Unformatted text preview: Laboratory 10
Simple Harmonic Motion
Any physical system governed by a restoring force, described (at least approximately) by a
quadratic potential well, and which executes periodic motion, can be described as a simple
harmonic oscillator. Simple harmonic oscillators occur in many ﬁelds, from engineering and
physics to geology and biology. Therefore, understanding this concept is critical to analyzing a
broad range of theoretical models describing real-world systems.
One of the simplest examples of a simple harmonic oscillator is a mass oscillating due to a spring’s
restoring force. A biochemical extension of this consists of two oxygen atoms (masses) oscillating
under the inﬂuence of the interatomic force (spring) between them. The atoms’ positions ﬂuctuate
through their equilibrium position as they are attracted to and repelled by each other. Although interatomic forces are generally complex, in the region near equilibrium they are well-characterized
by the linear relationship F = −k(x − x0 ) as long as the atoms’ excursions from their equilibrium
separation, x0 , are not too large. Both a mass in a mass-spring system and the oxygen atoms in the
biological system will undergo oscillatory motion at some natural frequency that is a function of
the system itself and does not depend on the amplitude of the oscillation.
During this laboratory, we will investigate simple harmonic motion through the measurement of
the natural oscillation frequencies of three different systems. Theoretical Background: Simple Harmonic Motion
Simple harmonic motion is sinusoidal motion that results when an object is subject to a restoring
force of the form
= −k ,
where is the object’s displacement from its equilibrium position - the position where it expex
riences no net force. This formula says that no matter which direction the displacement is in, the
restoring force will act to bring the object back toward its equilibrium position. A graph of force
vs. displacement for simple harmonic motion is shown in Figure 1.
Using Newton’s second law, = m , we obtain the equation of motion
m a = −k x,
85 (1) Figure 1: These two graphs are deﬁning features of simple harmonic motion. The force law is
linear with respect to the displacement, x, and the potential energy function is a parabola. You can
think of simple harmonic motion as a situation where a ball sitting at the bottom of the potential
energy “well” is displaced slightly. What would happen? It would roll back and forth periodically,
and its x-position would be a sine curve. where x is the object’s position and a is its acceleration. For those familiar with calculus, a is the
second derivative of x with respect to time, so
= −k x.
dt 2 All systems that undergo simple harmonic oscillation have equations of motion of this form. Solutions to this equation have the form
x(t ) = A sin(ω t + δ ),
where δ is a constant that depends on the time we set to be t = 0 (this is arbitrary), and
ω = k /m
is the angular frequency of the object’s oscillation. The angular frequency of the oscillation is related to the frequency, f (the number of oscillation
cycles the object undergoes each second) and the period, T (the amount of time it takes the object
to undergo one complete oscillation cycle) via the relations
T (2) The deﬁning feature of simple harmonic motion is the force law F = −kx. The potential energy
86 graph associated with this force law is a parabola (see Figure 1), and the curvature of this parabola
is deﬁned by k; if an object moves a distance x away from its equilibrium position, it gains an
amount of potential energy
U = kx2 .
Note that the force the object experiences at a displacement x is the negative slope of the potential
energy function at that point.
Interestingly, these properties are very closely related to similar concepts from uniform circular
motion. In fact, if one considers an object moving in uniform circular motion, its projection onto
the x or y-axis oscillates in simple harmonic motion (see Figure 2). From this perspective, the
angular speed of the object around the circle is equal to ω , the angular frequency of the simple
harmonic oscillation. y y v
x t Figure 2: The relationship between uniform circular motion and simple harmonic motion. The
wavy lines at right represent the y-position of the object that is moving around the circle, and the
colored dots correspond to its y-position at the similarly-colored points on the circle. Experimental Background: Mass Hanging From A Spring
Consider a mass hanging from a “massless” spring of spring constant k, as shown in Figure 3. Figure 3: Schematic diagram of a mass hanging from a spring. 87 Let y = 0 be the point at which the spring is in equilibrium; then the equation of motion for the
m ay = −k y − m g,
which looks suspiciously like Equation 1 except for the extra g term on the right. This makes this
equation a little more difﬁcult to solve, but the end result is that
y = A sin(ω t + δ ) − mg
k (4) The mass therefore oscillates in simple harmonic motion about the equilibrium point y = mg/k.
