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[answered] Lab 10 The Harmonic Oscillator In your future, you will enc


Physics Pre Lab questions. The Lab manual is attached and is needed to answer some of the questions. I really need help and need it asap.?


Lab 10

 

The Harmonic Oscillator

 

In your future, you will encounter the harmonic oscillator repeatedly in different, seemingly unrelated, contexts and in nearly every physics course. This lab serves as an introduction to the

 

?simple? harmonic oscillator. Introduction

 

There are two common examples of harmonic oscillators in introductory mechanics: the mass on

 

a spring, and the pendulum. These are, in fact, equivalent systems. We will study the pendulum

 

because the harmonic oscillator equation arises without recourse to an empirical relation (Hooke?s

 

law), and because the pendulum demonstrates how the harmonic oscillator equation appears as

 

an approximation to a more complex exact equation (a common route to the harmonic oscillator

 

throughout physics). It might appear that the pendulum is more complicated because the motion

 

appears in the (x, y) plane rather than in a single coordinate x(t), but the pendulum is formally

 

also motion in one dimension; a single (angle) coordinate ?(t) completely describes the motion.

 

This experiment explores several of the essential features of the harmonic oscillator: a characteristic

 

frequency of oscillation independent of the amplitude, and the inevitable decrease in the amplitude

 

with time (damping). Furthermore, you will observe the small deviations from the harmonic

 

oscillator behavior at large oscillation amplitudes. Theory

 

Harmonic Oscillator Equations

 

The harmonic oscillator equation occurs repeatedly in physics, here as an instance of Newton?s

 

second law and its consequences. For one-dimensional motion of a mass attached to a spring

 

2

 

obeying Hooke?s law, m ddt2x = ?kx, or

 

d2 x

 

= ?? 2 x

 

(1)

 

2

 

dt

 

k

 

(with ? 2 = m

 

), where equation (1) is considered the standard form of the harmonic oscillator

 



 

equation. For the simple pendulum of mass M and length l, the torque equation dL

 

= ? gives

 

dt

 



 



 

d

 

2 d?

 

Ml

 

= ?M gl sin ?.

 

(2)

 

dt

 

dt

 

Solving for d2 ?

 

dt2 reveals the equation of motion for a pendulum,

 

d2 ?

 

= ?? 2 sin ?,

 

dt2

 

1 (3) with g

 

(4)

 

?2 = .

 

l

 

Equation (3) has an analytic solution in terms of elliptic functions, which might be useful for us

 

if they weren?t incredibly complex. For our purposes of avoiding elliptic functions, we will use the

 

Taylor expansion of sine:

 

?3

 

sin ? = ? ?

 

+ ... ? ?.

 

(5)

 

6

 

For sufficiently small angles, we recover the harmonic oscillator,

 

d2 ?

 

= ?? 2 ?.

 

2

 

dt (6) For larger angles, we can seek approximate solutions to equation (3), keeping only the first two

 

terms in equation (5). There exist established techniques that are generally easier than finding

 

suitable approximations for the elliptic functions. Even numerical integration of equation (3) is

 

often more convenient than dealing with elliptic functions.

 

Solutions of Differential Equations

 

Let?s consider the solution of equation (1). Equation (1) is called a differential equation, and

 

because you probably have not yet taken a course in techniques for solving differential equations,

 

you might be surprised to learn that guessing a solution is a legitimate technique for finding

 

its solution. You already know enough calculus to realize that the derivative of the exponential

 

function returns another exponential, and the second derivatives of the sine and cosine functions

 

return the same functions. A line or two of algebra will confirm that

 

x(t) = A sin(?t) + B cos(?t) = C sin(?t + ?) (7) (with ? = 2?? = 2?/T ) will solve equation (1) for any choice of A and B (or C and ?). The second

 

form is likewise a solution, and a few trigonometric identities will show that the two forms are

 

interchangeable. The notation used in equations (1) and (3) was deliberately chosen to introduce

 

?, the angular frequency of the oscillation. Although one often talks of the period T or frequency

 

? of oscillation, physics rarely deviates from the notation of equation (7) in terms of angular

 

frequency. Equation (1) is the standard form of the harmonic oscillator equation, and the solution

 

is an oscillation with angular frequency ?.

