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Experiment 11 Simple Harmonic Motion

Questions How are swinging pendulums and masses on springs related? Why are these types of problems so important in Physics? What is a spring’s force constant and how can you measure it? What is linear regression? How do you use graphs to ascertain physical meaning from equations? Again, how do you compare two numbers, which have errors? Note: This week all students must write a very brief lab report during the lab period. It is due at the end of the period. The explanation of the equations used, the introduction and the conclusion are not necessary this week. The discussion section can be as little as three sentences commenting on whether the two measurements of the spring constant are equivalent given the propagated errors. This mini-lab report will be graded out of 50 points Concept When an object (of mass m) is suspended from the end of a spring, the spring will stretch a distance x and the mass will come to equilibrium when the tension F in the spring balances the weight of the body, when F = - kx = mg. This is known as Hooke's Law. k is the force constant of the spring, and its units are Newtons / meter. This is the basis for Part 1. In Part 2 the object hanging from the spring is allowed to oscillate after being displaced down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives for the acceleration of the mass:

Fnet = m a or

The force of gravity can be omitted from this analysis because it only serves to move the equilibrium position and doesn’t affect the oscillations. Acceleration is the second time- derivative of x, so this last equation is a differential equation.

To solve: we make an educated guess:

Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A, which indicates that A is the initial distance the spring stretches before it oscillates. If friction is negligible, the mass will continue to oscillate with amplitude A. Now, does this guess actually solve the (differential) equation? A second time-derivative gives:

Comparing this equation to the original differential equation, the correct solution was chosen if w2 = k / m. To understand w, consider the first derivative of the solution:

−kx = ma a = −

k m

⎛

⎝ ⎜⎜⎜⎜

⎞

⎠ ⎟⎟⎟⎟ x

d 2x dt 2

= − k m

x x(t) = A cos(ωt)

d 2x(t) dt 2

= −Aω2 cos(ωt) = −ω2x(t)

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Integrating gives We assume the object completes one oscillation in a certain period of time, T. This helps set the limits of integration. Initially, we pull the object a distance A from equilibrium and release it. So at t = 0 and x = A. (one-quarter of the total period later) the object passes the equilibrium (x=0) position.

This integration yields: Canceling the A’s and w’s

and evaluating these limits gives:

or, and adding 1 to both sides gives cos [ (wT / 4) ] = 0

The cosine is zero only when its argument is p/2 radians. Hence this last equation implies that

and rearranging gives

Finally, it is clear that w is indeed the angular frequency of the object as it oscillates up and down. Earlier it was found that w2 = k/m. Putting these two relations for w together yields:

or

This last equation is only valid if the mass of the spring is negligible compared to m. In the case of a massive spring, the actual mass that oscillates includes a portion of the mass of the spring - the upper part of the spring is stretched by the lower part. This suggests that an “effective mass” meff should be added to m to give:

or after squaring both sides

Writing this last equation this way emphasizes the equation is of the form of y = Ax + B, if y is taken as T2 and x is taken as the mass m. Therefore, a graph of T2 versus mass will yield a straight line with a physically meaningful slope and y-intercept.

v(t) =

dx(t) dt

= −Aωsin(ωt)

dx = Aω sin(ωt)dt t=0

t=T /4

∫x =A x =0

∫

−A = Aω

cos(ωt) ω

⎡

⎣ ⎢

⎤

⎦ ⎥

0

T /4

−1 = cos(

ωT 4

)−cos(0)

−1 = [cos

ωT 4

⎛

⎝ ⎜⎜⎜⎜

⎞

⎠ ⎟⎟⎟⎟ −1]

ωT 4

= π 2

ω = 2π T

2π T

⎛

⎝ ⎜⎜⎜⎜

⎞

⎠ ⎟⎟⎟⎟

2

= k m

T = 2π

m k

⎛

⎝ ⎜⎜⎜⎜

⎞

⎠ ⎟⎟⎟⎟

1/2

T = 2π

m + m eff

k

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

1/2

T 2 =

4π2

k m +

4π2

k m

eff

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Procedure Part 0 Evaluations and Assessment 1) Reminder: After this experiment, find some time to use Canvas to fill out the end-of-

semester course evaluation. 2) (Note: This task may be converted into a Canvas quiz or it may be something entirely

different. As of this writing, things are in flux.) Instructors will reserve 35-40 minutes to administer a diagnostic exam, which is part of the Department’s year-to-year assessment of the Physics 1 lecture class (PHY 1001). Please take the time to answer the 30 questions as best you can. This exam is only for students who are taking PHY 1001 concurrently (and for the first time) with this laboratory course. If you completed PHY 1001 before this semester, you are exempt from this assessment. You are also exempt if you obtained transfer or AP credit for Physics 1 and are only taking the laboratory course to obtain the fifth, required credit hour. Please follow these guidelines. Thank you for your cooperation.