This makes sense because when the mass is in equilibrium, the spring will be slightly stretched to
counteract the force of gravity. The angular frequency of oscillation for this system is
ω = k/m.
Problem 10.1 What y value do you obtain when you set the force of the
spring, −k y, equal to the force of gravity, −m g, which is what occurs when
the mass is in equilibrium. Does this resemble anything in the equations
Problem 10.2 What advantage do we gain by measuring k via the period
and not the displacement of the pendulum bob from equilibrium? In other
words, why is our measurement of k more precise if we use the period? Experimental Background: Simple Pendulum
One example of a situation in which an object undergoes simple harmonic motion is the simple
pendulum, which is shown in Figure 4. Figure 4: Diagram of a simple pendulum. The pendulum bob has mass m, the distance from the
pivot point to the center of mass of the bob is l , and the angular distance of the pendulum from the
vertical is θ . (Note that θ changes as the pendulum moves.) 88 The term “simple pendulum” refers to a point mass hanging from a massless rod, which swings
back and forth when it is displaced from its equilibrium position (straight down). We will analyze
the motion of the simple pendulum by considering the torque generated by gravity about the pivot
τ pivot = I α = × .
where I = m l 2 is the bob’s moment of Inertia about the pivot, α is the angular acceleration of the
pendulum, and × = −m g sin θ .
Solving equation 5 for α , we obtain α =− g
l If θ is small, we can make the approximation1 that
sin(θ ) ≈ θ ,
which leads to (6) g
α ≈ − θ.
l (7) There is a direct parallel between Equation 7 and Equation 1, so we can see that in the case of the
Problem 10.3 The angular frequency of oscillation in simple harmonic
motion usually depends on two terms: one related to the restoring force
and another related to the inertia of the oscillating mass. In the case of the
simple pendulum, does g represent the restoring force or the inertia? How
about l ?
One interesting feature of this equation is that it shows us that, provided oscillations are small
(we can make the approximation shown in Equation 6), the frequency of oscillation of the pendulum does not depend on the mass of the pendulum bob or on the amplitude of the pendulum’s
Problem 10.4 As the length of a pendulum’s string goes up, does the
angular frequency of its oscillation go up or down? Does the period of its
oscillation go up or down?
3 5 can be seen most easily by examining the Taylor series expansion of sin θ = θ − θ + θ − . . . and letting
θ → 0. Which term goes to zero most slowly?
1 This 89 Problem 10.5 Does ω depend on how far back you pull the pendulum bob
initially, provided you can still make the approximation in Equation 6? If
you give the pendulum a “kick” as you let it go, will this affect ω ? Why or
A photograph of the experimental setup for this part of the laboratory is shown in Figure 5. LoggerPro will record the times at which the pendulum bob passes through the photogate and use them
to calculate the period of the pendulum’s oscillation. Figure 5: A picture of the apparatus used for the simple pendulum experiment. Experimental Background: Ball in Dish
A ball rolling in a circular dish with a spherically-rounded bottom is another example of a system
that undergoes simple harmonic oscillation. Its motion is slightly more complicated than that of
the simple pendulum, but we can analyze it in a similar way.
First, consider the approximation shown in Figure 6, where the bowl bottom is treated as a plane
inclined at a shallow angle θ , where θ also happens to be the ball’s angular displacement from the
bottom point of the dish.
The linear acceleration of the ball down the “plane” is given by the equation of motion
m at = Ff − m g sin(θ ),
90 (9) Figure 6: The more complicated curvature of the bowl bottom is approximated as a simple inclined
plane. where Ff is the force of friction. The force of friction also produces a torque on the ball, which
causes it to undergo a constant angular acceleration α . The magnitude of the torque the ball
experiences is given by
τ = I α = −r Ff ,
where I is the ball’s moment of inertia and r is the ball’s radius. As long as the ball rolls without
slipping, its linear and angular acceleration are related by
at = α r. (11) Combining Equations 9, 10, and 11 yields
at = − g sin(θ )
1 + I /mr2 (12) Problem 10.6 Can you derive Equation 12 using Equations 9, 10, and 11?
You should know how to do so.
