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Chapter 8: Problem 23

A wind turbine is initially spinning at a constant angular speed. As the wind's strength gradually increases, the turbine experiences a constant angular acceleration of \(0.140 \mathrm{rad} / \mathrm{s}^{2} .\) After making 2870 revolutions, its angular speed is \(137 \mathrm{rad} / \mathrm{s}\). (a) What is the initial angular velocity of the turbine? (b) How much time elapses while the turbine is speeding up?

Short Answer

Expert verified
Initial angular velocity \(\omega_i\) is calculated by solving \(\omega_f^2 = \omega_i^2 + 2\alpha\theta\). Time \(t\) is found using \(t = (\omega_f - \omega_i)/\alpha\).

Step by step solution

01

Understand the problem

We need to find the initial angular velocity and the time it takes for a wind turbine to speed up from its initial to a final angular velocity while experiencing constant angular acceleration. The given information includes a constant angular acceleration of \(0.140 \, \text{rad/s}^2\), a final angular speed of \(137 \, \text{rad/s}\), and the fact that the turbine makes 2870 revolutions as it speeds up.
02

Convert revolutions to radians

First, convert 2870 revolutions into radians. Since 1 revolution equals \(2\pi\) radians, we have:\[\theta = 2870 \times 2\pi = 5740\pi \text{ radians.}\]
03

Apply the kinematic equation

We use the angular kinematic equation that relates angular displacement \(\theta\), initial angular velocity \(\omega_i\), final angular velocity \(\omega_f\), and angular acceleration \(\alpha\):\[\omega_f^2 = \omega_i^2 + 2\alpha\theta\]Plug in the values: \(\omega_f = 137 \, \text{rad/s}\), \(\alpha = 0.140 \, \text{rad/s}^2\), and \(\theta = 5740\pi\):\[137^2 = \omega_i^2 + 2 \times 0.140 \times 5740\pi\]
04

Solve for initial angular velocity

Rearrange the equation from Step 3 to solve for \(\omega_i\):\[\omega_i^2 = 137^2 - 2 \times 0.140 \times 5740\pi\]Calculate \(\omega_i\):\[\omega_i = \sqrt{137^2 - 2 \times 0.140 \times 5740\pi}\]
05

Set up the time calculation

We use another kinematic equation to find the time \(t\):\[\omega_f = \omega_i + \alpha t\]Rearrange to find \(t\):\[t = \frac{\omega_f - \omega_i}{\alpha}\]
06

Solve for time

Plug the known values and the calculated \(\omega_i\) into the time equation:\[t = \frac{137 - \omega_i}{0.140}\]
07

Perform calculations to find \(\omega_i\) and \(t\)

Calculate the specific values from the previous steps to find the initial angular velocity and the total time for acceleration.

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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 revolves around a particular point or axis. It is often expressed in radians per second (rad/s). In this context, we are interested in finding the initial angular velocity of a wind turbine.
  • The turbine's initial angular velocity tells us how quickly it was rotating before the wind caused it to speed up.
  • Knowing the initial and final angular velocities, as well as the angular acceleration, allows us to calculate how quickly the turbine speeds up.
  • Angular velocity is a vector quantity, which means it has both magnitude and direction.
When dealing with problems like this, it's helpful to remember that initial angular velocity (\( \omega_i \)) is part of key equations that relate to rotational motion. Finding it involves working backward from given data on final angular speed (\( \omega_f \)) and angular acceleration (\( \alpha \)). By reconstructing the steps, we can calculate the initial angular velocity.
Angular Acceleration
Angular acceleration (\( \alpha \)) is the rate of change of angular velocity over time. It tells us how quickly something speeds up or slows down its rotational speed. For our exercise, the wind turbine experiences a constant angular acceleration.
  • Constant angular acceleration means that the rate at which the angular velocity changes remains the same throughout the motion.
  • In the problem, the given angular acceleration is \(0.140 \, \text{rad/s}^2\).
  • Angular acceleration can be used to find initial velocity, time, and various points of rotational motion.
To compute the initial angular velocity, the constant angular acceleration is used in conjunction with other known variables, such as the end velocity and angular displacement (in radians) to form the kinematic equation. Thus, it plays a central role in calculating the dynamic motion of objects like the turbine.
Revolutions to Radians
To solve kinematics problems involving rotation, we often need to convert units from revolutions to radians. This is crucial because angles in rotational equations like the kinematic equations need to be expressed in radians rather than revolutions.
  • One full revolution is equivalent to \(2\pi\) radians.
  • To convert revolutions to radians, we multiply the number of revolutions by \(2\pi\).
  • In the exercise, the turbine makes 2870 revolutions, which equals \(5740\pi\) radians.
Converting revolutions to radians allows us to use standard kinematic equations for rotational motion. It provides a consistent unit for analyzing the rotational movement and ensures accurate calculations for initial velocities and the time required for such changes in motion.
Kinematic Equations
Kinematic equations are essential tools when analyzing the motion of objects. When applied to rotational motion, they help us link angular displacement, angular velocity, angular acceleration, and time. The equations serve as the foundation to solve the wind turbine problem.
  • The key kinematic equations for rotation mirror those for linear motion but involve angular components, like angular displacement (\(\theta\)), initial angular velocity (\(\omega_i\)), final angular velocity (\(\omega_f\)), angular acceleration (\(\alpha\)), and time (\(t\)).
  • In this scenario, the kinematic equation \(\omega_f^2 = \omega_i^2 + 2\alpha\theta\) is used to relate the known variables to find the initial angular velocity.
  • Another equation, \(\omega_f = \omega_i + \alpha t\), helps determine the time it takes for the turbine to accelerate.
These equations are versatile and allow us to determine unknown quantities by relating them to known values, making them indispensable for solving problems related to angular motion.

