Download Lab Writeup Springs and SHM

Springs and Simple Harmonic Motion
LBS 164L
OBJECTIVES:
1. To understand Hooke’s law (system under a restoring force) and to use various
methods to determine the force constant of a spring.
2. To investigate the simple harmonic motion of a spring and a pendulum, and to
understand the restoring force in both cases.
APPARATUS:
Spring, string, masses, digital timer, and stand with attached measuring stick.
THEORY:
Many forces vary with position. That is, they can grow stronger or weaker as the position
of the particle undergoing the force changes. One such example is the force exerted on a
mass attached to a spring. As the particle is moved away from the attached spring, the
spring will exert more force to restore it to the equilibrium position (where the spring and
attached mass are not in motion). The same force will resist the compression of the
spring. In both cases (compression and expansion of the spring), the restoring force
increases as the distance the particle is moved from equilibrium increases. The
importance of the spring system is that it is encountered in our daily lives in many ways
(i.e. shock absorbers, analog wristwatches, grandfather clocks, and garage doors) as well
as being used as a model for describing chemical bonds. In fact, practically all forces can
be described as spring forces when the displacements are kept small.
Hooke’s Law can very well approximate the force exerted by a spring:
F(x) = -kx
where x is the displacement from the equilibrium length of the spring, F(x) is the position
dependent restoring force exerted by the spring on the particle, and k is the force constant
of the spring (what is the negative sign showing us?). This is an approximation in that
Hooke’s Law is the limiting case for small displacements.
When a mass is added to the end of a spring and hung vertically, the force of gravity pulls
the mass down until the force of the spring is pulling upwards with the same force. By
knowing the mass (M) attached and measuring the displacement (x) from the equilibrium
position, the force constant can be calculated by:
Mg
k
x
where g is the acceleration of gravity. The new equilibrium position established with the
mass attached can be used to investigate Simple Harmonic Motion by applying a small
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perturbation (a displacement in this case) to the position of the mass. If the mass is
displaced a small amount (say a few centimeters), then the spring will accelerate the mass
toward the equilibrium position. Since this acceleration leaves the mass some velocity at
the equilibrium position, it does not stop there, but instead continues until the force of the
spring decelerates it to a turning point. At the turning point the mass has a velocity of
zero and the spring starts to accelerate the mass downward again. This process continues
until friction and heat loss drain the energy, and the mass and spring system settle into the
equilibrium state it was in before the perturbation.
The above mass-spring system is in Simple Harmonic Motion. A system in Simple
Harmonic Motion is one where the displacement varies with time according to the
relationship:
x  Acos( t   )
where A, , and  are constants of motion. The constant  is the angular frequency and
is defined as
2
  2 f 
T
where f is the frequency of vibration and T is the period.  has units of rad/s and f has
units of Hz. For the spring system we are investigating, the motion can be characterized
by
k

M
and the force constant can be determined by k = M2. But springs are not the only
devices that move in simple harmonic motion. The first study done concerning Simple
Harmonic Motion was that of Galileo on the Simple Pendulum. Galileo found that the
frequency (or period) for the oscillation of a pendulum was dependent only upon the
length of the pendulum and not on the mass. This discovery lead directly to more
accurate clocks. This relationship can be summed up as:
g
.

L
where L is the length of the pendulum and g is the gravitational constant.
Procedure:
Exploring the Simple Pendulum
1. Tie some string to a weight holder, tie the other end of the string to the stand, and
measure the length of the string-holder system. Make sure that the string is not so long
that the holder hits the table.
2. Choose around 100g of mass and add it to the holder. Pull the mass and string to the
side and release it. As the mass crosses the stand (zero), start the timer. Stop the timer
when the mass has completed five full cycles (past zero five times going in the same
direction).
3. Repeat step 2 using another mass. For this string length, calculate the frequency of
these pendulums. What do you find that is interesting?
4. Repeat steps 1-3 with a different length string.
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5. When did the frequencies change (when adding mass or changing string length or
both)? Does this agree with theory? Using the theory above, calculate the expected
frequencies for the experimental conditions and compare your experimental results to the
calculated.
Obtaining k from the force of gravity (Hooke’s Law)
1. Obtain a spring, a selection of fifty gram masses, and a weight holder if they are not
already at your station. Hang the spring from the stand with a weight holder attached.
Adjust the measuring stick on the stand so that the bottom of the weight holder is at zero.
2. Measure the displacement with at least three different masses added. The maximum
displacement should be at least 6 cm.
4. Repeat step 2 at least two more times.
5. Using MS Excel, make a table to enter your data. For each mass, determine the best
value of the displacement and its uncertainty. Calculate the force constant, k, and the
uncertainty for each of the measurements of k. To find the uncertainty in k, assume each
mass added has a value of 50.00 ± 0.03 g.
5. Using Kaleidagraph, make a plot of mass vs. displacement. Add error bars as needed
and then do a linear best-fit line for the data. Display the equation showing the errors.
What does the slope give you from the line? What should the intercept be?
Obtaining k from the frequency (Simple Harmonic Motion)
1. To your spring, add a holder and a fifty gram mass if they are not already attached.
Allow the system to become motionless. Adjust the measuring stick so that the bottom of
the holder is zero.
2. Displace the bottom of the holder 3 or 4 centimeters and release.
3. As you release the mass, start the timer.
4. Stop the timer when the holder passes through the bottom point after five complete
cycles.
5. Repeat the above at least two more times with this mass.
6. Using a four different masses, repeat steps 1-5.
7. In MS Excel, make a table and enter your data. Calculate the best value for the time
and the uncertainty in time for each mass. Use the relation between the period, the spring
constant and the mass to calculate  and k. Note that the mass must now include the
holder which has m = 20.0 ± 0.3 g.
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8. Make a plot in Kaleidagraph of M vs. 2 from your data. Display the equation of the

best-fit line showing the errors. What does this slope give you?
Extra Credit Project:
If you have time, see what happens to the spring constant k if two springs are hooked
together. Does it get bigger or smaller? Can you explain why?
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