Simple Harmonic Motion (last edited December 30, 2011)

Dr. Larry Bortner

 Learning Objectives

Determine the period of an oscillating mass-spring system from plots of x vs. t and v vs. t.

Find the spring constant statically and dynamically.

(Calculus-based) Show how much damping affects the frequency of oscillation.

 Purpose

To find the spring constant both by verifying Hooke’s Law and by analyzing the oscillatory motion of the loaded spring.

 Background

Understanding oscillations can go a long way towards making sense of a wealth of physical phenomena such as the wiggle of an atom, the spin of a galaxy, MRI’s, earthquakes, Doppler radar, the temperature of stars, cell damage from radiation, and the collapse of a bridge.

 

In the first quarter of physics labs, you investigated uniform circular motion, which is one type of oscillation.  If you observe this motion perpendicular to the plane of the circle (from the side instead of from the top), the object appears to be going back and forth instead of round and round. This is simple harmonic motion.

 

A swinging pendulum also undergoes simple harmonic motion, as long as the swing isn’t too large. (The motion for a larger swing is still harmonic but no longer simple.)

 

If you have a weight hanging from a spring, and you pull it down and let it go, the weight starts bobbing up and down. (If you tilt your head, it’s going back and forth.) Any guesses as to what kind of motion this is? Simple harmonic! In this experiment we look at both the static stretch of the spring and the oscillatory motion and use these observations and analyses as a model of all simple harmonic motion.

Hooke’s Law Theory

Let’s say we have mass attached at the lower end of a spring hanging vertically from a support. The nonmoving mass is in an equilibrium position. For analysis purposes, put the origin of the y-axis here. Now move the mass to a new position y by stretching or compressing the spring. Hooke’s law says that the spring produces a restoring force F that is proportional to the distance from equilibrium and in a direction that will return the mass to the equilibrium position. In equation form,

 

(1)              

 

where k is the spring constant and the minus sign indicates the direction. (We’re dealing with 1D vectors.) From Newton’s 3rd law, this restoring force is equal and opposite the stretching force,

 

(2)             

 

and if we plot Fstretch vs. y, we’ll get a line with a slope equal to the spring constant.

 

If we lift the mass up from its equilibrium position and let it go, what is the position of the mass at a given time after letting it go? Eq. 1 leads us to a second order differential equation to solve, definitely beyond the scope of this course. But we can come up with the correct expression for y(t) without the high powered mathematics.

Circular Motion

Figure 1

Figure 1 Linear and circular quantities.

 

Let’s review the relationships between linear and circular quantities in 2D and the motion of a point object of mass mpoint going around in a circle of radius r at a constant speed v. Set the origin of an xy coordinate system at the center of the circle as in Fig. 1. There are two different pairs of numbers that tell us exactly where the object is: one pair are the coordinates x and y, the other are the radius r and the angular displacement (angle) θ as measured in a counterclockwise direction from the x-axis. Alternatively, these pairs of numbers also unambiguously describe the position vector rP.

 

Define the following quantities:

s is the distance a point in circular motion travels along the circular path

v is the speed of this point, or how fast s is changing

a is the centripetal acceleration required to change the velocity so that the object keeps going in a circle

r is the radius of the circle

θ  is the angle as measured from some fixed reference point on the circle

ω is the angular frequency or angular velocity, that is, how fast the displacement θ is changing

 

The linear quantities s, v, and a are related to the circular quantities r, θ, and ω in the following manner:

(3)               

 

Projection along y-axis  

The vertical or y components of the vectors  (refer to Fig. 1) are

 

(4)            

 

From Newton’s second law, the component of the force that causes motion in the y direction is

 

(5)             

 

Eq. 5 is the same form as Eq. 1, with the spring constant identified as

 

(6)              

 

If the circular motion is uniform, ω is constant and θ = ωt +π/2 assuming we start at the maximum y (θ must be in radians). Call this maximum y an amplitude A and the expressions for y and v as functions of time become

 

(7)               

 

Eqs. 7 describe not just the y position and velocity of a mass going around in a vertical circle but also the y position and velocity of a mass bobbing up and down at the end of a spring. That is, these are the equations of simple harmonic motion. Any time a force acting on an object can be expressed in the form of Eq. 5, simple harmonic motion occurs whenever the object is nudged out of equilibrium and released.

 Two Ways to Find k: Equations to Verify

Recall that the period T is related to the angular frequency by

 

(8)                

 

In the experiment you do today, the spring has a nonzero mass which has to be taken into account in Eq. 6. A more advanced analysis gives the additive correction factor mc to be

 

(9)                

 

In this experiment, the load on the spring is the mass m placed on a mass hanger of mass mh. We have to replace the ideal mass mpoint on a massless spring in Eq. 6 with the total effective mass m+mh+mc.

