# PHY 123 Lab 9 - Simple Harmonic Motion

(updated 11/17/16)

The purpose of this lab is to study simple harmonic motion of a system consisting of a mass attached to a spring. You will establish the relationship between period, mass, and spring constant.

If you need the .pdf version of these instructions you can get them here.

## Equipment

• air track (you must check that it is level) with pulley (make sure it spins freely) and string for hanging weights
• beam balance for weighing objects
• glider (you must weigh yours) with flag for photogate
• weights (with masses marked on them, but you may check them!)
• two springs (you must measure their spring constants)
• photogate
• interface box
• computer (with pre-programmed software) and LCD monitor

## Introduction

In this lab you will study the phenomenon of simple harmonic motion – sinusoidal oscillation as a function of time – for a system of mass(es) + spring(s). The physical basis of these oscillations is Hooke's Law: the force exerted on a mass by a spring is proportional to, and opposite to the direction of, the displacement of the mass-spring system from its equilibrium position. Before you measure the oscillation frequency as a function of the mass being oscillated, you will need to measure the (combined) spring constant of the two springs you will use. Finally, you will determine if total mechanical energy – kinetic plus potential energy – of the mass(es) + spring(s) system is conserved.

To prepare for this lab, you should review the material in Ch. 9, Periodic Motion, of the online course notes and/or in Chap. 14 of Knight, Jones and Field (2nd ed.), (abbreviated KJF2), which is the optional, recommended textbook for PHY 121. In Secs. 14.1-14.5 and 14.6 pay particular attention to the worked-out examples.

## Part I: Measurement of the Spring Constant

In this part you need to measure the combined spring constant $\:\:k$ (that appears in Hooke's Law, Eq. 7.1) of two springs. One end of the springs will be attached to a glider on your air track, and the other ends will be attached to one end of the track. The reason for mounting the two springs together is that you will use these same two springs in Parts II and III of this lab. For a one-dimensional system (motion along a line), Hooke's law reads

$\Large F=\;-kx\:\:.$
(7.1)

The force $\:\:F$ is supplied by the weight of a mass hanging on the string; see Fig 2 and Eq. 7.2. Assume that the weight of the string is negligible. Moreover, because of the procedure described below, the mass of the string does not affect your measurement of $\:\:k$. (Can you figure out why this is so?) The weight $\:\:W$ of the mass hanging on the string is

$\Large W=mg\:\:.$
(7.2)

This creates the tension force in the string that pulls on the glider and stretches the springs. You will hang various weights from the string and measure the displacement of the glider from its equilibrium position, which you should define as the position when a 100 gram weight is hanging.

Before starting you must ensure your air track is level, just as you did in previous experiments using the air track. Also, make sure your pulley spins “freely” with minimal friction. Spin it with your finger. You can “hear” if it's spinning freely. Attach a piece of string to the glider, lay it on the pulley, and hang a 100 gram mass from the free end of the string as shown above. Record the position of your glider for this 100 gram mass and define this as your equilibrium position. Be sure that the string is able to move freely on the pulley and does not “scrape” anything else. Now add 20 grams, and repeat the measurement. Adding 20 grams more each time, repeat this procedure three more times for a total of four measurements. Make sure you are measuring and recording as your data the difference between the position for each value of hanging mass and the equilibrium position. Enter the values for each the additional mass $\delta m$ , the corresponding additional displacement $\:\:\delta x$, and the corresponding additional force $\:\:\delta F$ into Table 1 on your worksheet.

Calculate for each of the four sets of numbers the ratio of incremental changes $\:\:{\Large k =\frac{\delta F}{\delta x}}$ (which give the slope of a force-vs-displacement curve) and enter them into Table 1.

Use your data to calculate the average value of $\:\:k$ and its absolute uncertainty $\:\:\Delta k$; use $\:\:{(\rm max \: – \: \rm min)/2}$ for the uncertainty. Write them on your worksheet.