 

You might think that the first form of equation (7) is redundant, that sin(?t) or cos(?t) is a

 

satisfactory solution, but that is not quite correct. You should recall that the general equation of

 

motion for motion with constant acceleration,

 

d2 x

 

= a,

 

dt2 (8) 1

 

x(t) = x0 + v0 t + at2 .

 

2 (9) has the general form 2 This solution includes two arbitrary constants, the initial velocity (v0 ) and position (x0 ). The

 

general solution to a second-order differential equation (one with second derivatives, such as acceleration) always has two free constants, often taken as initial velocity and position. The two forms

 

of equation (7) are equivalent, the choice is that of convenience in particular cases, and the two

 

constants may be chosen to match any initial velocity and position.

 

Experimental tests of these equations range from basic to thorough. The most basic test for a

 

pendulum would be simply to measure the period and confirm that it corresponds to the ? of

 

equation (3). An additional test would be to confirm that the period did not depend on the

 

initial amplitude over some range of amplitudes. The most complete test would be to measure the

 

?displacement? ?(t) and confirm that it is described by equation (7).

 

The Damped Harmonic Oscillator

 

Before proposing experimental tests, however, it is important to ask whether equations (1) and

 

(3) are adequate representations of physical oscillators. The solution in equation (7) continues

 

to oscillate as t ? ?, whereas real oscillators always slow down and eventually stop (x ? 0 as

 

t ? ?). To explain the slowing, an additional force term is required in the equation: a nonconservative force like friction that will remove energy from the oscillations. Equation (1) may be

 

modified to

 

dx

 

d2 x

 

= ?? 2 x ? ? ,

 

(10)

 

2

 

dt

 

dt

 

where the new term is proportional to velocity. This is mathematically convenient because the new

 

term is always in the direction opposite to the velocity and therefore always does negative work,

 

it extracts energy from the system (friction is a constant dissipative force, but it changes direction

 

when the velocity changes sign and cannot be introduced analytically to give a simple equation like

 

equation (10)). For mechanical systems, equation (10) represents viscous (fluid) damping rather

 

than the more usual frictional losses, but the justification for using the more manageable form of

 

equation (10) is that for weak damping (?  ?), the effect of frictional losses can be adequately

 

represented by an appropriate choice of ?. Equation (10) also applies to the pendulum (for small

 

angles) with x(t) replaced by ?(t) as in equation (3). Note that the solution in equation (7) does

 

not satisfy equation (10). Nevertheless, equation (10) is a special type of differential equation, the

 

coefficients are all constants. A function, like an exponential, whose derivative returned the same

 

function times a constant would seem to be a possible solution. The answer is indeed as simple

 

as it seems, provided you use complex numbers. Assuming a solution of the form x(t) = Ae?t ,

 

equation (10) requires

 



 

? 2 + ? 2 + ?? Ae?t = 0.

 

(11)

 

This equation has a trivial solution when A = 0, or if ? is a root of the quadratic ? 2 + ? 2 + ?? = 0,

 

solving for ?. The roots are

 

r

 

p

 

 ? 2

 

?? ? ? 2 ? 4? 2

 

?

 

?

 

?

 

= ? i ?2 ?

 

?? =

 

= ? ? i?r ,

 

2

 

2

 

2

 

2 3 where the latter expressions assume ? > ?2 , that the damping is not strong. For weak damping,

 

p

 

(?  ?), ?r = gl ? ?. In the zero damping limit, ?? = ?i?. Note that if ? is complex, |?| = ?.

