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Chapter 9: Problem 10

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. (a) Find the angular acceleration in \(\mathrm{rev} / \mathrm{s}^{2}\) and the number of revolutions made by the motor in the \(4.00-\) interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

Short Answer

Expert verified
Angular acceleration is \(-1.25 \text{ rev/s}^2\). Revolutions made: 23.32; additional time to rest: 2.664 s.

Step by step solution

01

Understand the problem

We need to find the angular acceleration and the number of revolutions made during deceleration, and then calculate the additional time required for the fan to completely stop.
02

Convert units

Convert the initial and final angular velocities from revolutions per minute to revolutions per second. Initial angular velocity: \( \omega_i = \frac{500 \text{ rev/min}}{60 \text{ s/min}} = 8.33 \text{ rev/s} \)Final angular velocity: \( \omega_f = \frac{200 \text{ rev/min}}{60 \text{ s/min}} = 3.33 \text{ rev/s} \)
03

Calculate angular acceleration

Use the formula for constant angular acceleration: \( \alpha = \frac{\omega_f - \omega_i}{t} \)Substitute the values:\( \alpha = \frac{3.33 \text{ rev/s} - 8.33 \text{ rev/s}}{4.00 \text{ s}} = -1.25 \text{ rev/s}^2 \)
04

Calculate the number of revolutions during the interval

Use the formula for the number of revolutions during uniform acceleration:\( \theta = \omega_i t + \frac{1}{2} \alpha t^2 \)Substitute the known values:\( \theta = 8.33 \times 4 + \frac{1}{2} \times (-1.25) \times 4^2 \)\( \theta = 33.32 - 10 = 23.32 \text{ revolutions} \)
05

Calculate the time to rest

Use the formula for time to reach a velocity of zero with constant acceleration:\( t = \frac{-\omega_f}{\alpha} \)Since the fan moves from 3.33 revolutions per second to rest:\( t = \frac{-3.33}{-1.25} \approx 2.664 \text{ seconds} \)Add the time already elapsed (4.00 seconds) to find total time:Total time = 4.00 + 2.664 = 6.664 seconds.

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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 something is rotating. In the context of this exercise, it refers to how quickly the electric fan's blades were spinning. The unit typically used is revolutions per minute (rev/min) or revolutions per second (rev/s). Here, the fan starts at an angular velocity of 500 rev/min, which we convert to 8.33 rev/s. It's important to convert units depending on your calculation needs to ensure consistency.

The final angular velocity after some time is 200 rev/min, also converted to 3.33 rev/s. These values help us describe how quickly the fan slows down over time.
Angular Acceleration
Angular acceleration is the rate at which angular velocity changes over time. It's a bit like how acceleration tells you how speed changes for a car. For the fan's motion, we calculated the angular acceleration using the formula: \[ \alpha = \frac{\omega_f - \omega_i}{t} \]where \(\omega_f\) is the final angular velocity, \(\omega_i\) is the initial angular velocity, and \(t\) is the time over which the change occurs.

Substituting our values, the angular acceleration \(\alpha\) came out to be \(-1.25 \text{ rev/s}^2\). A negative value indicates the fan is slowing down. This means every second, the fan is reducing its angular velocity by 1.25 rev/s.
Uniform Motion
Uniform motion is a term used when an object moves with a constant speed. While the fan in this problem doesn't maintain a constant speed throughout, the type of deceleration it exhibits can be considered uniform. This is because the angular acceleration, or deceleration in this case, is constant (\(-1.25 \text{ rev/s}^2\)).

A constant rate of change in angular velocity simplifies calculations. You can predict the future state of an object knowing that it will continue to slow down or speed up at the same rate. This principle allowed us to find both the remaining time for the fan to come to rest and the total number of revolutions completed during deceleration.
Revolutions
Revolutions refer to the number of complete spins around a central axis. In the case of our fan, determining the number of revolutions helps us understand the extent of the fan's motion before coming to rest.

We calculated this using the formula for uniformly accelerated motion: \[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]By substituting the initial angular velocity \(\omega_i\), time \(t\), and angular acceleration \(\alpha\), we found that the fan made approximately 23.32 revolutions during the 4-second interval. Revolutions are crucial in understanding rotational dynamics as they provide a tangible measure of movement over a period.

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Most popular questions from this chapter

A uniform 3.00 -kg rope 24.0 \(\mathrm{m}\) long lies on the ground at the top of a vertical cliff. A mountain climber at the top lets down half of it to help his partner climb up the cliff. What was the change in potential energy of the rope during this maneuver?

How \(I\) Scales. If we multiply all the design dimensions of an object by a scaling factor \(f,\) its volume and mass will be multiplied by \(f^{3}\) (a) By what factor will its moment of inertia be multiplied? (b) If a \(\frac{1}{48}-\) scale model has a rotational kinetic energy of \(2.5 \mathrm{J},\) what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

A sphere with radius \(R=0.200 \mathrm{m}\) has density \(\rho\) that decreases with distance \(r\) from the center of the sphere according to \(\rho=3.00 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}-\left(9.00 \times 10^{3} \mathrm{kg} / \mathrm{m}^{4}\right) r .\) (a) Calculate the total mass of the sphere. (b) Calculate the moment of inertia of the sphere for an axis along a diameter.

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 slender rod with length \(L\) has a mass per unit length that varies with distance from the left end, where \(x=0\) , according to \(d m / d x=\gamma x,\) where \(\gamma\) has units of \(\mathrm{kg} / \mathrm{m}^{2}\) . (a) Calculate the total mass of the rod in terms of \(\gamma\) and \(L .\) (b) Use Eq. \((9.20)\) to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Use the expression you derived in part (a) to express \(I\) in terms of \(M\) and \(L .\) How does your result compare to that for a uniform rod? Explain this comparison. (c) Repeat part (b) for an axis at the right end of the rod. How do the results for parts (b) and (c) compare? Explain this result.
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