SHM: Simple Harmonic Motion
Back and Forth and Back and Forth. . .
Cameo
Any motion that repeats itself regularly is known as harmonic or periodic motion. The pendulum in a grandfather clock, molecules in a crystal, the vibrations of a car after it encounters a pothole on the road, and the rotation of the earth around the sun are examples of periodic motion.
In this unit we will be especially interested in a type of periodic motion known as Simple Harmonic Motion, which is often called SHM. SHM involves a displacement that changes sinusoidally in time. It is always caused by an unbalanced (restoring) force that is proportional to the displacement.
In this unit, you will explore the mathematical significance of the phrase "displacement that changes sinusoidally in time". You will also study the mathematical behavior of two classical systems that undergo SHM the mass on a spring and the simple pendulum.
SHM is so common in the physical world that your understanding of its mathematical description will help you understand such diverse phenomena as the behavior of the tiniest fundamental particles, how clocks work, and how pulsars emit radio waves. Pendula and masses on springs are merely two examples of thousands of similar periodic systems that oscillate with simple harmonic motion.
Some Characteristics of Oscillating Systems
All systems that oscillate with simple harmonic motion have two characteristics. A force that is directed toward the equilibrium position and a force that is proportional with the displacement from the equilibrium position. Other systems that oscillate have forces that approximate these conditions.
In describing the similarities between the oscillating systems it would be useful if we all used a common vocabulary. The three terms used most often in describing oscillations are the following:
Useful Definitions for Oscillating Systems
Period The time it takes for the oscillating particle to go through one complete cycle of oscillation. This is commonly denoted by the capital letter T.
Frequency The number of cycles the oscillating particle makes in one second. This is denoted by the symbol f or the Greek letter small n (pronounced nu) in most textbooks. The units of frequency are hertz where 1 hertz = 1 cycle/sec. The unit hertz is abbreviated Hz.
Amplitude The maximum displacement of the oscillating particle from its equilibrium position. Following the convention in this course, displacement is usually measured in meters. The symbol for Amplitude is often the capital letter A. Other letters such as Xmax or Ymax are also used. In the case of the simple pendulum, the amplitude is usually measured in radians and denoted by qmax.
Oscillating Spring
Set up equipment as in diagram.
Diagram of the setup for the Graphical Observation of the Motion of a Mass on a Spring.
Pull down on your spring to obtain a good amplitude. (Somewhere between a small displacement and one that stretches the spring so much that it remains permanently distorted.) Let the mass go. As you watch the mass oscillating on the spring you can see the mass going from a maximum displacement to no displacement and then to a maximum displacement in the opposite direction. What do you expect a graph of this motion to look like?
(a) What would a graph of the mass' distance from the motion detector vs. time look like? Sketch your prediction.
(b) Set up the motion software to record a distance vs. time graph for 10 seconds. If possible, set the data rate for the maximum feasible number of points/second. Use the motion detector to measure the equilibrium position of the mass and record.
(c) Give your mass approximately the same amplitude you gave it for your casual observations. Sketch the graph of position vs time.
(d) Label the sketch of your observed graph in part (c) as follows:
"1" at the beginning of a cycle and "2" at the end the same cycle
"A" on the points on the graph where the mass is moving away from the detector most rapidly.
"B" on the points on the graph where the mass is moving toward the detector most rapidly.
"C" on the points on the graph were the mass is standing still.
"D" where the mass is farthest from the motion detector.
"E" where the mass is closest to the motion detector.
(e) Use the analysis feature of the software to read points on the graph and find the period, T, of the oscillations.
(f)Find the frequency of the oscillations, n, from the graph.
(g) Use the analysis feature again to find the equilibrium distance and the amplitude , Xmax, of the oscillations. Show your computations.
Note: At this point, be sure to save your data as a file on your disk. You will need it for the next couple of activities!
Velocity Graph for a Mass-Spring System
(a) At what displacements from equilibrium is the velocity of the oscillating mass a maximum? A minimum? At what displacements is the velocity of the mass zero? (For instance, is the velocity a maximum when the displacement is a maximum? is zero? or what?)
(d) Arrange to display the distance, velocity and acceleration graphs on the same screen using the MBL motion software. Sketch the actual velocity/acceleration on the graphs with a solid line of a different color.
What is Simple Harmonic Motion?
Simple Harmonic Motion is any periodic motion in which the displacement varies sinusoidally in time. In other words, either a sine or cosine function, which both have exactly the same basic shape, can be used to describe the displacement as a function of time. SHM motion occurs whenever the restring force is proportional to the displacement and is directed toward the equilibrium point. To be more exact, a general sinusoidal equation that describes the displacement x(t) as time goes on can be given in the form
where A is the amplitude or (maximum displacement from equilibrium) of an oscillating mass, w is its angular frequency, and f is its phase angle.
Angular frequency (w):
w = 2pn (rad/s)
where w has units of rad/s and n is the frequency of oscillation in hertz
Phase Angle (f):
f± Arc Cosine (x(0)/Xmax). The phase angle is the angle in radians needed to determine the value of the displacement, x(0) of the oscillating system when t=0 s.
This phase angle is positive (+) when the initial velocity of the mass is negative and it is negative when the initial velocity of the mass is positive.
Was the Motion Harmonic?
In this unit we would like you to demonstrate that, within the limits of experimental uncertainty, the actual motion of a mass on the end of a spring undergoes a sinusoidal oscillation, which can be represented by the Simple Harmonic Motion Equation
Consider the data you recorded earlier with the MBL motion detector for displacement of a mass on a spring as a function of time. How closely can the data be represented by a cosine function? This is the acid test for ideal simple harmonic motion.
