Download Lab #11: Simple Harmonic Motion of a Linear Oscillator

Physics 211: Lab
Simple Harmonic Motion of a Linear Oscillator
Reading Assignment:
Chapter 16 – Section 1- Section 9
Introduction:
Imagine a point P that oscillates back and forth with simple harmonic motion. The period, T, the
frequency, f, and the angular frequency, , of point P are defined using the following equations.
T  1/ f
  2f
The linear motion of point P as a function of time is sinusoidal in nature. Assume that the position
x= 0 is defined as the central location of the motion.
P
X= -Xmax
X= 0
X = +Xmax
The position, velocity, and acceleration of point P can be described as a function of time as follows:
x(t )  xmax cos(t   )
v(t ) 
d ( x(t ))
 xmax sin( t   )
dt
a(t ) 
d (v(t ))
  2 xmax cos(t   )
dt
Notice that a(t) is related to x(t) and can be rewritten as follows:
a(t )   2 x(t )
In other words, the acceleration is not constant. It, too, varies in a sinusoidal manner.
The dynamics of simple harmonic motion are governed by Newton’s Second Law and can be
written as:
Fnet (t )  ma(t )
or, substituting for a(t):
Fnet (t )  m 2 x(t )
Since both m and  are constants, notice that the Net Force, Fnet, is proportional to the position, x,
of point P. The negative sign signifies that the direction of the Net Force is opposite the direction of the
position. In other words, the Net Force, Fnet, is a restoring force because it attempts to bring point P back
to its equilibrium position.
As a result, it should follow that any Net (Restoring) Force that varies as a function of position, x,
should cause simple harmonic motion. Such a system is called a linear oscillator, since the force varies
linearly with the position. A spring force acting on a mass, described by Hooke’s Law, satisfies this
condition:
© 2004 Penn State University
Physics 211R: Lab – Simple Harmonic Motion
of a Linear Oscillator
Fspring  kx
In fact, it has been shown that a mass oscillating at the end of a spring (with no frictional forces)
experiences pure simple harmonic motion. By combining the above two equations, an equation for the
theoretical value of the angular frequency of the frictionless system is obtained:
Fspring  Fnet
 kx  m 2 x

k
m
(for a system that is not damped)
When describing simple harmonic motion the term used for  is “angular frequency” instead of “angular
velocity”. This often confuses students because the concepts are similar. There are many other examples of
simple harmonic motion. (Read Section 16-5 and 16-6 of the text for an analysis of torsion pendulums,
simple pendulums, and physical pendulums.)
Damping:
In the real world, most oscillations are eventually “damped” due to friction or viscous forces. A
damping force causes the amplitude (and therefore, energy) of a given system to decrease over time. In the
position equation, x(t), for a damped simple harmonic oscillator, xmax , is a decreasing function, not a
constant. The angular frequency,  of the damped system is less than the natural angular frequency, ,
that the system would have if it was not damped. (Read Section 16-8 for a specific example of a damping
force.)
Driven Oscillations:
Many systems that oscillate experience a driving force in addition to a restoring force. A driving
force is a periodic force (with a unique period, frequency, and angular frequency of its own) that effectively
pushes or pulls an object away from the equilibrium position. However, if the angular frequency of the
driving force, d, is approximately the same as the natural angular frequency, , of the system, then the
system experiences resonance.
  d
When resonance occurs, the amplitude, xm, of the system reaches a maximum.
There are many examples of resonance in everyday life. Engineers must be extremely careful not
to design a structure that has a natural frequency that matches a potential driving force. The Tacoma
Narrows Bridge disaster is an example of such an error. Troops usually march in step with each other as
they march in formation. However, whenever crossing a bridge, they are taught to break step, just in case
their cadence matches the natural frequency of the bridge. A wine glass has a natural frequency of
vibration that can be heard if the top of the glass is moistened and rubbed. When a singer with a strong
voice sings a note at the very same pitch (frequency), the sound waves in the air drive the glass to vibrate
back and forth. If the force by the air molecules is strong enough and acts long enough, the glass will
break. Musical instruments, a child on a swing, the motor in a car, etc. all exhibit resonance under certain
conditions.
Most real examples of simple harmonic motion involve all three types of forces simultaneously: a
restoring force, a damping force, and a driving force. Depending upon the nature of these forces, the
resulting motion is sometimes more complicated than simple sinusoidal motion. In addition, most systems
exhibit more than one natural frequency. The analysis of these more complicated systems is an advanced
topic of study in physics. However, the basics of simple harmonic motion remain the same.
© 2004 Penn State University
Physics 211R: Lab – Simple Harmonic Motion
of a Linear Oscillator
Physics 211: Lab
Simple Harmonic Motion of a Linear Oscillator
Goals:






