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PHY 133 Lab 8 ­ Simple Harmonic Motion [Stony Brook Physics Laboratory Manuals]
Stony Brook Physics Laboratory Manuals
PHY 133 Lab 8 - Simple Harmonic Motion
The purpose of this lab is to study simple harmonic motion of a system consisting of a mass attached to a spring, and to establish the
relationship between the period T, mass m, and spring constant k.
Equipment
air track with pulley at end
1 - 100g mass
4 - 20g masses
string
glider with metal tab
2 springs
interface box
photogate
computer
digital scale
Introduction
This experiment explores the phenomenon called Simple Harmonic Motion (SHM) for a mass-spring system. SHM is a type of
periodic or oscillatory motion in which the restoring force acting on the mass m is directly proportional to the displacement x of the
mass, and acts in the opposite direction. Hence, Newton's second law becomes:
F net = ma net = m
d 2x
dt 2
= − kx ⟹ d 2x
dt 2
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= −
()
k
m
x = − ω 2x
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where k is the spring constant (or “stiffness”) of the spring, and ω is the angular frequency of the periodic motion. To better
understand the object's motion, we solve this differential equation for expressions of its displacement, velocity, and acceleration as
functions of time:
x(t) = Acos(ωt)v(t) =
dx
dt
= − ωAsin(ωt)a(t) =
d 2x
dt 2
= − ω 2Acos(ωt)
Because these expressions are sinusoidal (cosine or sine functions), they repeat after some finite time or period, T. (See the image
below, in which the sinusoidal motion of an oscillating mass on a spring is traced out through time.) Hence, the simple harmonic
motion is periodic.
ω=
√
k
m
ω = 2πf =
2π
T
√
⟹ T = 2π
m
k
These results also show that as the displacement increases, the velocity decreases, and vice-versa. This exchange occurs due to the
transfer between the two types of energy in the system: translational kinetic energy and spring potential energy. When the mass is at
maximum displacement x max = A, it has a velocity of v = 0 m/s, and all the energy is spring potential energy. Conversely, when the
mass passes through the equilibrium point at x = 0 m, its velocity is maximum at v max = ωA, and all the energy is translational
kinetic energy. Hence, the total mechanical energy is conserved in this (ideally) frictionless system, and the fraction of this total which
is kinetic energy or potential energy oscillates between the two.
ME = KE + PE =
ME = PE max =
ME = KE max =
1
2
1
2
1
2
mv 2 +
kx max 2 =
mv max 2 =
1
2
1
2
1
2
kx 2
kA 2
mω 2A 2
In this lab, you will investigate the nearly-ideal simple harmonic motion of a glider (the mass m) attached by springs of spring
constant k. Though the air track reduces friction significantly, it does not remove it entirely, so some damping of the motion will
occur. Despite this, you should be able to verify the periodic nature of the mass-spring system, as well as its conservation of
mechanical energy.
Part I: Measurement of the spring constant k
In this first part, you will need to experimentally determine the spring constant k of the springs you'll be using for the remainder of
the experiment. To do this, you'll attach two springs to a glider, place it on the air track, and hang a mass tied to the other end via a
string. By measuring the displacement of the glider from its equilibrium position (defined as its position when the 100 g mass is
attached) while increasing the hanging mass, you will be able to calculate a value for the spring constant using Hooke's Law, which
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says that the restoring force of a mass-spring system is linearly proportional to, and in the opposite direction of, the displacement of
the mass from equilibrium:
F = − kx
(The reason for connecting two springs is that these two springs are used in Part II and Part III of this lab. If you use only one spring
in Part I, your results will be incorrect for Parts II/III.)
The hanging mass m supplies a weight equal to its gravitational force upon the string, which transfers this force to the glider. On the
other side of the glider, the springs stretch enough so that the restoring force equals out the string tension on the other side, until the
net force is 0, and the glider stops. Hence, we can write an expression for the net force on the glider:
F net = T + F spring = mg + ( − kx) = mg − kx = 0 ⟹ mg = kx ⟹ k =
mg
x
Before you start collecting data, ensure that your air track is level by adjusting the screw underneath it until the glider does not move
much on its own. Attach a piece of string to the glider, pass it over the pulley, and hang the 100 g mass at the free end. Record the
position of your glider along the air track with the 100 g weight suspended, and call this x 0. This is your “equilibrium position.” Now,
add a 20 g mass to the free end, record the new position x, and find the displacement δx = x − x 0 due to this additional mass
δm = 20 g. Repeat this process for additional masses of 40 g, 60 g, and 80 g, for a total of four measurements. Make sure you are
taking the difference between your position readings x and the equilibrium position x 0 for your displacement δx values. Enter the
values for the additional mass δm, the additional displacement δx, and the additional force δF = δ(mg) = (δm)g in your notebook,
as in the table below. For each measurement, calculate the corresponding spring constant value k =
ˉ and for its uncertainty, use: σ ˉ =
average value k,
k
k max − k min
2
δF
δx
=
( δm ) g
δx
. Then, find the
. For Parts II/III, whenever you need to use the spring constant k
for calculations, use this average value and its uncertainty.
