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Chapter 9: Problem 93

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\(^2\).

Short Answer

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

Step by step solution

01

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.
02

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.
03

Time Interval

The time interval \( t \) during which the change in angular velocity occurs is given as 10 seconds.
04

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: \[ \alpha = \frac{\omega_f - \omega_i}{t} = \frac{16\pi - (-28\pi)}{10} = \frac{16\pi + 28\pi}{10} = \frac{44\pi}{10} \text{ rad/s}^2.\]
05

Choose the Correct Answer

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

While riding a multispeed bicycle, the rider can select the radius of the rear sprocket that is fixed to the rear axle. The front sprocket of a bicycle has radius 12.0 cm. If the angular speed of the front sprocket is 0.600 rev/s, what is the radius of the rear sprocket for which the tangential speed of a point on the rim of the rear wheel will be 5.00 m/s? The rear wheel has radius 0.330 m.

A uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 4.00 kg, while the balls each have mass 0.300 kg and can be treated as point masses. Find the moment of inertia of this combination about an axis (a) perpendicular to the bar through its center; (b) perpendicular to the bar through one of the balls; (c) parallel to the bar through both balls; and (d) parallel to the bar and 0.500 m from it.

A uniform, solid disk with mass \(m\) and radius \(R\) is pivoted about a horizontal axis through its center. A small object of the same mass \(m\) is glued to the rim of the disk. If the disk is released from rest with the small object at the end of a horizontal radius, find the angular speed when the small object is directly below the axis.

A thin uniform rod of mass \(M\) and length \(L\) is bent at its center so that the two segments are now perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments meet and (b) the midpoint of the line connecting its two ends.

A thin, light wire is wrapped around the rim of a wheel as shown in Fig. E9.45. The wheel rotates about a stationary horizontal axle that passes through the center of the wheel. The wheel has radius 0.180 m and moment of inertia for rotation about the axle of \(I =\) 0.480 kg \(\cdot\) m\(^2\). A small block with mass 0.340 kg is suspended from the free end of the wire. When the system is released from rest, the block descends with constant acceleration. The bearings in the wheel at the axle are rusty, so friction there does \(-\)9.00 J of work as the block descends 3.00 m. What is the magnitude of the angular velocity of the wheel after the block has descended 3.00 m?
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