Displacement vs. Time : Experimental
(a) Refer to the data you reported before and use it to find the amplitude Xmax and the angular frequency w associated with the motion you recorded for the mass on the spring.
Xmax = w =
(b) Use equation 1 to show mathematically that the phase angle is given by f = ± Arc Cosine (x(0)/Xmax). Explain why the ± sign is needed.
(c) Use the definition of the phase angle along with the value of the displacement at time t=0 and the values of w and A you determined to calculate the value of the phase angle, f.
(d) In order to compare the theoretical and experimental displacements with each other, you should do the following:
Simple Harmonic Motion for the Mass & Spring
Theoretical Confirmation of SHM for a Spring-Mass System
You should be able to show that a sinusoidal motion will occur for an oscillating mass-spring system if:
(1) the one dimensional force exerted by the spring on a mass has the form F= -kx where k is the spring constant, and
(2) Newton's second law holds.
Using these assumptions you can show mathematically that the following equation will hold:
where
f = the phase angle indicating the displacement, x(0), at t = 0 s
x(t) = displacement of the spring from equilibrium at time t
Xmax = amplitude of the system ,i.e. its maximum displacement
w = the angular frequency of oscillation given by 
where k is the spring constant and m is the oscillating mass
Displacement vs. Time : Theoretical
(a) Show that the equation of motion (F = ma when written out) of a mass m attached to a spring of force constant k is given by
Hint: (1) What are the definitions of instantaneous velocity and acceleration in one dimension? (2) x(t) is really just x expressed in a form that reminds us that x changes with t.
Note: This type of equation, which occurs frequently in physics, is known as a differential equation.
(b) Show (with the help of a calculus student) that if
then equation x(t)= Xmax cos (wt + f) satisfies the equation
A Mathematical Model for the Mass-Spring System.
You have confirmed the fact that the theoretical equation describing a mass-spring system is given by
where
f = the phase angle indicating the displacement , x(0) at t = 0 s
x(t) = displacement of the spring from equilibrium
Xmax = amplitude of the system i.e. its maximum displacement
w = the angular frequency of oscillation given by 
where k is the spring constant and m is the oscillating mass
In the next activity you will construct a model to explore the behavior of mass-spring systems for different values of the four parameters Xmax, f, k, and m. You will use Interactive Physics.
To construct your mathematical model you should do the following:
Summarize you findings of how the amplitude, phase, mass, spring constant affect the period.
The Simple Pendulum
What Does the Period of a Pendulum Depend On?
When a mass suspended from a string is raised and released it oscillates. The oscillating motion of a simple pendulum has been used throughout history to record the passage of time. How do clock makers go about constructing a pendulum with a given period? Why does a pendulum oscillate? What factors affect its period? For the following activity, you will need:
Factors Influencing Pendulum Period
(a) Watch the oscillation of a pendulum carefully. Sketch the forces on the bob when the pendulum is at its maximum angular displacement and at zero displacement
(b) Explain why the pendulum oscillates back and forth when the bob is lifted through an angle qmax and released.
Increase
Decrease
No effect
Factor
I
D
N
Reasons
Correct?
(d) Play with the pendulum factors and see which factors obviously matter. Summarize your findings in the space below and indicate in the table above which of your predictions were correct.
(e) Design an experiment to check the possible dependence of the period of a pendulum on its length in a much more careful quantitative way. You should design your experiment to be as accurate as possible with your time measurements. Use additional pages to describe your experiment and summarize your data and conclusions. Hint: Be sure to use some very short lengths and some much longer lengths for your pendulum so your longest is at least 8x more than your shortest.
(f) How do your results compare with your prediction?
Should the Pendulum Really Undergo SHM?
In your theoretical consideration of the mass-spring system, you showed mathematically that, if the restoring force is proportional to the displacement but opposite in direction, then one would expect to see the mass undergo simple harmonic motion. The restoring force for the spring has the form F=-kx. To what extent does the restoring force for a simple pendulum which oscillates at a small angle of displacement have a similar mathematical form to that of the mass on a spring? In this next activity you will derive the equation of motion for a simple pendulum. This equation is very similar to the equation of motion of the mass-spring system, and so it will be clear that the simple pendulum ought to undergo a simple harmonic motion in which its period of motion is independent of the mass of the pendulum bob.
The Pendulum Equation of Motion
(a)What is the restoring force on the pendulum bob as a function of m, ag, and the displacement angle q?
(b) The small angle approximation: The value of q in radians and the value of sinq are quite close to each other for small values of q. Use a spreadsheet or scientific calculator to find the angle in radians for which q and sinq vary from each other by 1%. Use three decimal places in your calculations.
(c) Calculate the value in degrees of the angle you have just calculated in radians.
(d) Show that, if the maximum angle through which the pendulum swings is small enough so that q = sinq (say to within about 1%), then the restoring force can be expressed (to within 1%) by F =magq.
(e) Refer to the solution of the spring-mass equation of motion to write down the solution to the pendulum equation and show why its solution is given by
Hint: In the pendulum equation of motion the term q plays the role of x in the mass-spring equation and the term (mag/L) plays the role of the spring constant k.
(f) Show that if the period, T, of a mass-spring system is given by the equation
then the period of the pendulum ought to be given by
(g) How does this expression for the period compare to the one which you found experimentally in Activity 28-7 (e)?
(h) Many people are surprised to find that the period of a simple pendulum does not depend on its mass. Can you explain why the period of a simple pendulum doesn't depend on its mass? Hint: Can you explain why the acceleration of a falling mass close to the surface of the earth is a constant regardless of the size of the mass?
Checklist of skills to obtain during this section
1 Analyze and draw graphs of d-t,v-t and a-t for SHM
2 Find frequency and period from equation d^2x/dt^2 = -kx