δm (kg)
δx (m)
δF (N)
k=
δF
δx
(N/m)
.020
.040
.060
.080
Part II: Measurement of the period T of glider-spring system
In this part, you will measure the period of the oscillatory motion of the glider about its equilibrium position, due to the springs from
the previous part. Contrary to the first part, in this section, you will attach one spring to each side of the glider. Recall that
the period T of a mass-spring system with mass m attached to a spring with spring constant k is given by:
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T = 2π
√
m
k
You should note that when we have the springs attached to opposite ends of the glider, as in this part, the k in this equation is the
sum of the spring constants of each spring. This sum is what we measured in the first part of the lab! (Try working this out in your
own, to verify that the k value in the “two springs on one side” and “one spring on each side” cases are identical.)
To prepare this setup for data collection, do the following:
Remove the string from the glider, and measure the mass M of the glider using the digital scale in the lab room. Record this
value, assuming negligible uncertainty (σ M ≈ 0).
Measure the mass of each of your springs, and add them together to find their total mass. Call this quantity m springs, and
assume it has negligible uncertainty (σ m
).
springs
Attach one spring to each end of the glider, and place the glider on the air track. Hook the free end of each spring to the two
ends of the air track, as in the diagram below.
Position the photogate perpendicular to the metal tab, and centered so that the beam hits the middle of the tab when the
glider is at rest at its equilibrium position, as in the diagram below. Connect the photogate output wire to the input of the
interface box labeled “DIG/SONIC 1.”
On the computer, double-click the Desktop icon labeled “Exp7_Period.” A “Sensor Confirmation” window should appear,
and select “Photogate,” then click “Connect.” A LoggerPro window with a spreadsheet (having columns labeled “Time,”
“State,” and “Period”) on the left and an empty Period vs. Time graph on the right should appear.
Check that the photogate is positioned properly by passing the glider through the photogate, and seeing the red LED flashes
on/off when the photogate beam is blocked/unblocked by the metal tab.
Note: For this part of the experiment, since you are measuring the period of one complete oscillation using LoggerPro,
there is no d value to input into the User Parameters list for this section!
Now, for collecting data, proceed with the following:
With the glider centered at the photogate at its equilibrium position, carefully displace it by ~10 cm to either side. Hit the
green “Collect” button on the LoggerPro window, and once the “Waiting for data…” text appears, release the glider.
The glider should now oscillate about its equilibrium position without coming to a stop too quickly. If it does come to rest in
a short time, you should tell your lab instructor/TA so that they can adjust your setup or replace your glider to reduce the
source of friction. Allow the glider to pass through its equilibrium position several times, then click the red “STOP” button
on the LoggerPro window. You should see the spreadsheet fill with values, and an approximately constant value of period
across time on the plot. If you do not see this, repeat the trial.
Take the average of several (~3-5) period values near the beginning of your trial, and call this value your measured period
T measured (assuming negligible uncertainty).
Repeat this procedure four more times, adding a 20 g mass onto the glider for each new trial. You should position the
additional masses nearer to the center of the glider, to distribute the weight evenly on all sides of the glider. If you do not do
this, friction between the track and glider will severely impact your results!
Record the values of total mass (glider mass + additional mass) and period for each trial into your lab notebook.
For each trial, calculate the expected period value predicted by theory, T theory using equation (9) above. (The mass m in that
equation should be the total mass (glider + additional mass) for each of your trials.) For the uncertainty σ T
in each
theory
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value, use the uncertainty propagation relations in the uncertainty guide applied to equation (9), using your value for σ kˉ from
Part I, and no uncertainty in the mass. You should find that:
σT
theory
Ttheory
=
1 σk
2 k
.
Looking at equation (9), you may be questioning its accuracy because it only includes the total mass of the oscillating object (glider +
additional masses). It assumes an “ideal” system, with negligible spring masses. However, during your trials, you may have noticed
that the springs were oscillating too, so some energy of the motion must be put towards their motion as well. This should increase
the duration of the periodic motion above its theoretical value. In the 19th century, physicist Lord Rayleigh analyzed the influence of
the spring masses, and derived a more accurate expression for the period of the object's (the glider, with mass M) oscillatory motion:
√( ) √ (
M+
T improved = 2π
m springs
3
≈ 2π
k
M
k
1+
m springs
6M
)
(
Comparing this expression of T improved to that of T theory in equation (9), we see that it is greater by a factor of 1 +
m springs
6M
)
. For
your five trials from this part, find the percentage difference between your calculated T theory and T improved values. This should
convince you that using equation (9), which neglects the spring masses, is adequate for estimating the period of the oscillatory
motion.
