Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

(III) $\quad(a)$ Show that the total mechanical energy, $E=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2},$ as a function of time for a lightly damped harmonic oscillator is$E=\frac{1}{2} k A^{2} e^{-(b / m) t}=E_{0} e^{-(b / m) t}$where $E_{0}$ is the total mechanical energy at $t=0$ . (Assume $\omega^{\prime}>b / 2 m . )(b)$ Show that the fractional energy lost per period is$\frac{\Delta E}{E}=\frac{2 \pi b}{m \omega_{0}}=\frac{2 \pi}{Q}$where $\omega_{0}=\sqrt{k} / m$ and $Q=m \omega_{0} / b$ is called the quality factor or $Q$ value of the system. A larger $Q$ value means the system can undergo oscillations for a longer time.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

a. $$E_{0} e^{\frac{b t}{m}}$$b. $$\frac{2 \pi}{Q}$$

11:49

Shital Rijal

Physics 101 Mechanics

Chapter 14

Oscillators

Motion Along a Straight Line

Motion in 2d or 3d

Periodic Motion

University of Michigan - Ann Arbor

Simon Fraser University

McMaster University

Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

02:18

In physics, an oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The oscillation may be periodic or aperiodic.

06:04

The frequency of a damped …

03:29

The energy of a linear har…

10:26

A three-dimensional harmon…

09:36

The total energy of a part…

05:15

An undamped harmonic oscil…

03:56

(II) Obtain the displaceme…

05:29

Show that the time rate of…

05:58

06:05

Show that Schrödinger'…

this problem on the topic of oscillators, we want to find the total mechanical energy as a function of time for a lightly damned harmonic oscillator and we want to show that it is given by the expression that is provided. We also want to show the fraction of total energy lost period is given by the second expression that is provided. Now for the lightly damped harmonic oscillator, we have b squared To be much less than four mm. Kay. Which means that be squared over for m squared is less. It's a lot less than K over. Em, which means that our angular frequency we call it omega prime is approximately the resonance frequency omega not now. We assume also that the object starts to move from maximum displacement. And so the displacement X is equal to a note E to the minus Bt divided by to m times co sign of omega prime times T. And the velocity V is simply the X DT. You're the time derivative of of the expression above for the displacement. And so this is minus B over to em times A, not E to the minus Bt over to M. Call sign make a prime times t minus omega prime A. Not E to the minus Bt over to em times the sine of a mega prime times T. And so we can write this to be approximately minus omega not times a, not E to the minus Bt over to em times the sine of omega prime times T. Now that we have expressions for X and V, we can write the energy E to be a half. Okay, X squared plus a half M V squared. And so this can be written as a half. Okay, A not squared E to the minus Bt over. Em call sign squared omega prime T plus a half M omega North squared a north squared E to the minus Bt over. Em, I'm sine squared of omega prime times T. And so we can write this as a half. Okay, A not squared E to the minus B T over em James, cole, sine squared omega prime times T less than half times K A not squared E to the minus Bt over. Em I'm sine squared omega prime T. And we can see that this simplifies quite significantly to half K A not squared E to the minus Bt over em since Coulson square theater plus Sine square theater should give us one and this is simply E note e to the minus Bt over em as required. Next for part B. We want to show that the fractional energy that has lost superior delta U over E is two pi be over em omega nought or to pi over Q. Now the fractional loss of energy during one period is as follows. Not again that we use the approximation that be over to em. There's a lot less than the resident frequency omega not Which is two pi over the period T. And so this means that be times T over. Em is much less then four pi, which means that Bt over em Is less than one. And so the energy changed out to E. Is equal to the energy E minus the energy E. At time T plus a period capital T. And so this is equal to the note mm to the minus B. T. Or M. As we've shown above minus not E to the minus B into T plus the period capital T. Over em. And so we can write this as E note E. To the minus B. T. Over. Em multiplied by one minus E. To the minus mm T capital T over em. And so the change in energy over the total energy, which is the fractional loss of energy delta E of E is E, not E to the minus Bt over em into one minus E, the minus B capital T over em. And all of this divided by E note E to the minus Bt over em. So we can see that this simply becomes one minus E, the minus B capital T over em, which we can write as an approximation of one minus one minus Bt over em which is simply Bt of em and Bt over em is B times two Pi divided by um, omega not, which is simply two Pi divided by the quality factor Q as required.

View More Answers From This Book

Find Another Textbook

Numerade Educator

02:31

Drrive the formula for kinetic energy of a particle having mass m and veloci…

03:59

Rain is falling vertically with a speed of 30 metre per second a woman write…

01:39

an object of mass 50 gram has volume of 20 cm cube calculate the density of …

01:45

P,Q,R,and S are all various types of fibres. The fibres P and Q are obtained…

02:38

An astronomical telescope has an objective lens whose power is four diopter …

02:24

How can you separate the dye from black and violet inks?

02:36

You are provided with a piece of rock and common laboratory equipment. how w…