. Just as a heavier object requires more force to accelerate, a body with a larger moment of inertia requires more torque to change its angular velocity.
For the diver in our example, moment of inertia changes significantly based on her body position. With her arms and legs extended, her moment of inertia is 18 kg \(\cdot\) m\(^2\). When she tucks into a ball, it reduces to 3.6 kg \(\cdot\) m\(^2\). This change directly impacts her angular velocity by the conservation of angular momentum.

• Broad Body Position: Higher moment of inertia, lower angular velocity.
• Tucked Position: Lower moment of inertia, higher angular velocity.

Understanding the moment of inertia helps predict how a change in body configuration can influence rotational speed, which is crucial for athletes, engineers, and anyone dealing with rotational systems.

Angular Velocity

Angular velocity is the rate at which an object rotates or revolves around a point. It's typically measured in radians per second (rad/s). In this scenario, the diver's change in body shape impacts her angular velocity.
Initially, when the diver has a higher moment of inertia, her angular velocity is lower, which aligns with her broad body position. When she tucks, reducing her moment of inertia, her angular velocity increases as she spins faster.
The relationship between moment of inertia \( I \) and angular velocity \( \omega \) is evident from the conservation of angular momentum:

• \( L = I \cdot \omega \)

This relationship shows that if the moment of inertia decreases (as when tucking), the angular velocity must increase to maintain a constant angular momentum in the absence of external torques. This principle allows divers and gymnasts to control their rotational speeds during maneuvers by simply adjusting their body's shape.

Rotational Dynamics

Rotational dynamics deals with the motion of objects that rotate and how external forces and torques affect their motion. It encompasses concepts like moment of inertia and angular velocity.

• Conservation of Angular Momentum: In a closed system with no external torques, angular momentum \( L \) is conserved. This means that any change in the moment of inertia will result in an inverse change in angular velocity to keep \( L \) constant.
• Torque and its Effects: Torque is the rotational equivalent of force. It can change an object's angular velocity by altering its moment of inertia over time, but in our example, the absence of external torques keeps \( L \) constant.

In our exercise, understanding these dynamics is crucial for computing how the diver changes her speed and number of rotations. By tucking, she effectively manipulates her rotational dynamics to maximize her revolutions, showcasing a real-world application of these principles.