a) Place all your answers on the Scantron answer sheet. Do NOT write on the exam. b) Clearly print your name, the date and the section number on the answer sheet. c) Clearly print AND bubble-in your student number on the answer sheet. d) Fill your chosen oval on the answer sheet fully using a dark pencil. e) Avoid erasing. This often makes it impossible for the Scantron machine to count

your answer sheet. f) Lab instructors must remove the exam from the room after students have finished.

3) As you may recall, part of the total grade in this course is determined by the instructor’s evaluation of student participation, engagement and preparation. This week, no specific guidelines are provided to suggest how the experiment might be performed in an equitable way to involve all members of the lab group. Instructors will be watching to determine to what degree this instructional goal is met.

Part 1 Static Investigation 1) Suspend the spring in front of a vertical meterstick, attach a weight hanger to the end of

the spring and place about 400 grams of mass on the hanger. 2) Sight along the bottom of the flat mass hanger and measure the distance the spring

stretches. Make five additional measurements using more mass of equal increments. Don’t stretch the spring to more than three times its unloaded length.

3) Plot a graph of weight (not mass) on the y-axis and displacement on the x-axis. Fit the

data points with a straight line using the method of least squares (i.e. linear regression). See Appendix D for relevant details. Display the equation of the fit on the graph and enlarge the graph according to the number of significant figures available. A graph of

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data with two significant figures should cover half a page; data with three significant figures justifies a full-page graph.

4) Use the instructions in the last part of Appendix D to calculate the propagated error in the

slope and y-intercept. Question: The slope is the spring constant (now named: k1). Theoretically, what should be the y-intercept? Hint: Note the similarity between the two equations in the left column of Figure 1.

mg = -kx

y = Ax + B

y = A x + B

Figure 1. The two equations graphed in Parts 1 and 2. Part 2 Dynamic Investigation 1) Here, the goal is to measure the period of oscillations, T, for each weight you used in Part

1. First, measure the spring’s mass. 2) Use Logger Pro and an ultrasonic motion sensor to measure and display a graph of the

mass hanger’s position vs. time. Sometimes the mass hanger’s base is too small of a target for the motion sensor. You can either elevate the motion sensor by placing on a lab stool or tape an index card to the bottom of the hanger to reflect more of the ultrasonic pulses. In either case, always keep the wire cage over the motion sensor to protect it from accidental falling masses.

3) Use the Examine command in the Analysis menu to record the time coordinate of the

peak in one oscillation graph. Then repeat this procedure for a later nth oscillation. Subtract the two times to obtain the time interval and then divide by n. Amplitudes of 3 - 4 cm are adequate.

4) Plot a second graph of T2 on the y-axis and mass on the x-axis. Use Excel to produce the

graph and fit the data with a straight line. Display the equation for the line on the graph and again calculate the error in the slope and y-intercept.

5) This graph is a plot of the equation in the upper right cell of Fig. 1. Use Fig. 1 and

determine the physical meaning of the graph’s slope and y-intercept. 6) Calculate a value for the spring constant: k2 and compare it to k1 found in Part 1.

a) Find the relevant error propagation formulas in Appendix C for the mathematical operations used to compute k2. Use these formulas to find the error in k2.

T2 =

4π2

k m +

4π2

k meff

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b) Calculate d the difference between the two k’s, and the propagated error in d. Recall, sd is the propagated error from a subtraction, so sd = [ sk1 2 + sk2 2 ]1/2. Compare d and sd to see if the two k’s agree within experimental error.

7) If the mass of the spring is negligible, then Graph #2’s y-intercept should be

approximately zero. Question: Is the y-intercept zero given the random error in the y- intercept as determined by the linear regression?

8) Reminder: all written work must be submitted for grading no later than 5PM on the last

day of classes.