Determine the natural period, frequency, and angular frequency of a linearly oscillating system.
Observe the relationships between the position, velocity, and acceleration of an object undergoing
simple harmonic motion.
Observe the resulting motion of a damped simple harmonic oscillator.
Verify the conditions necessary for a driven oscillating system to achieve resonance.
Observe the resulting motion of a driven simple harmonic oscillator.
Apply the concepts of resonance and damping to the shocks of a car.
Equipment List:
1.2 meter track with clamp
3-point stand with vertical rod
Spring (k = 3.85 N/m)
Cart (500 grams)
Ultrasonic motion detector
20 volt DC Harrison power supply
Voltage Sensor
Mechanical Oscillator/Driver
Activity 1: Analysis of the Simple Harmonic Motion of a Cart / Spring System
There are two purposes for this activity. The first is to determine the natural frequency of a
cart/spring system. The second is to compare the position, velocity, and acceleration graphs of an object
undergoing simple harmonic motion.
Setting Up the Graphs:
1. Set up Data Studio™ to read the data collected by the ultrasonic motion detector located at the base of
the track. Check to make sure that the motion detector is oriented towards the cart and is sending out a
narrow beam signal.
2.
Create a graphing window that contains the following graphs: Position vs. Time, Velocity vs. Time,
and Acceleration vs. Time.
3.
Attach the 500 gram cart to the spring (k = 3.85 N/m) located on the inclined track. Using the ruler
along the side of the track, make note of the equilibrium position of the cart.
4.
Using the ruler located along the length of the track, estimate the distance from the motion detector to
the closest end of the cart when the cart is at rest in the equilibrium position. Record this value in the
table below.
Distance of Cart from Motion Detector at Equilibrium
(meters)
5.
Stretch the spring some initial length (but do not over stretch the spring!) and then release the cart
from rest. Press Record to collect data for approximately five cycles of simple harmonic motion, and
then press Stop.
© 2004 Penn State University
Physics 211R: Lab – Simple Harmonic Motion
of a Linear Oscillator
6.
Notice that the Position vs. Time graph does not oscillate about the origin of position (the x-axis) of
the graph. Why not? What does Data Studio™ define as “Position”? How could this graph be altered
so that “Position” was redefined as “the location of the cart measured from equilibrium”?
7.
Using the Experiment Calculator, create a new calculation called “Position from Equilibrium”. State
the calculation formula in the table below.
Calculation Name
Position From Equilibrium
8.
Short Name
X
Units
meters
Formula
In the graphing window, change the plot of the Position vs. Time to Position from Equilibrium vs.
Time by clicking on the y-axis icon. Notice that the graphs differ by a vertical shift. The Position
from Equilibrium vs. Time graph should oscillate around the x-axis.
Determine the Angular Frequency of the System:
9.
Using a method of your choice, measure the amount of time that it takes the cart to complete 1 cycle.
10. From your measurement, record and/or calculate the period, frequency, and angular frequency of the
resulting simple harmonic motion of the cart/spring system in the table below.
Period – T
(s)
Frequency – f
(Hz)
Angular Frequency – 
(rad/s)
11. Determine the theoretical value for the natural angular frequency, , of the cart (mass) and spring
system. Explain your calculation. (See the Introduction or Section 16-3 in the text. Note: This
calculation assumes that the system is frictionless.)
(Theoretical)
Explanation of Calculation
Natural Angular Frequency - 
(rad/s)
12. What is the value of the % error between the experimental value of and the theoretical value of 
(assumed to be “true”)? Clearly show your calculations. Is the experimental value (which is slightly
damped) less than the theoretical value (which assumes that it is not damped, and therefore “natural”)
as predicted?
Comparing the Graphs of Simple Harmonic Motion:
13. Analyze the relationships between position, velocity, and acceleration by answering the following
questions:
Note #1: Recall that the motion detector defines the positive direction to be pointing up the incline.
Note #2: The Analyze Tool located in the bottom left hand corner of the graphing window is a
helpful resource for analyzing the graphs. This tool looks like a cross hair.



What is the location and direction of the cart when the velocity is at a maximum or minimum
value?
What is the location of the cart when the velocity is equal to zero?
When the acceleration of the cart is at a maximum or minimum value, what is the velocity and
location of the cart?
© 2004 Penn State University
Physics 211R: Lab – Simple Harmonic Motion
of a Linear Oscillator
Activity 2: Damped Simple Harmonic Motion
The purpose of this Activity is to observe the behavior of an object undergoing damped simple
harmonic motion.
1.
Stretch the spring some initial length (but do not over stretch the spring!), press Record, and then
release the cart from rest. Allow the cart to oscillate back and forth until it comes completely back to
rest before pressing Stop.
2.
Copy the graphing window into the Template by using “paste special”. Paste it as if it was a picture.
3.
Answer the following questions regarding the damped motion of the cart / spring system:

How does the period of the resulting simple harmonic motion change over time as damping
occurs? Support your reasoning by analyzing data from the graphs. Clearly explain your analysis.