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Most popular questions from this chapter

An auto race takes place on a circular track. A car completes one lap in a time of \(18.9 \mathrm{s}\), with an average tangential speed of \(42.6 \mathrm{m} / \mathrm{s}\). Find \((\mathrm{a})\) the average angular speed and (b) the radius of the track.

Suppose you are driving a car in a counterclockwise direction on a circular road whose radius is \(r=390 \mathrm{m}\) (see the figure). You look at the speedometer and it reads a steady \(32 \mathrm{m} / \mathrm{s}\) (about \(72 \mathrm{mi} / \mathrm{h}\) ). Concepts: (i) Does an object traveling at a constant tangential speed (for example, \(\left.v_{\mathrm{T}}=32 \mathrm{m} / \mathrm{s}\right)\) along a circular path have an acceleration? (ii) Is there a tangential acceleration \(\overrightarrow{\mathbf{a}}_{\mathrm{T}}\) when the angular speed of an object changes (e.g., when the car's angular speed decreases to \(4.9 \times 10^{-2} \mathrm{rad} / \mathrm{s}\) )? Calculations: (a) What is the angular speed of the car? (b) Determine the acceleration (magnitude and direction) of the car. (c) To avoid a rear-end collision with the vehicle ahead, you apply the brakes and reduce your angular speed to \(4.9 \times 10^{-2} \mathrm{rad} / \mathrm{s}\) in a time of 4.0 s. What is the tangential acceleration (magnitude and direction) of the car?

A racing car travels with a constant tangential speed of \(75.0 \mathrm{m} / \mathrm{s}\) around a circular track of radius \(625 \mathrm{m}\). Find (a) the magnitude of the car's total acceleration and (b) the direction of its total acceleration relative to the radial direction.

An automobile, starting from rest, has a linear acceleration to the right whose magnitude is 0.800 m/s2 (see the figure). During the next 20.0 s, the tires roll and do not slip. The radius of each wheel is 0.330 m. At the end of this time, what is the angle through which each wheel has rotated?

A rider on a mountain bike is traveling to the left in the figure. Each wheel has an angular velocity of \(+21.7 \mathrm{rad} / \mathrm{s},\) where, as usual, the plus sign indicates that the wheel is rotating in the counterclockwise direction. (a) To pass another cyclist, the rider pumps harder, and the angular velocity of the wheels increases from \(+21.7 \mathrm{to}+28.5 \mathrm{rad} / \mathrm{s}\) in a time of \(3.50 \mathrm{s}\) (b) After passing the cyclist, the rider begins to coast, and the angular velocity of the wheels decreases from \(+28.5 \mathrm{to}+15.3 \mathrm{rad} / \mathrm{s}\) in a time of \(10.7 \mathrm{s}\) Concepts: (i) Is the angular acceleration positive or negative when the rider is passing the cyclist and the angular speed of the wheels is increasing? (ii) Is the angular acceleration positive or negative when the rider is coasting and the angular speed of the wheels is decreasing? Calculations: In both instances, (a) and (b), determine the magnitude and direction of the angular acceleration (assumed constant) of the wheels.
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