If we consider θ , the ball’s angular displacement from the bottom of the dish, as our natural
coordinate, and we use the relationship
at = (R − r) αθ , 91 (13) where αθ is the ball’s angular acceleration about the center of curvature of the dish, we obtain
(R − r)αθ = − gsin(θ )
1 + I /mr2 (14) Note that αθ means something very different from α , which is the ball’s angular acceleration about
its own center of mass.
If we again make the approximation that sin(θ ) ≈ θ and compare our result to Equation 1, we
(1 + I /mr2 ) (R − r)
as the angular frequency of the ball’s oscillation. Recall from Laboratory 6 that
fI ≡ I
mr2 is a unitless number between zero and one called the geometric factor.
Recall, however, that the result in equation (15) was attained by an approximation. The approximate result is very close to exact in the limit that the ball’s radius is much smaller than the dish’s
radius of curvature. We can get an exact expression for ω by using a slightly different approach. It
turns out that we need to make two simple modiﬁcations to our above formulation.
First, we replace Newton’s second law for forces in Equation 9 with Newton’s second law for
τ net = τ f + τgravity ,
which states the net torque on the ball equals the sum of the torques, i.e. the torque generated
by friction plus the torque generated by gravity. It is important to recall that in order to calculate
torque, you must specify a pivot point in order to determine the lever arm of the applied force. In
theory, you can choose any point you wish as the pivot point. However, for harmonic motion, there
is usually a particular choice in pivot point that proves much more useful than any other point.
For the simple pendulum, the most useful pivot point is the location where the rod is ﬁxed to the
ceiling. For the ball-in-dish apparatus, the pivot point is the center of curvature of the dish. Using
the deﬁnition for torques introduced in Equation 1 of lab 8, we can rewrite Equation 16 as
Fnet (R − r) = Ff R − Fg (R − r) sin θ , (17) where Fnet = m at , the net force acting on the ball from Equation 9, Fg = m g, the force of gravity,
and Ff is the friction force (see Figure 7). Notice that Equation 17 is just Equation 9 where,
instead, each term is multiplied by its respective lever arm attained by choosing the dish’s center
of curvature as the pivot point.
The ball also undergoes rotational motion about its own center of mass due to the presence of static
92 friction between the ball and the dish. We must re-examine what it means to “roll without slipping”
in the case of motion on a curved surface. Rolling without slipping means that, as the ball rolls,
the contact point between the ball and dish travels the same arc length along the ball’s surface as it
does along the dish’s surface (see Figure 7). Using the formula for arc length, we get ∆β r = ∆θ R.
In a small time interval ∆t we have
R −→ ωβ r = ωθ R ,
∆t (18) which relates ωβ , the angular velocity of the rotation of the ball, to ωθ , the angular velocity of the
oscillation. Furthermore, since there is a net torque on the ball, the angular velocities will change
by a small increment ∆ω over a small increment of time ∆t , so
R −→ α r = αθ R ,
∆t (19) and we’ve found the appropriate expression relating the angular acceleration of rotations for the
ball, α , to the angular acceleration of oscillatory motion, αθ .
pivot (R-r) r R rg
Ff contact pt Fg Force Diagram Angle Relations Figure 7: Left: The force diagram for the ball in the dish. The force of gravity originates from the
ball’s center of mass while friction acts at the contact point between the dish and ball. The lever
arms for these two forces are rg = R − r and r f = R, respcectively. Right : A sphere rolling in
a spherical dish, demonstrating the relation between the two different angular displacements, ∆θ
and ∆β . The rolling without slipping condition says that the two red arcs in the diagram are of
equal lengths, which means that β and θ change at different rates as the ball rolls without slipping
along the dish’s surface. The mathematical relation between these two angles is given in Equation
Finally, combining Equations 17, 19, and 13, one can arrive at the expression
(R − r)αθ = − g sin θ
R 2 .
I 1 + m r2 93 R−r (20) If you compare Equation 14 with equation 20, you will see that the only difference is the factor of
(R/(R − r))2 multiplying the geometric factor in the denominator of Equation 20. This is the factor
that appears when the curvature of the dish is taken into account. This factor effectively increases
the inﬂuence that rolling motion of the ball has on ω . We could also interpret the result obtained
from the “planar approximation” in Equation 13 as the limiting case where the radius of the ball is
much less than the radius of curvature for the dish, i.e. r R. In the limit that the dish’s radius is
very large, then locally the dish’s surface looks very much like a plane to the ball, and the planar
approximation is recovered.
