/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 A rotating flywheel has moment o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 37

A rotating flywheel has moment of inertia \(12.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\) for an axis along the axle about which the wheel is rotating. Initially the flywheel has \(30.0 \mathrm{~J}\) of kinetic energy. It is slowing down with an angular acceleration of magnitude \(0.500 \mathrm{rev} / \mathrm{s}^{2} .\) How long does it take for the rotational kinetic energy to become half its initial value, so it is \(15.0 \mathrm{~J} ?\)

Short Answer

Expert verified
Calculate the values in each step to answer the question.

Step by step solution

01

Represent Given Information

Let's denote the moment of inertia as \(I = 12.0\) kg\(\cdot m^2\), initial kinetic energy as \(K_{initial} = 30.0 J\), final kinetic energy as \(K_{final} = 15.0 J\), and angular acceleration as \(\alpha = 0.500 rev/s^2\). Note that the angular acceleration needs to be converted to radian measure, which gives \(\alpha = 0.500 \cdot 2\pi rad/s^2\).
02

Calculate Initial Angular Speed

Remember that the rotational kinetic energy is given by the formula \(K = \frac{1}{2}I\omega^2\), where \(\omega\) is the angular speed. Solve this equation for the initial \(\omega\) using the initial kinetic energy value: \( \omega_{initial} = \sqrt{\frac{2K_{initial}}{I}}\).
03

Calculate Final Angular Speed

Do the same as in Step 2, but this time use the final kinetic energy to solve for the final angular speed. This gives: \( \omega_{final} = \sqrt{\frac{2K_{final}}{I}}\).
04

Calculate Time

Use the equation for angular speed \(\omega_{final} = \omega_{initial} - \alpha t\) where \(t\) is the time. Re-arrange the formula to isolate \(t\) and substitute the calculated values of \(\omega_{initial}\), \(\omega_{final}\) and \(\alpha\) to get \(t = \frac{\omega_{initial} - \omega_{final}}{\alpha}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Moment of Inertia
When dealing with rotating objects, understanding the concept of "Moment of Inertia" is crucial. Moment of inertia measures how difficult it is to change the rotational state of an object. It is often referred to as the rotational equivalent of mass for linear motion.
  • Moment of inertia depends on both the mass of the object and the distribution of that mass relative to the axis of rotation.
  • The formula for moment of inertia for a point mass is given by: \[ I = mr^2 \]where \(m\) is the mass and \(r\) is the distance of the mass from the axis of rotation.
  • For complex objects, like wheels or cylinders, the calculation involves integrating over their volume.
Moment of inertia is a key factor in determining how an object will behave when a torque is applied. In our exercise, the flywheel's given moment of inertia is \(12.0 \text{ kg} \cdot \text{m}^2\), which indicates how the mass is distributed along its axis.
Angular Speed
Angular speed is the rate at which an object rotates or revolves around an axis. It provides information about how quickly an object is changing its angle as a function of time. Angular speed is usually measured in radians per second (rad/s) and is a central concept in understanding rotational motion.
  • The relationship between linear velocity (\(v\)) and angular speed (\(\omega\)) is given by: \( v = \omega r \), where \(r\) is the radius.
  • In terms of rotation, angular speed can be affected by different factors like torque and moment of inertia.
  • For the flywheel problem, the initial and final angular speeds can be calculated using the kinetic energy formula for rotation: \[ \omega = \sqrt{\frac{2K}{I}} \]
Understanding angular speed helps determine how fast the energy in the rotating system is changing and is key to solving for time intervals during which certain conditions are met, like halving the kinetic energy.
Rotational Kinetic Energy
Rotational kinetic energy refers to the energy possessed by a rotating object due to its motion. Just like linear kinetic energy, rotational kinetic energy depends on the movement and mass distribution of the object.
  • It is calculated using the following formula: \[ K = \frac{1}{2} I \omega^2 \]where \(I\) is the moment of inertia and \(\omega\) is the angular speed.
  • This equation highlights the two key components of rotational motion: moment of inertia and angular speed.
  • In exercises like the flywheel problem, understanding rotational kinetic energy helps determine how energy shifts as the system slows down or speeds up.
For the flywheel, knowing the initial and final rotational kinetic energies allows us to calculate changes in angular speed as the system loses energy. The decrease from \(30.0 \, \text{J}\) to \(15.0 \, \text{J}\) illustrates how the system's energy and angular velocity are related.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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 rev/s \(^{2}\) and the number of revolutions made by the motor in the 4.00 s 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)?

A uniform sphere made of modeling clay has radius \(R\) and moment of inertia \(I_{1}\) for rotation about a diameter. It is flattened to a disk with the same radius \(R .\) In terms of \(I_{1},\) what is the moment of inertia of the disk for rotation about an axis that is at the center of the disk and perpendicular to its flat surface?

About what axis will a uniform, balsa-wood sphere have the same moment of inertia as does a thin-walled, hollow, lead sphere of the same mass and radius, with the axis along a diameter?

An electric turntable \(0.750 \mathrm{~m}\) in diameter is rotating about a fixed axis with an initial angular velocity of \(0.250 \mathrm{rev} / \mathrm{s}\) and a constant angular acceleration of \(0.900 \mathrm{rev} / \mathrm{s}^{2}\). (a) Compute the angular velocity of the turntable after \(0.200 \mathrm{~s}\). (b) Through how many revolutions has the turntable spun in this time interval? (c) What is the tangential speed of a point on the rim of the turntable at \(t=0.200 \mathrm{~s} ?\) (d) What is the magnitude of the resultant acceleration of a point on the rim at \(t=0.200 \mathrm{~s} ?\)

A wheel of diameter \(40.0 \mathrm{~cm}\) starts from rest and rotates with a constant angular acceleration of \(3.00 \mathrm{rad} / \mathrm{s}^{2}\). Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship (a) \(a_{\mathrm{rad}}=\omega^{2} r\) and (b) \(a_{\mathrm{rad}}=v^{2} / r\)
See all solutions

Recommended explanations on Physics Textbooks

Force

Read Explanation

Astrophysics

Read Explanation

Wave Optics

Read Explanation

Space Physics

Read Explanation

Solid State Physics

Read Explanation

Waves Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.