## Part II: Measurement of the Period of Glider/Spring System

In this part you will measure the period $\:\:T$ for oscillation of the mass(es) + spring(s) system that uses the two springs from Part I. Taking $\:\:M$ as the mass of the (glider + small masses-along-for-the-ride), the period of a mass-spring system is, ideally, given by

$\Large {T_{\rm theor}=2\pi \sqrt{\frac{M}{k}}}$
(7.3a)

A first question that probably occurs to you is what value of $\:\:k$ to use. The $\:\:k$ you measured in Part I had both springs “on one side” of the glider. Now, in Part II, you have one spring on each side of the glider. Though it may not be immediately obvious to you, the value of $\:\:k$ for the “one spring on each side” case is the sum of the spring constants for the two springs. This is just what you measured in Part I. (Can you work out why this is true?)

Remove the string from the glider, measure the mass of the glider with the beam balance, and record it on your worksheet. You will also need to know the combined mass of the two springs. Your TA will have that combined mass written on the blackboard. Enter that value on your worksheet along with the ratio, as a decimal number, of the (mass of the two springs) over the (mass of the glider).

Because of the relatively high resolution and reasonable accuracy of the beam balance, you may ignore the uncertainty in your measurements of mass using it.

Attach the glider to the two springs and place the glider on the air track so that the ‘metal flag’ atop the glider is centered at the equilibrium position.

• Connect the photo gate output to the interface box by plugging its cable into the top socket (labeled “DIG/SONIC 1”) of the black interface box (“LabPro”).
• Turn on the computer and check the system by following these instructions: Double click the icon “Exp7_Period”. A window with a spreadsheet on the left (with columns for “Time”, “State”, and “Period” and a graph for “Period” vs. Time) comes up. On top is a window “Sensor Confirmation”.
• Click “Connect”, and again “Connect”.
• Test the photo gate: block the photogate beam with your finger and see the red light on the cross bar of the photogate turn on.
• Click OK.

You are ready for data taking now.

• Place a 20 gram weight on top of the glider. Click the green “Collect” icon on the PC terminal, displace the glider by ~ 10 cm and let go. Collect a few data points and click STOP. You see a plot with a constant period as time goes on on the graph. Take a value for the measured period Tmeas and enter it into Table 2 on your worksheet.
• Repeat the above three times, each time adding an additional 20 grams on top of the glider. Record the values of the total mass (glider and additional mass combined) and the periods into Table 2 of your worksheet.

Calculate the theoretical period $\:\:T_{\rm theor}$ using Eq. (7.3a).

An interesting detail: If you think about it carefully, Eq. (7.3a) cannot be correct since it includes only the mass $\:\:M$ of the (glider + mass(es)-along-for-the-ride). The two springs are moving, too, and Eq. (7.3a) neglects their masses. However, if you were watching the oscillations carefully, you would see that the “oscillation velocity” at each moment of time was not uniform along the length of each spring, so whatever correction is needed for the masses of the springs, one should not expect it to include the “full mass” of the springs. Lord Rayleigh, and others afterward, analyzed the influence of the masses of the springs on the oscillation frequency of a (spring + mass) system and derived an expression for the “effective mass” $\:\:m^*$ of the spring that should be added to $\:\:M$ in Eq. (7.3a). (The derivation is shown here for a vertical (spring + mass) system; the correction for your case of “horizontal” springs is the same. From the animation at that web site, you can verify that the “oscillation velocity” varies along the length of the spring.) From that derivation $\:\:m^*$ is equal to 1/3 of the combined mass $\:\:m_{\rm springs}$ of the two springs, the value of which the TA gave you.

Therefore, an improved theoretical expression for the period of oscillation is

$\Large{T_{\rm improved}=2\pi \sqrt{ \frac{\Large( M + \frac{m_{\rm springs} } { 3 }\Large) } { k } } }\:\: = \:\: 2\pi \sqrt{\frac{M}{k}}\:\Large(1 + \frac{m_{\rm springs}}{6M} \Large) \:\:.$
(7.3b)

From Eq. (7.3b) we see that $\:\:T_{\rm improved}$ is greater than $\:\:T_{\rm theor}$ by a factor $\:\:\large{(1 + \frac{m_{\rm springs}}{6M})}$. For at least one of your measurements of the period $\:\:T$, compare the percentage difference in the predictions of $\:\:T$ from Eq. (7.3b) versus Eq. (7.3a). This should show you that using Eq. (7.3a), which neglects the masses of the springs, is adequate for your data.

