Problem 71 - College Physics

Question

The eel is observed to spin at 14 rev \(/ \mathrm{s}\) clockwise, and \(10 \mathrm{~s}\) later it is observed to spin at 8 rev \(/\) s counterclockwise. What is the magnitude of the eel's average angular acceleration during this time? A. \(6 / 10 \mathrm{rad} / \mathrm{s}^{2}\) B. \(6 \pi / 10 \mathrm{rad} / \mathrm{s}^{2}\) C. \(12 \pi / 10 \mathrm{rad} / \mathrm{s}^{2}\) D. \(44 \pi / 10 \mathrm{rad} / \mathrm{s}^{2}\)

Step-by-Step Solution

Answer

D. \(44\pi / 10 \mathrm{rad} / \mathrm{s}^{2}\)

Step 1: Understand the Problem

The problem requires us to calculate the magnitude of the average angular acceleration of the eel as it changes its spin from 14 revolutions per second clockwise to 8 revolutions per second counterclockwise in 10 seconds.

Step 2: Initial Angular Velocity

The initial angular velocity of the eel is 14 rev/s clockwise. In calculations, clockwise rotations are usually taken as negative, so \( \omega_0 = -14 \times 2\pi \text{ rad/s} \).

Step 3: Final Angular Velocity

The final angular velocity is 8 rev/s counterclockwise, which is taken as positive: \( \omega_f = 8 \times 2\pi \text{ rad/s} \).

Step 4: Calculate the Change in Angular Velocity

The change in angular velocity \( \Delta \omega \) is given by the formula: \( \Delta \omega = \omega_f - \omega_0 \). Plug in the values: \( \omega_f = 8 \times 2\pi \) rad/s and \( \omega_0 = -14 \times 2\pi \) rad/s. Therefore, \( \Delta \omega = (8 - (-14)) \times 2\pi \text{ rad/s} = 22 \times 2\pi \text{ rad/s} \).

Step 5: Calculate the Average Angular Acceleration

The average angular acceleration \( \alpha \) is given by \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta t = 10 \) seconds is the time interval. Substituting the values: \( \alpha = \frac{22 \times 2\pi}{10} \text{ rad/s}^2 = \frac{44\pi}{10} \text{ rad/s}^2 \).

Step 6: Conclusion

The magnitude of the average angular acceleration is \( \frac{44\pi}{10} \text{ rad/s}^2 \), which matches option D.

Key Concepts

Rotational Motion

Rotational motion is a type of movement where an object spins around a central point or axis. This is just like the way Earth rotates around its axis. In rotational motion, just like linear motion, there are concepts of velocity, acceleration, and displacement but in a circular form. An important aspect of rotational motion is how objects move at different points relative to the axis. Closer to the axis means they move slower, and farther means faster. This is crucial when analyzing problems in physics as it helps us understand how different parts of a spinning object move over time. Rotational motion is also described using angles instead of distances like meters, which is why we use radians, a natural unit for measuring angles in physics.

Angular Velocity

Angular velocity is like how fast something spins, but it's more about the angle change over time rather than distance. It tells us how quickly an object rotates and in which direction. - **Measured in:** Radians per second often - **Clockwise vs. Counterclockwise:** In physics, clockwise angular velocity is often considered negative, while counterclockwise is positive. This helps in calculations to know the direction of rotation. Angular velocity is calculated based on how many radians an object turns in a given time. It’s a fundamental concept for solving physics problems involving spinning or rotating objects like wheels or in our example, eels! Understanding this helps us figure out the changes an object undergoes when its spin speed changes over a period of time.

Physics Problem Solving

Solving physics problems involves breaking the problem into smaller, more manageable parts. This allows us to see the relationship between different quantities and use them to find a solution. Here’s a simple approach: 1. **Identify what is given:** Look for values, directions, and time durations that are clearly stated. 2. **Determine what is needed:** Find out what you need to calculate, like acceleration or final velocity. 3. **Use relevant equations:** Choose equations that relate all the quantities given in the problem. 4. **Check your work:** After solving, revisit the problem to ensure the solution makes sense and follows logically from the data provided. In our example, we need to calculate the angular acceleration by using the changes in angular velocity over time. The step-by-step breakdown makes it so much easier to follow.

Average Acceleration Calculation

Average angular acceleration is a way to express how the rotational velocity changes over a certain period of time. The concept is built around understanding how an object's speed changes as it spins. Here’s how to calculate it: The formula used is: \[\alpha = \frac{\Delta \omega}{\Delta t}\]where:

• \( \Delta \omega \) is the change in angular velocity
• \( \Delta t \) is the change in time

This tells us how much the velocity has increased or decreased each second. For complex motions, understanding average acceleration helps in predicting future positions and speeds of rotating objects. In our eel example, we calculated how the eel's spin changed over a 10-second interval, enabling us to determine its average angular acceleration as \( \frac{44\pi}{10} \text{ rad/s}^2 \). Knowing how much the velocity changes per second is crucial for understanding and predicting the motion in rotational movements.