 

Eq. 2 becomes (using a negative force since the direction is downward; the mass hanger and the mass of the spring have no effect on stretching from an arbitrary zero position)

 

(10)              

 

Eq. 6 with Eq. 8 transforms to (assuming the oscillating mass is m+mh+ mc)

 

(11)              

 

If Hooke’s law holds, plotting the 1D vector weight −mg vs. the 1D vector position y in eq.10 gives a straight line with a slope equal to the spring constant and a y-intercept of zero. Similarly, if the motion is simple harmonic, plotting 2m vs. T2 in eq.11 gives a line with a slope of k and a y-intercept of −4π2(mh+mc). (You can extract an experimental value of mc from the intercept.)

 Damped Harmonic Motion Theory (General Physics)

In our model of oscillatory motion, we have failed to take into account friction (damping). In the real world, the bobbing of the spring eventually dies down to zero velocity. The typical damping force, called viscous damping, is opposite the movement and proportional to the speed. The total force F acting on the total mass m' = m+mh+mc becomes

 

(12)                  

 

where b is the damping constant. Eq. 12 leads to another differential equation the solution of which we’ll just state:

 

(13)                  

 

Comparing this to the expression for the position in Eq. 7, we see that it is the same except that the amplitude is decaying in time instead of being constant. The damping, or slowing down, changes Eq. 11 and leads to a different frequency. The relationship between the period T, the load m, and the spring constant k becomes

 

(14)                 

 

Recall that ω=2π/T. As above, we denote m' = m+mh+mc and we can rewrite Eq. 11 as

 

(15)                  

 

where the zero subscript represents the undamped case. To calculate this, m must be in kilograms. Eq. 14 then becomes

 

(16)                 

 

The uncertainty operator u{} introduced in Appendix 2: Propagation of Uncertainty is essentially the same thing as the difference operator Δ(). From rule 4 in the Lab References, u{ω02}=2 ω0 u{ω0}. Combining this with Eq. 16, we can show that for the linear frequency f

 

(17)                 

 

The causes of damping in the system you are investigating today are more complex than viscous forces, leading to a linear damping. But we still use the viscous model in the analysis.

 Procedure

You need the following items:

*        Conical brass spring hanging from support.

*        Mass hanger with attachment for connecting to a string.

*        Mass set.

*        Rotary motion sensor to measure position (u{y}=0.005 m), pulley, and string.

*        Computer and Science Workshop interface.

*        Balance.

 Hooke’s Law Experiment


 

1.   Measure the mass of the conical spring :

a.   Take the spring off the hook at the top and lower it straight down until the mass hanger rests on the table. The string should stay on the rotary motion sensor pulley at the top and the regular pulley at the bottom. Please do not remove the mass hanger from the string.

b.   Remove the spring from the mass hanger.

c.   Record the mass of the spring.

d.   Record mh = 53.3 g.

e.   Return the spring to the support with the smaller diameter part at the top and reattach the mass hanger.

2.   Click on Data Studio Experiments> Second Quarter>Simple Harmonic Motion. A DataStudio window appears with two graphs for time-dependent position and velocity and a digital readout for position.

3.   Set up a data table to record the mass m in grams added to the mass holder and the position y (the stretch) in meters. Your first point will be (0,0).

4.   Click on the DataStudio Start button in the upper left corner of the window. An instantaneous readout of the mass holder position appears in the digital display and traces start on both the graphs.

5.   Supporting the mass holder, slip on a 20-g weight. Ease the mass holder down until the spring is fully supporting it. Record the added mass (20 g) and the resulting stabilized position from the digital readout. Ignore the graphs for now. Do not press Stop.

6.   Repeat the preceding step for the following masses (in grams), in this order: 40, 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 400, 300, 200, 100, and 0. This gives you a total of 18 points for Hooke’s law. When adding or taking off mass, hold the mass holder at its previous position until the change has been made. Then slowly let the mass find its equilibrium position. (Note: If you click on the Stop button before you’ve recorded all your points, you have to start over. Total bummer.)

7.   Click on the Stop button after you return to zero mass.

 Simple Harmonic Motion Experiment

Figure 2

Figure 2 DataStudio window for this experiment.

 

 


8.     Set up a data table with six columns labeled as m (g), Run #, t0 (s), t0+ (s), n, and tn (s). m is the mass in grams added to the mass holder.

a.     The Run # is a number generated from DataStudio.

b.     t0 is the time of the first discernable position peak.

c.     t0+ is the next largest time that the cursor can read.

d.     The number of complete oscillations is n, and

e.     tn is the time of the n+1st peak. We will eventually need the period of oscillation for each mass (which we won’t be measuring with a stopwatch).

9.   Maximize the graph display (see Figure 2).

10.  Put 50 g on the mass hanger on the spring. Click on the Start button when the mass and hanger have stopped moving.

11.  Lift up the mass hanger until it barely rests on the spring, then release it.

12.  The mass starts oscillating. Let it bob up and down until the motion dies out. Click on the Stop button. Record the mass that has been added to the mass hanger, and the Run #.

13.  At the top left of the two graphs there are some icons for measurement and display. Click in the Position graph then click on the Scale to Fit icon, the first one in the row. This fits onto the graph all the data that you recorded. Repeat this for the Velocity graph.