The uncertainty in $\:\:T_{\rm theor}$ can be calculated from the the uncertainty in $\:\:k$ you estimate in Part I. From Eq. (7.3a) and Eq. (E.8) of Uncertainty, Error and Graphs, the Lab 1 manual, if you take into account only the uncertainty of the spring constant $\:\:\Delta k$ you can find that $\:\:\Delta T_{\rm theor}=(T_{\rm theor})\,(\frac{1}{2})\,(\frac{\Delta k}{k})$.

Use the plotting tool to plot $\:\:T_{\rm theor}$ versus $\:\:T_{\rm meas}$, including error bars for $\:\:T_{\rm theor}$. Write your slope and its uncertainty on your worksheet.

x axis label (include units):
y axis label (include units):
Check this box if the fit should go through (0,0).
(Don't include (0,0) in your list of points below, it will mess up the fit.)
What kind of uncertainties are you entering below?
x1: +/-    y1: +/-
x2: +/-    y2: +/-
x3: +/-    y3: +/-
x4: +/-    y4: +/-

## Part III: Potential Energy in the Spring

In this part you will test for conservation of energy in the system of mass(es) + spring(s). You will see if the maximum potential energy in the spring at maximum displacement from equilibrium,

$\Large PE_{\rm max}=\frac{1}{2}kx_{0}^2\:\:,$
(7.4)

equals the maximum kinetic energy, which occurs when the glider goes through the equilibrium point:

$\Large KE_{\rm max}=\frac{1}{2}mv_{0}^2\:\:.$
(7.5)

The setup is the same as in Part II, but you use a different computer file to take the data.

• Double click the icon “Exp7_Velocity”. A window with a spreadsheet on the left (having columns for “Gate Time”, “Velocity 2”, and “State” and a graph of “Velocity 2” and “Gate Time” vs. “Time”) comes up. On top is a window “Sensor Confirmation”.
• Click “Connect”, again “Connect”.
• Measure the width $\:\:w$ of the flag on top of the glider and enter it in your worksheet, along with an estimate of the uncertainty in your measurement.
• Click Data→User Parameters: the window “User Parameters” comes up and should show:
Name:Value:Units:Places:Increment:Editable:
• Displace the glider by ~ 30 cm from equilibrium and enter the glider displacement $\:\:x_{0}$ from its equilibrium on your worksheet, along with an estimate of the uncertainty in your measurement.
• Click the green “Collect” icon on the PC screen and release the glider. Record a few velocities as the glider goes backwards and forwards through the photogate on the PC screen. You should see a slowly falling linear graph due to the gradual slowdown of the glider due to friction. Enter the first velocity as the maximum velocity $\:\:v_{0}$ on your worksheet. Calculate its uncertainty. You should neglect the uncertainty in the time, meaning that the relative uncertainty in the velocity is the same as the relative uncertainty in the flag width, i.e., $\:\:{\Large \Delta v=v\;\frac{\Delta w}{w}}$.
• Calculate the maximum potential energy originally stored in the spring using Eq. (7.4) and write it on your worksheet. Also calculate the uncertainty $\:\:{\large \Delta PE_{\rm max}=2PE_{\rm max}\frac{\Delta x_{0}}{x_0}}$ and write that on your worksheet as well.
• Calculate the kinetic energy the system has when it is moving with its maximum velocity $\:\:v_{0}$ using Eq. (7.5) and write it on your worksheet. Also calculate the uncertainty $\:\:{\large \Delta KE_{\rm max} = 2KE_{\rm max}\frac{\Delta v_{0}}{v_0}}$ and write that on your worksheet.