Otterbein University Department of Physics
Physics Laboratory 1500-12
EXPERIMENT 1500-12
HOOKE’S LAW AND SIMPLE HARMONIC MOTION
NAME:
INTRODUCTION
Oscillatory systems are extremely common in nature. Examples abound, from the
vibrations of molecules to the beating of the heart. The motion of a mass on a spring is
one of the simplest forms of oscillatory motion, and is called simple harmonic motion.
PART A: MASS AND SPRING
At your table you have a set of masses and a spring suspended from a beam. Springs will
generally obey Hooke’s law, F = -kx, unless they are stretched to extremes, where ‘x’ is
the linear extension from an equilibrium position, and ‘F’ is the additional force applied
after the equilibrium has been established. By measuring ‘x’ and ‘F’, we can determine
the stiffness of the spring, ‘k’, often referred to as the spring constant. We can stretch the
spring by hanging a mass from its end. Measure the resulting elongation of the spring,
and determine the weight of the mass, starting a table that lists the force as a function of
the elongation. Keep adding mass in reasonable increments. Find the spring constant by
putting your data in a force-versus-elongation plot on the graph paper below. Label the
axes; include units. The spring constant is the slope of the graph.
k = _______
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Otterbein University Department of Physics
Physics Laboratory 1500-12
PART B: RELATIONSHIP TO SIMPLE HARMONIC MOTION
Attach a 500g metal cylinder to the spring and start the motion of the mass-spring system
by releasing the mass when it is pulled a few centimeters down from the equilibrium
position. First, make a plot of the position of the mass as a function of time with t=0
corresponding to the instant the mass is released. Show several periods of oscillation.
Label both axes with the appropriate numerical values based on your observations.
Now that we know what happens in nature, let's see what physics has to say about the
motion. We are looking for a mathematical description of the position of the metal
cylinder as a function of time. We know that an elastic spring force is acting on it, so we
have one side of Newton's second law, F = –kx(t). The other side (ma) can be viewed as a
function of the position, too, as the acceleration is the second derivative of the position
with respect to time, so we have
F= –kx(t) = ma = m(d2x/dt2)
(Eq. 1)
This is a differential equation, i.e. an equation whose solution is an entire function, x(t),
not a number. From our observations above, we suspect a sinusoidal behavior of the
position, so we make the ansatz, i.e. the educated guess
x(t) = A cos(2πt/T + φ)
(Eq. 2)
Here, A is the amplitude of vibration, T is the period, i.e. the time for one oscillation, and
φ is a fixed phase constant that can be determined by where you start the clock (t=0).
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Otterbein University Department of Physics
Physics Laboratory 1500-12
Verify that this ansatz is correct by plugging it into (Eq. 1). The ansatz has three
parameters (A,T, φ), while (Eq.1) has two (m, k). The ansatz will be correct only if these
two sets of parameters are related in a specific way. Find the relationship for the period T.
What does physics say about A and φ?
Let the mass-spring system oscillate ten times and measure the period with a stop watch.
T = ________
(one oscillation)
How does your calculation above compare to your result?
Determine the phase constant φ:
φ =__________
Draw a new curve on the plot above (using colored pencil or a dashed line). This time let
t =0 correspond to the moment the mass is passing
down through the equilibrium position:
φ' = __________
Repeat again, but this time let t = 0 correspond to the moment the mass is passing down
about half-way between the highest point and the equilibrium point: φ’’ = __________
We see that the phase constant changes, depending on our choice of the starting time.
You might also have noticed that you can freely choose the amplitude, yet the period of
the oscillation is fixed. The reason is that the period is determined by the specs of the
physical system (mass and spring constant), whereas the amplitude and the phase
constant do not follow from any physics, they are arbitrary integration constants. There
are two integration constants, because the harmonic equation (Eq. 1) is a second order
differential equation.
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Otterbein University Department of Physics
Physics Laboratory 1500-12
PART C: MODIFYING THE OSCILLATOR.
Start the mass oscillating again, but this time only displace it about half the distance you
used earlier. Predict if the period becomes larger, smaller, or stays the same: __________
Measure the period of the oscillator: T’ = ________________
Try starting with a bigger displacement. Predict if the period becomes larger, smaller, or
stays the same: __________
Measure the period of the oscillator now: T’’ = _________________
Did the experiments turn out how you predicted? ___________
Double the mass attached to the spring. Predict if the period becomes larger, smaller, or
stays the same: __________
Measure T2m =______________________
Is this what you expected? ___________
Trade springs with one of the other groups. Using a mass of 700g, set the spring
oscillating.
Measure the period: Tother spring = _________________________
Using the period and the mass, find the spring constant for this spring.
kother spring = _____________________
What did the other group measure the spring constant to be? How closely does it match
your measurement? Which way to determine k is easier: Part A or Part C?
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Otterbein University Department of Physics
Physics Laboratory 1500-12
PART D: THE SIMPLE PENDULUM
You have a golf ball attached to a string of length L at your table. Draw a free-body
diagram of the forces acting on it when it is at its maximum displacement to the right of
the equilibrium. Label all of the components of the forces, and call the angle between the
string and the vertical direction θ.
As you see, the restoring force accelerating the ball towards equilibrium is F= -mg sinθ.
For small angles, sin θ is close to θ itself, so F ≈ -mg θ. On your diagram, observe that the
arc length is x ≈ L θ with θ in radians. Using θ = x/L we arrive at F = -(mg/L) x. This
situation is very similar to the spring where F = -kx is the restoring force.
From this, derive the period of oscillation for the simple pendulum at small angle and
state what it does and does not depend on.
Calculate the period, T, of your pendulum, by measuring the length of the string and
whatever else you need.
T = _________________________
Using the stopwatch, time ten oscillations and compare the experimental to the calculated
value.
Plot the position of the golf ball as a function of time for at least four periods. Start time
at the instance when the ball is at one of the maximal displacements. Below this plot,
graph the velocity of the golf ball as a function of time, and below that the acceleration as
a function of time. Make sure to draw axes and label them – with units. Align the vertical
axes so that the same points in time are exactly below each other in the three diagrams.
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Otterbein University Department of Physics
Physics Laboratory 1500-12
Position
Velocity
Acceleration
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Otterbein University Department of Physics
Physics Laboratory 1500-12
PART E: DAMPED HARMONIC MOTION
Copy the first plot from part D into the first picture below. Now, tape a piece of paper to
the golf ball. Set the pendulum in motion. Plot the position of this pendulum.
Position(undamped,from c)
Position(damped)
Which properties of the sinusoidal function are changing, and which stay the same?
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