Download Lab 12 PERIODIC MOTION

Lab 12: Periodic Motion
Name: __________________________
Goals:
- Measure the basic characteristics of periodic systems (amplitude, period, frequency) and verify their dependence on the
physical properties of the system (mass, length, etc.).
- Analyze periodic systems in terms of energy.
Basic properties of oscillating systems:
Any system that repeats its behavior on a fixed time interval is called periodic. The period of the system is the time interval on
which it repeats its behavior.
The reciprocal of the period is called the frequency – frequency measures “oscillations per second” instead of “seconds per
oscillation”. The units of frequency are Hertz (Hz).
Periodic systems oscillate about a point of stable equilibrium, which we usually just call the equilibrium position. The periodic
motion is the result of a restoring force – a force which always points toward the equilibrium position.
For simple periodic systems, the maximum displacement from the equilibrium position is the same on each side of the
equilibrium position. This maximum displacement from equilibrium is called the amplitude of the system.
The simple harmonic oscillator:
A simple harmonic oscillator is an oscillating system guided by Hooke’s Law: F = kx . That is, the force is proportional to the
displacement from the equilibrium position. This kind of oscillation leads to a sinusoidal position function for the oscillating
mass.
The period of the simple harmonic oscillator is independent of the amplitude. For an oscillator with mass m and spring constant
k, the period is given by the formula:
T  2
m
k
When we plot the position of the SHO as a function of time, we obtain a simple sinusoidal function (a sine/cosine wave with a
phase shift that depends on when we start collecting the data):
position (x):
period
time(t)
amplitude
We can see the amplitude and period on this plot: the amplitude is the distance from the “midline” to a crest or trough, and the
period is the distance from peak-to-peak or trough-to-trough.
It is also useful to analyze the SHO from an energy perspective. At any particular moment, the oscillator has some
displacement, x, from the equilibrium position, and it also has a speed v. The energies associated with these parameters are
given by the familiar formulas:
PEspring 
1 2
1
kx and KE  mv 2
2
2
We can view the behavior of the SHO as a continuous transfer of energy between these two systems.
A note about vertical spring-mass systems:
The simple harmonic oscillator consists of a mass attached to a Hooke’s Law spring. In this lab we will hang the spring vertically,
but the only effect of gravity is that it sets the location of the equilibrium position. The weight of the hanging mass can be used
to compute the spring constant, but gravity does not have to be considered in the analysis of the motion. The argument goes
like this:
s
s
m
x
m
Unstretched spring before we
attach the mass. The spring sits
at its equilibrium position.
New equilibrium position after the mass is
attached. At this location, the upward spring
force is equal to the downward force of
gravity: ks  mg .
Now if we stretch the spring by an
additional distance, x, the upward force
exerted by the spring is k ( s  x) and the
downward force exerted by gravity is mg .
The net force is upward is given by the
expression: Fnet  k ( s  x)  mg , but we
use the fact that ks  mg to obtain:
Fnet  kx ; i.e., it behaves like an SHO with
the equilibrium position at length s.
Thus, for a vertical spring-mass system we can still say F = kx , where x is measured from the new equilibrium position.
Everything we know about the simple harmonic oscillator follows as before.
The Simple Pendulum:
The simple pendulum consists of a light string (or “massless rod”) attached to a small mass. The system is allowed to swing back
and forth in periodic motion: the tangential component of the gravitational force is the restoring force here – it always points
toward the equilibrium position for the mass.
L


L
Fg
 mg sin 
Fg
tan
 mg sin 
tan
mg
mg
With the simple pendulum, our amplitude is now an “angle amplitude” – telling us the maximum angular displacement max
from the equilibrium (straight down) position. The angular displacement is nearly sinusoidal as a function of time, and this
approximation is better for smaller amplitudes. The approximate period of the simple pendulum is given by the formula:
T  2
L
g
Notably, the period depends only on the length of the pendulum, not the mass!
We can also analyze the simple pendulum from the energy perspective. Now, the potential energy is gravitational, so the
formulas are:
1
PE grav  mgy and KE  mv 2
2
Just like the simple harmonic oscillator, the motion of the pendulum can be understood as a continuous exchange of energy
between the potential and kinetic systems.
Simple Harmonic Oscillator
1. Set up a hanging spring/mass system by using a stand secured with a C-clamp or table clamp, an extension rod, and a meter
stick. Tape a cardboard square to the mass, so the position sensor can get better echoes. You will use a 200 g mass for the
actual oscillations.
cardboard square
Position sensor.
2. Use two 100 g masses to measure the spring constant of the hanging spring. Remember, we have to use a change in force
and a change in length to get the spring constant ( F  k x ), because the spring is not “ideal” when its coils are touching.
D F = __________ N
D x = __________ m
k
__________ N/m
3. Hold the 200g mass at +5 cm (5cm above the equilibrium point) and release it from rest. Take position and velocity data
using the position sensor. Each person should print out their position and velocity graphs and note the following numbers
by referring to the raw data:
maximum height : _____________ m
minimum height: _____________ m
maximum speed: _____________ m/s
period: ____________ s
4. Repeat step 3, but start with an initial position of +10 cm instead of +5 cm.
maximum height : _____________ m
minimum height: _____________ m
maximum speed: _____________ m/s
period: ____________ s
5. How does the period change as you change the amplitude of the harmonic oscillator?
6. Compute the period of this oscillator by theoretical means: T  2
m
. Compare to the average of the two experimental
k
values.
Ttheoretical  __________ s
% difference = ______________
7. Use your position data for the 10 cm oscillations to find the maximum potential energy stored in the spring .
PEmax  __________ J
8. Use your velocity data for the 10 cm oscillations to find the maximum kinetic energy stored in the mass.
KEmax  __________ J
9. According to the theory, the maximum kinetic and potential energies should be the same. Explain why the maximum kinetic
energy came out less than the maximum potential energy.
Simple Pendulum Part 1: Changing the Mass
1. Set up a simple pendulum by using 1m of string.
L=1 m
2. Measure the period of oscillation by using 20 oscillations for the following masses:
m  20 g ,
T  ________ s
m  50g
T  ________ s
m  100g
T  ________ s
m  500g
T  ________ s
3. Compute the period for this pendulum by theoretical means: T  2
L
g
Ttheoretical  ______________
4.
According to the theory, the period should not depend on the mass, but you should notice that the period did change in a
systematic way as you added more mass. Explain how the period changed and why.
Simple Pendulum Part 2: Changing the Length
1. Now set up the small pendulum (with the spherical bob). Find the period for the following lengths:
Length
10 cm
20 cm
30 cm
40 cm
50 cm
60 cm
70 cm
80 cm
90 cm
100 cm
Period
2. To analyze the dependency of period on length, we’ll perform a graphical analysis of the data using a log-log plot. The
purpose of the log-log plot is to linearize the data in order to draw a best-fit line, because the relationship between T and L
is non-linear (T is proportional to L ).
We start with the given relationship between period and length and then derive a linear relationship:
T  2
L
g
 2 

L
 log(T )  log  2
  log
  log 
g
 g 

 log(T ) 
 L
 2 
1
log( L)  log 

2
 g 
Now, if we view log(T ) as our vertical variable, and log( L) as our horizontal variable, we should obtain the plot of a line
 2 
with slope 1/ 2 and y-intercept log 
.
 g 
Make a log-log plot for your data using Excel, and verify that it has the correct slope. Print out your scatterplot – including
the best fit line and its formula in the graph.