 

The general solution is then

 

x(t) = Aei?+ t + Bei?? t ,

 

(12)

 

where A and B are also complex numbers. The only

 

question is how to interpret this mathematical solution

 

of the equation. The question is quickly answered by

 

noting that the original equation (10) has only real

 

coefficients. Imaginary parts must both be zero and

 

therefore the solutions separate as the <e {x(t)} and

 

=m {x(t)} each being solutions. There are several alternative approaches at this point, but the most direct

 

is to set B = 0, write the constant A = |A|ei? , and

 

choose <e {x(t)} as the solution

 

x(t) = |A|e??t/2 cos (?r t + ?) , (13) Figure 1: A representative plot of equation (13). a suitable general solution with two arbitrary constants.1 A representative example of equation

 

(13) is shown in Figure 1, an oscillation with an exponentially decreasing amplitude.

 

  i(?+ t+?)

 

The velocity associated with equation (13) is <e dx

 

,

 

which

 

is

 

v(t)

 

=

 

<e

 

i?

 

|A|e

 

, and

 

+

 

dt

 

simplifies to

 

v(t) = ?|A|?e??t/2 sin(?r t + ? + ?)

 

(14)

 

(with ?+ = ?ei? ), which has the same form as equation (13). It oscillates at the same frequency

 

?r and damps at the same exponential rate. The phase of v(t) is shifted from x(t) by ?/2 (90? ,

 

cosine to sine) with an additional contribution from damping ?, which is small for weak damping.

 

Lab 10 is designed to prove that the motion of the pendulum has the form of equation (13), with

 

a weak exponential decay ? (to be determined), and an oscillation at ?r ? ? from equation (3)

 

that remains constant.

 

General Pendulum Motion

 

General pendulum motion is quite complex, unfortunately one would have to replace the string

 

used in these experiments with a rigid bar to explore motion with ? > ?/2, and friction would

 

play a larger role in the experiment. When sin(?) ? ? is no longer an adequate approximation,

 

the name changes from the simple harmonic oscillator to the nonlinear harmonic oscillator. You

 

will not begin to analyze nonlinear oscillators until your next mechanics course, but you can look

 

for nonlinear effects in this experiment. The easiest effect to measure is the small change in the

 

1 There is a slight subtlety here. In equation (7), both A and B had to be kept to provide two arbitrary constants

 

in the solution. In this case, the complex A and B provide four constants so that there are two for the two solutions,

 

<e {x(t)} and =m {x(t)}. Because we only need one solution with two constants, we arbitrarily, but conveniently,

 

throw two away by choosing B = 0. 4 period of oscillation for large-amplitude oscillations. The result of a page or two of algebra is an

 

approximation for the period,

 



 



 

1 2

 

(15)

 

T = T0 1 + ?0 ,

 

16

 

where T0 is the period for small oscillations and ?0 is the initial angular amplitude. Procedure and Analysis

 

For this experiment, you will need:

 

?

 

?

 

?

 

?

 

?

 

? Tall pipe stand

 

Camera mounted on a short pipe stand

 

Aluminum cylinder on string attached to protractor assembly

 

Black viewing background

 

Meter stick

 

Photogate with small mounting stand Part 1 - Procedure

 

You will take several data sets to analyze various features of the solution to the harmonic oscillator.

 

First, you will take a short movie to establish that the motion is indeed harmonic, that

 

?(t) ? x(t) = A sin(?t + ?). (16) Mount the protractor assembly with the aluminum bob on the tall pipe stand and mount the

 

camera on the short pipe stand for horizontal viewing. Set up a black viewing background behind

 

the bob. After the apparatus is properly adjusted, take a movie, beginning with the pendulum in

 

equilibrium. Displace it ? 20? in one direction and release it. Continue recording through at least

 

two periods of oscillation.

