Problem 93 - University Physics with Modern Physics

Question

The eel is observed to spin at 14 spins per second clockwise, and 10 seconds later it is observed to spin at 8 spins per second counterclockwise. What is the magnitude of the eel's average angular acceleration during this time? (a) 6/10 rad/s\(^2\); (b) 6\(\pi\)/10 rad/s\(^2\); (c) 12\(\pi\)/10 rad/s\(^2\); (d) 44\(\pi\)/10 rad/s\(^2\).

Step-by-Step Solution

Answer

(d) \( \frac{44\pi}{10} \) rad/s\(^2\).

Step 1: Understand Angular Velocity

The eel initially spins at 14 spins per second clockwise. So, the initial angular velocity \( \omega_i \) is \( 14 \times 2\pi \) rad/s (since 1 spin = \( 2\pi \) radians). Additionally, in physics, clockwise rotations are typically considered negative. Thus, \( \omega_i = -28\pi \) rad/s.

Step 2: Final Angular Velocity Calculation

The eel spins at 8 spins per second counterclockwise 10 seconds later. Thus, the final angular velocity \( \omega_f \) is \( 8 \times 2\pi \) rad/s. Since counterclockwise rotations are considered positive, \( \omega_f = 16\pi \) rad/s.

Step 3: Time Interval

The time interval \( t \) during which the change in angular velocity occurs is given as 10 seconds.

Step 4: Calculate Average Angular Acceleration

Angular acceleration is calculated as the change in angular velocity divided by the time interval. So, the average angular acceleration \( \alpha \) is: \[ \alpha = \frac{\omega_f - \omega_i}{t} = \frac{16\pi - (-28\pi)}{10} = \frac{16\pi + 28\pi}{10} = \frac{44\pi}{10} \text{ rad/s}^2.\]

Step 5: Choose the Correct Answer

From the calculations, the magnitude of angular acceleration is \( \frac{44\pi}{10} \) rad/s\(^2\), matching option (d).

Key Concepts

Angular Velocity

Angular velocity is a measure of how fast an object rotates or spins. It tells us the speed of rotation and is typically expressed in radians per second. In this context, the eel's angular velocity changes over time as it spins at different speeds in both clockwise and counterclockwise directions.

• Initial angular velocity: It started spinning clockwise at 14 spins per second. Since clockwise is considered a negative direction in physics, the angular velocity is calculated as \( ext{-} 28\pi \) rad/s.
• Final angular velocity: 10 seconds later, the eel spins at 8 spins per second counterclockwise, translating to 16\pi rad/s.

This understanding of angular velocity is crucial in calculating angular acceleration, showing us how quickly the rotation is changing.

Radian

The radian is a unit used to measure angles. It's a more direct measure of angles than degrees, which simply divides a circle into 360 equal parts. A radian measures the angle created by taking the radius of a circle and wrapping it along the circle's edge. One full spin (or rotation) equals 2\( \pi \) radians, which is why we express spins in radians when calculating angular velocity.

• Understanding radians allows us to convert spins per second into a more mathematical unit, namely radians per second.
• A helpful point to remember is that one spin is equivalent to a rotation of 2\( \pi \) radians.

The radian makes calculations involving circular motion, like this eel's motion, much simpler and precise in the field of science and engineering.

Spins Per Second

The term 'spins per second' describes how many full rotations an object completes in one second. It is a basic measure of angular speed and is intuitively easy to understand, making it a useful starting point for calculations. In problems that involve complex rotational movements, understanding spins per second can help bridge the conceptual gap to more technical terms such as angular velocity.

• In the eel's problem, it starts with 14 spins per second clockwise, later changing to 8 spins per second counterclockwise.
• By converting these spins into radians per second, we can apply them in equations to find properties like angular velocity and acceleration. Each spin equals 2\( \pi \) radians, which provides the foundation for these conversions.

Utilizing spins per second effectively simplifies the process of understanding rotational dynamics before diving into deeper mathematical analysis.