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#[\ P#P#HOOKE'S LAW AND PERIODIC MOTION#XN\ PXP#
X0
xdIntroduction: In the previous lab you examined the phenomenon of elasticity for a simple
xspringmass system. With the system held at rest, masses were added to the bottom of a
xvertically suspended spring and the amount by which it stretched was measured. This property
X0
xdof elasticity is manifest in a wide variety of physical systems, ranging from rubber bands and
xsprings to earthquakes. These same physical systems also exhibit the phenomenon of periodic
x0motion. When the system is distorted from its original configuration, it tends to restore itself.
xThe motion of such a system is repetitive and the time interval over which the motion repeats
X? 0
xitself is called the temporal period. Systems that exhibit such motion are often called simple
X*
0
xdharmonic oscillators. The repetition of motion over a well defined path is called oscillatory
X0
x`motion. Motion that repeats itself in a well defined interval of time is called harmonic motion.
X0
xThe term simple refers to the fact that there is no friction. Examples of such motion include
xpendulum motion, oscillating springmass systems, the waves generated by earthquakes, the
motion of the planets, and binary star motion.
It is easily demonstrated that when an mass is suspended vertically from a spring and is
xthen pulled downward and released, the springmass system will oscillate in the vertical direction.
xTThe description of such motion will be the object of this lab experiment. You will examine how
xthe spring oscillates in the vertical direction as mass is added and subtracted. You will formulate
x|a mathematical description of the physical relationship between the period of motion and the
xmass added to the end of the spring. Your conclusions will be important not only in the realm
xof physics, but also in astronomy, where the period of revolution of a satellite around a planet
depends upon the mass of the planet. These points will be discussed in lecture and lab.
X0
xExperimental Apparatus: spring, vertical spring mount, set of different masses, stopwatch,
X0graph paper, and the computer software CricketGraph.
V0
X0
x|Implementation: With a known mass on the end of the spring, pull the mass down a short
xTdistance and release it. Be careful to pull the mass vertically downward as any sideways motion
x8will distort your measurements. Observe that the springmass system oscillates vertically and that
x<the motion seems to be harmonic. For different masses measure the average period and record
your data.
Observe that the time for one complete oscillation (what does this mean?) is very short.
xTo overcome this experimental difficulty, measure the time for ten complete oscillations.
xCalculate the average period and the standard deviation for each different mass configuration.
x0(Refer to the Appendix in the Lab I.) Plot the average period versus the total mass and examine
X#0
xdthe curve. Be sure to set the origin of your graph at (0, 0). Is it a straight line or a curve of
x@some recognizable form (refer to the graphs you generated in Lab I)? Can you represent the
experimental errors in the measurement of the period on the graph?
XT'0
xAnalysis and Conclusions: The analysis of this graph poses a different problem from that of
X=(0
xthe elasticity lab. Now you must determine some way to represent the data in a form that is
xsimple to interpret. At this point you must formulate an hypothesis that you can test or compare"()0*0*0*8("
xwith your experimental data. This is one of the most important aspects of experimental science.
X0
xIt is sometimes referred to as the hypotheticodeductive method. y~
yOb$#XN\ PXP#88FOOTNOTE TEX#A\ P/P# Í The debates in the Philosophy of Science over the notion of a single or unique scientific method often
yO*$begin with a discussion of this technique. #XN\ P
XP##KFOOTNOTE TEX#
#XP\ P6QXP# In essence you will make an
x"educated" guess as to which of the mathematical functions or curves that you have at your
xdisposal will give you the best description of your data. The notion of "best" is also crucial to
xthis process as sometimes the answer seems to lie somewhere in between two possible choices.
xDiscuss this with your partner and formulate some criteria to resolve this issue. After you have
formulated your hypothesis you must compare it with your data to test its validity.
HIn particular, one of the most common hypotheses that can be used to describe simple
X30
x,physical systems is the socalled powerlaw relationship. In this particular experiment the added
xmass (M) is the independent variable while the period of motion (T) is the dependent variable.
Consider a simple power law relationship between these given by
X0XX` ` Figure 1@T = kMn(#` `@"(#(1)
xwhere k is a constant and n is the power to be determined from the curvefitting process. Recall
xdthe properties of common logarithms and take the log of both sides of equation (1) to obtain
(Refer to the Appendix in Lab I.)
XQ0XX` ` log (T) = log (k) + n log (M)(#` `@"(#(2)
This is the equation of a straight line given by
X0XX` ` y = b + mx(#` `@"(#(3)
xwhere b (log k) is the yintercept, m (n) is the slope (the rise over the run) of the straight line,
x (log M) is the independent variable and y (log T) is the dependent variable.
X0
Using the transform operation under the Data menu transform your data into loglog
Xq0
xformat, plot it and use the simple curvefitting operation to fit your data to the equation of a
xTstraight line. The slope of this fit should answer the question as to which of the possible curves
xpgives the best or closest description of these data. Finally formulate a lawlike statement that
describes the relationship between the measured period and the mass at the end of the spring.
( s]
X 0` (##,(\ P6Q,P#8 January 1996#XP\ P6QXP#
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