Problem 32 - College 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 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) She then tucks into a small ball, decreasing this moment of inertia to 3.6 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) . 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

If the diver hadn't tucked, she would make 0.6 revolutions in 1.5 seconds.

Step 1: Understand Conservation of Angular Momentum

The key to solving this problem is the conservation of angular momentum. Angular momentum is conserved when no external torques act on the system. The expression for angular momentum is given by \( L = I \times \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. This means that \( I_1 \omega_1 = I_2 \omega_2 \), where the subscripts 1 and 2 refer to the diver's initial and tucked positions, respectively.

Step 2: Calculate Initial Angular Velocity

The moment of inertia when the diver is fully extended is 18 kg m². In the tucked position, her moment of inertia is reduced to 3.6 kg m², and she completes two revolutions in 1 second while tucked. First, calculate the angular velocity while tucked, \( \omega_2 \). Since there are 2 complete revolutions, \( \omega_2 = 2 \times 2\pi \) radians per second. There are \(2\pi\) radians in one revolution. So, \( \omega_2 = 4\pi \) rad/s.

Step 3: Use Conservation of Angular Momentum

With \( I_1 = 18 \) kg m² and \( I_2 = 3.6 \) kg m², use the conservation of angular momentum: \( I_1 \omega_1 = I_2 \omega_2 \). Substituting the known values: \( 18 \times \omega_1 = 3.6 \times 4\pi \). Solve for \( \omega_1 \), the angular velocity if she hadn't tucked: \( \omega_1 = \frac{3.6 \times 4\pi}{18} = \frac{4\pi}{5} \) rad/s.

Step 4: Calculate Number of Revolutions Without Tucking

Now, use the initial angular velocity \( \omega_1 \) to find how many revolutions she would make in the 1.5 seconds if she remained untucked. We have \( \omega_1 = \frac{4\pi}{5} \) rad/s. The total angular displacement \( \theta \) over 1.5 seconds is \( \theta = \omega_1 \times t = \frac{4\pi}{5} \times 1.5 \). Convert this angular displacement to revolutions by dividing \( \theta \) by \( 2\pi \): \( \text{Revolutions} = \frac{\theta}{2\pi} = \frac{(\frac{4\pi}{5} \times 1.5)}{2\pi} = \frac{3}{5} \).

Key Concepts

Moment of Inertia

Moment of inertia is a measure of how hard it is to change the rotational motion of an object. It's often compared to mass in linear motion. The larger the moment of inertia, the harder it is to start or stop something spinning. In this problem, the diver's position changes her moment of inertia.

• The open position has a high moment of inertia at 18 kg m², making her rotations slower.
• In the tucked position, the moment of inertia drops to 3.6 kg m², allowing her to spin faster.

This principle is similar to figure skaters pulling their arms in to spin faster during a jump. The same mass is easier to spin when it's closer to the axis of rotation. By tucking her arms and legs, the diver decreases her moment of inertia, thus increasing her spin speed. Understanding moment of inertia is crucial in analyzing any rotational motion.

Angular Velocity

Angular velocity refers to how fast something is spinning around a point or axis. It's like speed for rotating objects. In our example, it tells us how fast the diver rotates in the air. Let's break it down:

• When tucked, her angular velocity (\( \omega_2 \)) is given by the number of revolutions she makes, converted into radians per second. Since one revolution is \( 2\pi \) radians, two revolutions in one second give \( 4\pi \) rad/s. This angular velocity is quite high because she is spinning very fast.
• The initial angular velocity (\( \omega_1 \)), before tucking, is calculated using the conservation principle. The shift from a high moment of inertia to a low one allows us to determine how fast she would be spinning initially if she had not tucked.

Thus, angular velocity not only tells us the rotation speed but also connects dynamically with the moment of inertia to fulfill the conservation of angular momentum.

Revolutions

Revolutions tell us how many full circles a spinning object completes. For our diver, revolutions count her spins in the air from board to water. It's an important measure because it directly relates to her performance as seen by an audience.Here's an overview:

• In practice, she completes 2 revolutions when in a tucked position during a 1-second interval.
• The untucked scenario is hypothetical, calculated for 1.5 seconds. Determining the number of revolutions requires knowing the angular velocity, and dividing the total angular displacement by \( 2\pi \) gives the number of full circles or revolutions.

By understanding how revolutions relate to time and angular velocity, we can predict how the diver's actions and body positions influence her acrobatic spins.