14.  Ideally the plots should take up as much of the screen as possible. You may have taken a long time to start the oscillation or you may have continued to record data long after the oscillation stopped. In either of these two cases, you have to modify the time scale.

a.   Click on the Zoom Select button.

b.   Click and drag a window in the Position graph that includes all the peaks. The time axes of both graphs are locked and will expand to what you specified.

 

Figure 3

Figure 3 Using the Smart Tool to find the time of the zeroth peak.

 

 


15.  Click on the Smart Tool button. A crosshair icon appears with vertical and horizontal dotted lines going through it. The time and position coordinates of the crosshair are shown. As you click and drag this icon on the graphs, the lines move with it, and the coordinate pair changes accordingly. Click in the Velocity graph and again click on the Smart Tool icon to turn on this feature in this chart. (The vertical lines are locked.)

16.  Click and drag the crosshair icon in the Position graph so that the vertical line is on the initial position peak (number it as zero), as in Fig. 3. Note that this line intersects the velocity graph at the velocity axis (v=0). Record the time as t0.

17.  To get the uncertainty u{t} in this time measurement, move the crosshair the smallest amount that registers a change in the time.  Record this new time as t0+. The uncertainty, to be calculated later, is the difference of these two times. It may be different for each mass.

18.  Going to the right, count as many discernible peaks as possible and position the line of the Smart Tool on this peak. Record as n the total number of peaks minus one. Record the time as tn. The difference of the two times tn and t0 is the time for n periods. (No calculations yet.)

19.  Repeat steps 10-18 for the following masses in grams: 100, 150, 200, 250, 300, 350, 400, 450, 500. Use Step 11 only for 100 g. For the other masses, pull the spring down about 10 cm.

a.     You should have at least 11 Runs stored in DataStudio (1 for Hooke’s Law and 10 for oscillations). The Position and Velocity graphs may randomly switch places.

Note: Do not exit DataStudio until you are absolutely sure that you have copied or analyzed the data as required. [Be sure to read the Questions at the end of this write-up.] Definitely save your data in case of any malfunction or other oopsie. But the computers in GP 300 & 303 are the only ones outside the physics labs that can make sense of the file type that is stored. Definitely exit DataStudio at the end of class.

 

 Damped Harmonic Motion Experiment (General Physics)

20.  Assume that the damping constant is independent of the load.

21.  Click in the Position graph.

a.     Choose one of your Runs from the Data menu; double click on it. This displays the time and position data for that data set. Repeat this in the velocity graph.

b.     Record the mass.

c.     Follow steps 13-15.

d.     Use the Smart Tool to find both the time and position values for as many position peaks as possible in this one run. Record these values in a table. These data will be used to find the damping constant

22.  Click in the Position graph. Choose one of your Runs from the Data menu; double click on it. This displays the time and position data for that data set. Repeat this in the velocity graph. Record the mass.

23.  Follow steps 13-15.

24.  Use the Smart Tool to find both the time and position values for as many position peaks as possible in this one run. These data will be used to find the damping constant " \l 3

 Analysis

1.     Click on Excel Templates> Second Quarter> Simple Harmonic Motion> (the course you are taking College or General). This loads the template for this experiment.

 Hooke’s Law Analysis


2.     Fill in the data in the appropriate cells for Hooke’s Law. The positions are negative.

3.     Calculate the weight in newtons of the added mass on the mass hanger for each point. These are negative forces.

4.     Enter in the weight of the spring and calculate the theoretical mass correction factor for oscillations.

5.     Do a least squares fit with errors of weight vs. position (refer to Eq. 10).

6.     Include a linear trendline on the graph, with the equation displayed.    

 Simple Harmonic Motion Analysis

7.     Enter in the data from the oscillations.

8.     Calculate the time uncertainty u{t} (the difference between t0+ and t0), the period T, the error in this u{T}, T 2, 2m (in kilograms), and the error in T 2 (u{T 2}=2Tu{T}).

9.     Do a least squares fit with errors of 2m vs. T 2 (refer to Eq. 11).

10.  Include error bars for T 2 on the graph.

11.  Add a trendline with the equation displayed.

12.  Calculate the experimental mc and its error from the least squares fit for the oscillations.

13.  Compare the two experimental values of the spring constant from steps 4 and 8.

14.  Compare the value of mc from step 11 with the theoretical value from step 3.

 Damped Harmonic Motion Analysis (General Physics)

15.  Do an exponential least squares fit with errors and find the damping constant b. (Refer to Appendix 3.)

16.  How much (percentage) does the decay change the frequency for each of the 10 masses oscillated? Calculate Eq. 17 for each of these masses, using the experimental data for harmonic motion, k from step 8, mc from step 11, and b from step 14.

 Questions

1.     Where in the complete up-and-down period does the mass on the spring have the greatest and least speeds? Where does it have the greatest and least velocities?

2.     Propagate the error for T as calculated in this experiment.