Problem 10.7 Once again, we notice that part of our equation for the
angular frequency of oscillation depends on the restoring force, and part
depends on the ball’s inertia. Which parts of Equation 15 correspond to the
restoring force, and which to the ball’s inertia?
Problem 10.8 Does ω depend on the ball’s mass? Why or why not? Does
it depend on the ball’s radius? Why or why not? If so in either case, what
happens to ω when you increase the quantity in question (the mass or
Problem 10.9 If the ball slid within the bowl instead of rolling, how would
this change Equation 15? Compare your new Equation 15 to Equation 8 for
a simple pendulum.
Problem 10.10 As fI increases from zero to one, should ω go up or down?
A photograph of the experimental setup for the ball-dish experiment is shown in Figure 8. LoggerPro will record the times at which the ball passes through the photogate and use them to calculate
the period of its oscillation. Experiments
Experiment 1: Mass on a Spring
In our ﬁrst experiment today, we will use the angular frequency of oscillation for a mass on three
different springs to determine the force constants, k, of these springs. Our experimental setup will
look like that in Figure 9, with a mass and spring clamped above a photogate. We will use a nearly
identical setup for the simple pendulum in Experiment 2.
1. Hang a 150-gram mass (50-gram gold hanger plus 100-gram slotted mass) from one of your
three springs as shown in Figure 9, so that the mass is suspended just above the photogate
94 Figure 8: A ping-pong ball rolling in the dish used for the ball-dish oscillation experiment. Figure 9: The experimental setup for Experiment 1. We clamp one end of a spring above a
photogate, and attach a known mass to the other end. When the mass is displaced, it breaks the
photogate beam. LoggerPro will then record its period of oscillation. beam when the system is in equilibrium. Note that, because we will be using springs with
different spring constants, you will need to adjust the height of the photogate for each new
spring you try.
2. Set up LoggerPro to record ﬁve periods of the mass’s oscillation. You can do this using the
95 LoggerPro template pendulum.cmbl.
3. Pull down on the hanging mass so that it breaks the photogate beam. Release it and let it
bounce back and forth a few times. You need to make sure that the hanging mass breaks
the photogate beam twice - once on descent and once on ascent - so adjust the hanging mass
horizontally so that just its edge passes through the photogate beam.
4. Take data on ﬁve successive oscillation periods using LoggerPro.
5. Find the mean period of oscillation and convert it into an angular frequency of oscillation, ω .
6. Using your calculated ω and the known mass, m, ﬁnd k, the force constant of the spring.
7. Repeat steps 1-6 for the other two springs in your kit. All should have signiﬁcantly different
Problem 10.11 Your springs are labeled with colors: red, green, and blue.
Rank the springs in order of their spring constants, from smallest to largest.
How could you tell which spring had the largest spring constant if you had
nothing but the springs themselves?
Problem 10.12 How could you quantitatively measure the spring constant
of a spring if you had no equipment other than a 150-gram mass and a ruler?
Experiment 2: Simple Pendulum
In this experiment, we will show that the angular frequency of oscillation for a pendulum depends
only on its length and on the gravitational constant, g. Figure 10: The pendulum bobs used in Experiment 2. From left to right, they are: plastic, wood,
brass, and aluminum. 96 1. You have been provided with four different possible pendulum bobs to use for this experiment, all of which have the same geometry. They are: wood, brass, aluminum, and plastic.
A picture of the four bobs is shown in Figure 10. The center of mass of each cylindrical bob
is marked on it in pen. Choose one of the bobs and attach it to the end of a 1.5-meter piece
of black nylon string. Clamp this string onto the pendulum apparatus as shown in Figure 5.
Replace the word “material” on the spreadsheet with the type of material of the bob you’ve
chosen (see ﬁgure 10 for material types).
2. The length of the string from the pivot point to the center of mass of the bob () should
be somewhere between 0.4 and 1.2 meters. Measure this using the steel ruler provided and
record it on your template. Also weigh the bob and record its mass.