PercentageDifference(%) =
(
T theory − T improved
T improved
)
× 100
Now, to analyze your data and compare your experimentally-measured values of T measured to their theoretical predictions T theory,
you should plot T theory vs. T measured (including error bars for T theory only) using the web-based Plotting Tool. What value should
the slope of your plot be, ideally? Is your slope estimate consistent with this value within experimental uncertainty? What sources of
error might you be neglecting or underestimating in this part of the lab?
Part III: Conservation of energy in the glider-spring system
In this last part of the lab, you will test for energy conservation within this mass-spring system. Similar to the previous Conservation
of Energy experiment, the total energy within the system should remain constant, so long as there are no significant external forces
(like friction) acting on the system. In this case, however, the energy transfer will be from spring potential energy to translational
kinetic energy, and then back to spring potential energy, and so on.
Although we could use equation (4) above describe the total energy as the sum of the kinetic and potential energies at any given time,
we will simplify our analysis by choosing an initial point and a final point with only one type of energy each. The aim is to displace
the glider from equilibrium by some initial maximum displacement x max (equal to its amplitude A), release the glider, and measure its
final maximum velocity v max as it passes through its equilibrium point. Then, you should verify that the initial spring potential energy
of the system (equation (5)) was fully converted into the final kinetic energy of the system (equation (6)). We will again neglect the
mass of the springs for this part.
?
? 1
1
PE max = KE max ⟶ kx max 2 = mv max 2
2
2
The setup is identical to that used in Part II, so do not modify the system!
To collect your data for this part, do the following:
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Make sure that the metal tab on top of the glider is centered on the photogate, as in Part II, when the glider is at its
equilibrium position. Test the photogate by passing your finger through the beam, and check that the red LED flashes on/off
when the beam is blocked/unblocked.
On the computer, double-click the Desktop icon labeled “Exp7_Velocity.” A “Sensor Confirmation” window should appear,
and select “Photogate,” then click “Connect.” A LoggerPro window with a spreadsheet (having columns labeled “Gate
Time,” “Velocity 2,” “State”) on the left and an empty graph of “Velocity 2” and “Gate Time” vs. “Time” on the right should
appear.
Measure the width w of the metal tab (which should be ~5 cm wide) atop the glider. To input this value into LoggerPro,
under the “Data” tab, click “User Parameters.” On the row labeled “PhotogateDistance1,” enter your value of w (in meters
“m”), and adjust the “Places” and “Increment” values if necessary. Then, click “OK.”
Displace and hold the glider 30 cm from its equilibrium point, and record this initial maximum displacement value x max. For
its uncertainty, use σ x = 1 cm.
max
Click the green “Collect” button on the LoggerPro window, and once the “Waiting for data…” text appears, release the
glider. Notice the velocity values appearing in the LoggerPro spreadsheet as the glider passes forwards and backwards
through the photogate. After several passes, click the red “STOP” button on LoggerPro. You should see a slowly falling
linear plot of “Velocity 2” vs. “Time,” as the glider gradually slows down due to friction. Record the first velocity value in the
spreadsheet as your maximum velocity v max. For its uncertainty, you may assume that the photogate is precise so that there is
no uncertainty in the time of the glider passing through the photogate beam. So, you'd expect that:
σv
max
v max
=
σw
w
, just as in the
Conservation of Momentum lab!
Now, to check for energy conservation, calculate the maximum potential energy stored in the spring with the glider displaced (using
equation (5)), as well as the maximum kinetic energy of the glider on its first pass through its equilibrium position (using equation
(6)). If energy is conserved, these values should be equal. To determine if they are “equal” within experimental uncertainty, calculate
their uncertainties σ PE and σ KE . You may assume that there is negligible uncertainty in the mass of the glider and the spring
max
constant here, so that:
max
σ PE
max
PE max
=2
σx
max
x max
and
σ KE
max
KE max
=2
σv
max
v max
. Using the “overlap method” with these uncertainty ranges, is energy
conserved within experimental error? Regardless of your result, what might be some incorrect assumptions made about the system,
or unaccounted for sources of error, that may have influenced your results?
phy133/lab8simplemarmonicmotion.txt · Last modified: 2016/11/09 18:38 (external edit)
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