 

Part 1 - Analysis

 

Using ImageJ, measure the position of the bob at its equilibrium point when the pendulum is not

 

oscillating. This will provide the point x = 0 from which displacement x(t) is measured. Measure

 

x(t) for frames spanning somewhat more than one period2 and enter this data into the provided

 

spreadsheet. You may enter the time in physical units or in frame number and position in pixels.

 

The value of T will depend on the choice of time scale, but that has no effect on the result, which

 

is to establish whether x(t) has a sinusoidal time dependence. Enter the equilibrium position. You

 

may then enter values for the constants in equation (16) (A, ?, and ?), and adjust them for a good

 

fit to the data points on the plot. Although you will have chosen a good reference for x = 0, you

 

may also have to adjust the equilibrium position for best fit. To choose starting fitting parameters,

 

2 Since the analysis for the pendulum is only correct for small angles, you will use a small oscillation angle and

 

need only measure the horizontal displacement x(t) in pixels. 5 you can look at the plot of the data and infer A from the maximum and minimum values and the

 

period from where the plot crosses the x axis. From there, you will have to experiment to find the

 

best fit. As usual, the spreadsheet will calculate ?2 to assist with the fitting and assessment of the

 

quality of fit. Is the motion harmonic?

 

Part 2 - Procedure

 

To investigate other features of the motion, you will use different, more precise techniques. Remove

 

the camera and associated apparatus. Mount the photogate, center it on the pendulum bob in its

 

equilibrium position, and connect it to the computer through Capstone (as you did in the spring

 

experiment). For this experiment, you may provide Capstone with the size of the bob so the data

 

set will include both time and velocity for each pass through the gate. This velocity at the bottom

 

of the swing is the maximum |v| of the oscillation. Release the bob with a displacement of 40? to

 

45? and start the data acquisition. Continue until the amplitude is less than 10? (the measurement

 

stops when the amplitude decays to the bob size, but the interpretation fails earlier when the

 

amplitude is not significantly greater than the size of the bob).

 

Part 2 - Analysis

 

You may transfer the data to Excel or use Capstone?s built-in fitting function. If you use Excel

 

with the provided spreadsheet, be sure to adjust the data to conform to the format indicated there.

 

Make plots of v(t) and T (t).3 Because the timing is so accurate, you can obtain T for each period,

 

which are alternate times in the data sequence, your you could average over two or more periods.

 

Never compare crossing times in opposite directions (why not?). Compare your results with the

 

predictions of equations (4), (13), and (15). Although you have not made accurate measurements

 

of the amplitude, you have a rough measure of the initial displacement angle and an accurate

 

measure of v(t), the amplitude of velocity as it passes through x(0), which is strictly proportional

 

to the amplitude of oscillation. The exponential decay predicted by equation (13) is best judged

 

by using a log plot for v(t), but Capstone will fit an exponential decay quite well.

 

Remaining Analysis

 

Describe your conclusions regarding the validity of equations (6), (13), (14), and (15); find ? and

 

compare your measured period T for large oscillations with equation (15).

 

During this semester, you have used a digital camera and a digital timer (photogate) for mechanics

 

experiments. What are the advantages and disadvantages of each? Explain what determines the

 

choice in experimental design.

 

3 The notation here might be considered mathematically irregular. These are not the continuous functions x(t),

 

v(t), and ?(t) as used in equations (1), (3), (7), and (13). The data is a discrete set of tn , v(tn ), the times of gate

 

entry and the velocity at that time, which is a maximum in |v|, from which a discrete set of periods Tn may be

 

calculated. In practice, of course, all plots are based on discrete points, whether the data is intrinsically discrete

 

or considered to represent a continuous function. The only difference is whether the discreteness is intrinsic or

 

artificial for the purpose of plotting. 6 When you have completed the experiment and are ready to return the equipment for storage, wrap

 

the photogate cable the long way around the gate assembly, then complete your lab report and

 

have a nice summer break. Reference

 

Ohanian and Markert, Physics (Third ed.), chapter 15. 7

 


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