3. Use the same LoggerPro setup that you used in Experiment 1.
4. Set the photogate height so that the infrared light beam in the photogate is aligned with the
center of the bob at the bottom of the pendulum’s swing.
5. Record ﬁve successive periods of the pendulum’s oscillation. Copy and paste these into the
appropriate cells in your lab report.
6. Repeat steps 1-5 for ﬁve more different string lengths ranging from 0.4 to 1.2 meters. Try
to space your lengths equally over this interval. You will need to adjust the position of the
clamp holding the pendulum’s string so that the bob passes cleanly through the photogate in
7. Calculate the mean period for each of your six string lengths.
8. Convert your mean periods to angular frequencies using Equation 2.
9. Using Equation 8, calculate the theoretical values of ω for each of the six string lengths.
10. Remove your pendulum bob and choose another one made out of a different material. Record
the mass of this second bob in your lab report.
11. Repeat steps 1-9 for this new bob.
12. A graph will appear that plots ω 2 vs. 1/l for both pendulum bobs. Using the Excel Trendline
function, create linear ﬁts for these two plots. See Appendix A Section 2.4 for the procedure
and be sure to display the ﬁt equation on the graph. Use your ﬁt parameters to calculate an
experimental measurement of g for each of the two sets of data. 97 Problem 10.13 Does the angular frequency of oscillation depend on the
mass of the pendulum bob? How can you tell? Does it depend on the length
of the string? If so, does the frequency go up or down as the string length
Problem 10.14 If you used large pendulum displacements rather than
small ones, could you measure g accurately using this method? Why or
why not? If not, would you over- or under-estimate g?
Problem 10.15 If we were on the moon, would the angular frequency for
a given string length be larger or smaller than the one we observe here on
Earth? Why? What would this do to the slope of your plot?
Problem 10.16 If the string were quite heavy (as opposed to your experiment, where the string was assumed massless), how would ω be affected?
Assume that string length and total mass were constant.
Experiment 3: Ball-Dish Oscillator
In this ﬁnal experiment, we will compare measured oscillation frequencies for two balls in a dish
with the theoretical frequency from Equation 15. In the apparatus we will use today, the radius
of curvature of the dish, R, is approximately 18.0 inches. We will roll two different balls on its
surface. Both the solid steel ball and the ping-pong ball have diameters close to 1.50 inches, but
one is solid while the other is hollow, so their moments of inertia have different geometrical factors
1. Type your dish number into the appropriate cell on your lab report. The radius of curvature
of the dish should appear.
2. Use the same LoggerPro setup that you used in Experiments 1-2.
3. Record ﬁve successive periods of the ball’s oscillation using LoggerPro. You can do this
using the LoggerPro template ball and dish.cmbl. It may take a little practice to get the ball
to roll cleanly for ﬁve complete cycles without hitting the photogate. It will probably work
best to start the ball rolling about halfway from the center of the dish.
4. Record the periods of oscillation in your lab report.
5. Find the mean of your measurements. Using Equation 2, convert the mean period to an
angular frequency, ω .
6. Repeat steps 3-5 using the ping-pong ball.
7. Calculate theoretical angular frequencies of oscillation for the two balls using Equation 15. 98 Problem 10.17 Why do you observe a signiﬁcant difference in the periods
of the steel ball and ping-pong ball? Consult Equation 15 to help explain
Problem 10.18 How would the following sources of error affect your measured oscillation frequencies for a hypothetical ball-in-dish (if at all)? Write
a one-sentence explanation for each. Write another sentence describing
whether the source of error would affect one ball more than the other (solid
vs hollow sphere) and, if so, why. Note that you should assume the smallangle approximation still holds.
(a) The radius of the ball is slightly less than measured.
(b) You give the ball a slight push as you let it go, while the ball remains to
roll without slipping.
(c) The force of friction is not enough to prevent the ball from slipping,
however the energy losses to friction are negligible in comparison to
the total energy.
(d) The scale measures the ball to be more massive than it actually is. 99 ...
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This note was uploaded on 09/13/2011 for the course ECONOMICS 101 taught by Professor Gerson during the Spring '11 term at University of Michigan.
- Spring '11