How does the amplitude of the position change over time as damping occurs? (For example: Is it
constant? Is it changing linearly? Is it changing exponentially? Is it increasing? Is it decreasing?)

What does the change in amplitude suggest about the total energy loss of the system over time?
(i.e. What type of dissipating forces are likely acting on the system?)
Activity 3: Forced Oscillations & Resonance
The purpose of this Activity is to observe the behavior of an object undergoing forced (driven)
simple harmonic motion and to verify the conditions necessary for resonance to occur.
1.
Place the cart on the track so that it is attached to the spring and at rest in its equilibrium position.
2.
Important: Do not turn on the power supply until the following conditions have been satisfied:



The Voltage knob should be turned completely in the counter-clockwise position (its lowest value)
whenever the power supply is turned on and off. Note: There are two adjustment knobs. The
black one provides coarse-tuning, and the red one provides fine-tuning. Both should be turned
completely counter-clockwise.
The Current knob should be turned completely in the clockwise direction (its highest value). This
setting should not be changed throughout the entire experiment.
Check the connections between the power supply and the Oscillator/Driver to be sure that the Red
leads are attached to the + side of the power supply and the Black leads are attached to the – side
of the power supply.
3.
Set up the equipment:
 Set up Data Studio to read the data collected from the Voltage Sensor.
 Display the Voltage Sensor data on the Digits display window. (This window will display the
voltage supplied to the Mechanical Oscillator/Driver whenever Record is pressed.)
 Change the Digits display window from “0.0” to “0.00” so that more digits are displayed. This is
done by double- clicking anywhere on the display window.
 Turn on the 20-volt DC power supply.
 Slowly increase the voltage in the circuit that controls the Oscillator/Driver so that the
Oscillator/Driver begins to turn. Note that the Oscillator/Driver creates a periodic pulling motion
on the cart / spring system.
4.
Slowly increase the voltage in the circuit that controls the Oscillator/Driver so that the
Oscillator/Driver is caused to rotate at different frequencies. Using the value displayed on the Digits
display window, set the voltage of the system to the settings listed in the table below.
© 2004 Penn State University
Physics 211R: Lab – Simple Harmonic Motion
of a Linear Oscillator
5.
At each voltage setting (and therefore frequency), first allow the system to damp itself into a regular
pattern. Then, using Data Studio™, press Record to collect data on the resulting motion. At each
setting, determine the period, angular frequency, and amplitude. Place your data in the following table.
Voltage Setting
(volts)
1.5
2.0
2.5
3.0
3.5
4.0
Period
(sec)
Angular Frequency
(radians/s)
Amplitude
(meters)
6.
Use Excel™ to make a plot of Amplitude vs. Angular Frequency. (Place Amplitude on the y-axis
and Angular Frequency on the x-axis.) Label the axes.
7.
Copy and paste the Excel™ plot into the Template.
8.
Using the graph in Excel™, predict the approximate angular frequency of the Oscillator/Driver that
will make the cart/spring system “resonate”. Turn the Voltage knob to the corresponding value from
your graph, and then use the fine-tuning adjustments to find resonance. Ask your TA to help you if
you have trouble locating it.
9.
Once resonance is found, obtain a graph of the motion of the cart/spring system at this resonant
frequency: Start the cart from rest at its equilibrium position. Press Record and allow the cart to
oscillate back and forth until it reaches an apparent maximum amplitude limit for the given system.
Press Stop.
10. Remember:
supply off!
Turn the Voltage knob completely counter-clockwise before turning the power
11. Copy the graphing window of Position from Equilibrium vs. Time, Velocity vs. Time, and
Acceleration vs. Time into the Template by using “paste special”. Paste it as if it was a picture.
12. Determine the period, frequency, and angular frequency of the driven system at resonance and place
the values in the table below.
Resonant Period – T
(s)
Resonant Frequency – f
(Hz)
Resonant Angular Frequency – 
(radians/s)
13. Compare these values with the natural period, frequency, and angular frequency of the cart/spring
system determined in Activity 1. What condition is necessary to cause a system to resonate?
© 2004 Penn State University
Physics 211R: Lab – Simple Harmonic Motion
of a Linear Oscillator