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Chapter 9: Problem 10

At \(t=0\), a cooling fan running at \(200 \mathrm{rad} / \mathrm{s}\) is turned off and then slows down at a rate of \(20 \mathrm{rad} / \mathrm{s}^{2}\). Simultaneously (at \(t=0\) ), a second cooling fan is turned on and begins to spin from rest with an acceleration of \(60 \mathrm{rad} / \mathrm{s}^{2}\). (a) Find the time at which both fans have the same angular speed. (b) What is the angular speed of the fans at this time?

Short Answer

Expert verified
Both fans reach the same angular speed of 150 rad/s at 2.5 seconds.

Step by step solution

01

Define the problem variables

Let \( \omega_1(t) \) be the angular speed of the first fan, and \( \omega_2(t) \) be the angular speed of the second fan at time \( t \). Initially, \( \omega_1(0) = 200 \mathrm{rad/s} \) and \( \omega_2(0) = 0 \mathrm{rad/s} \). The deceleration of the first fan is \( \alpha_1 = -20 \mathrm{rad}/\mathrm{s}^2 \), and the acceleration of the second fan is \( \alpha_2 = 60 \mathrm{rad}/\mathrm{s}^2 \).
02

Write the equations of motion for each fan

For the first fan: \( \omega_1(t) = \omega_1(0) + \alpha_1 \cdot t = 200 - 20t \).For the second fan: \( \omega_2(t) = \omega_2(0) + \alpha_2 \cdot t = 0 + 60t \).
03

Set the angular speeds equal to find the time

Set \( \omega_1(t) = \omega_2(t) \). \( 200 - 20t = 60t \)Combine like terms:\( 200 = 80t \)Solve for \( t \): \( t = \frac{200}{80} = 2.5 \text{ seconds} \).
04

Find the angular speed at this time

Substitute \( t = 2.5 \) seconds into either angular speed equation:\( \omega_1(2.5) = 200 - 20 \times 2.5 \)\( \omega_1(2.5) = 200 - 50 \)\( \omega_1(2.5) = 150 \mathrm{rad/s} \).Since \( \omega_1(t) = \omega_2(t) \), \( \omega_2(2.5) = 150 \mathrm{rad/s} \) as well.

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Key Concepts

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

Angular Motion
Angular motion refers to the rotation of an object around a point or axis. In our case of the cooling fans, each fan rotates about a central axis, creating angular motion. This motion is described using angular velocity, which is measured in radians per second (rad/s).
Angular velocity tells us how fast the fan is spinning. In problems like this, we often use the symbol \( \omega \) to represent angular speed.
  • For the first fan, it starts at an initial velocity of 200 rad/s.
  • The second fan starts from rest, meaning its initial angular velocity is 0 rad/s.
The change in speed over time is affected by two factors: acceleration and deceleration, which we will explore next.
Deceleration
Deceleration is simply the slowing down of an object. For angular motion, this means the object's rate of spin decreases over time.
In our exercise, the first fan undergoes deceleration, slowing at a rate of 20 rad/s虏. This deceleration acts in the opposite direction to its initial angular velocity, causing the fan to gradually slow down.
Deceleration affects how quickly or slowly an object stops spinning. It's described using a negative angular acceleration value because it reduces the angular speed of the object. The formula for finding the new angular speed at time \( t \) can be expressed as:
\[ \omega(t) = \omega(0) + \alpha \cdot t \] where \( \alpha \) (here \(-20\,\mathrm{rad/s}^2\)) is the deceleration rate.
Acceleration
Acceleration, in the context of angular motion, refers to how quickly an object's spin increases. It is characterized by a positive change in angular velocity.
In the exercise, the second fan accelerates from a standstill with an angular acceleration of 60 rad/s虏.
A positive angular acceleration means the fan starts spinning faster as time progresses. Just like deceleration, the angular speed at any given time can be calculated using the formula:
\[ \omega(t) = \omega(0) + \alpha \cdot t \] where here, \( \alpha \) is a positive 60 rad/s虏, reflecting the fan's increasing speed.
Equations of Motion
Equations of motion are mathematical expressions used to describe the motion of objects. They help predict future positions or speeds of moving objects based on known quantities like initial speed or acceleration.
In this problem, the equations of motion for angular velocity are applied to both fans:
  • First fan: \( \omega_1(t) = 200 - 20t \)
  • Second fan: \( \omega_2(t) = 60t \)
Setting these equations equal helps us find when both fans reach the same angular speed, which happens at 2.5 seconds. At this moment, both fans spin at 150 rad/s.
Understanding equations of motion lets us solve complex problems involving rotation and enables us to predict future states of rotating systems.

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Most popular questions from this chapter

A circular saw blade \(0.200 \mathrm{~m}\) in diameter starts from rest. In 6.00 s, it reaches an angular velocity of \(140 \mathrm{rad} / \mathrm{s}\) with constant angular acceleration. Find the angular acceleration and the angle through which the blade has turned in this time.

An airplane propeller is \(2.08 \mathrm{~m}\) in length (from tip to tip) with mass \(117 \mathrm{~kg}\) and is rotating at \(2400 \mathrm{rpm}\) (rev/min) about an axis through its center. You can model the propeller as a slender rod. (a) What is its rotational kinetic energy? (b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to \(75.0 \%\) of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

Your car's speedometer works in much the same way as its odometer (see the previous problem), except that it converts the angular speed of the wheels to a linear speed of the car, assuming standard-size tires and no slipping on the pavement. (a) If your car has standard 24 -in.-diameter tires, how fast are your wheels turning when you are driving at a freeway speed of \(55 \mathrm{mph} ?\) (b) How fast are you going when your wheels are turning at 500 rpm? (c) If you put on undersize 20 -in.-diameter tires, what will the speedometer read when you are actually traveling at \(50 \mathrm{mph} ?\)

A solid copper disk has a radius of \(0.2 \mathrm{~m},\) a thickness of \(0.015 \mathrm{~m}\), and a mass of \(17 \mathrm{~kg}\). (a) What is the moment of inertia of the disk about a perpendicular axis through its center? (b) If the copper disk were melted down and re-formed into a solid sphere, what would its moment of inertia be?

a car is traveling at a constant speed on the highway. Its tires have a diameter of \(61.0 \mathrm{~cm}\) and are rolling without sliding or slipping. If the angular speed of the tires is \(50.0 \mathrm{rad} / \mathrm{s},\) what is the speed of the car, in SI units?
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