**Lab M5: Hooke’s
Law and the Simple Harmonic Oscillator**

Most springs obey Hooke’s Law, which states that the force exerted by the spring is proportional to the extension or compression of the spring from its equilibrium length.

(1) .

k is called the spring constant and is a measure of the stiffness of the spring. By inspection of Eq. 1, we see that in the mks system the units of k must be N/m. The minus sign indicates that the direction of the force is opposite to the direction of the displacement. The diagram shows a mass to the right of its equilibrium position (x=0). The displacement of the mass is (+) and the spring force is to the left (-). If the spring is compressed, then x is (-) and the spring force is (+).

Simple harmonic motion occurs whenever there is a restoring force which is
proportional to the displacement from equilibrium, as is the case here.
Assuming no frictional forces and *assuming that the spring is massless*, the equation of motion (ma = F) of a mass on
a spring is

(2) or .

The solution of this second-order differential equation is

(3) ,

where A and f are constants which depend on the initial conditions, the initial position and velocity of the mass. The period T, the frequency f, and the constant w are related by

(4) .

Thus the period T is given by

(5) .

A very important property of simple harmonic motion is that the period T
does __not__ depend on the amplitude of the motion, A.

If the mass is hung from
a vertical spring, it will still execute simple harmonic motion with the same period
(5), as we will now show. When the mass is hung from the spring, the spring is
stretched from its equilibrium length by the gravitational force on the mass,
mg, which we call the weight. The equilibrium displacement of the mass under
the influence of the force of gravity (down) and the force from the spring (up)
is x_{equil} = mg /k. Now that there are two
forces acting on the mass, the equation of motion becomes

(6) .

We now perform a change of variable by introducing a new coordinate y = x - mg /k. Since mg /k is a constant, differentiating y twice produces . Also, mg - kx = -ky, therefore eq’n (6) becomes

(7) ,

which is exactly the same as eq’n (2) except we have changed the name of the position coordinate from x to y. Since the equation of motion is the same, the solution is the same (3), with the same period (5).

A real spring has mass, a fact which we have ignored so far. A mass m on a
real spring with mass m_{spring} oscillates
more slowly than predicted by (5), since the spring has to push its own mass
about as well as the mass m. However, the theoretical expression (5) can be
corrected by replacing the mass m with an *effective mass*, consisting of
m plus some fraction of m_{spring}.

(8) ,

where f is some fraction (f < 1), which depends on the exact shape of the spring. For the spring used in this lab, the fraction f has been determined experimentally to be

(9) f = 0.281 ± 0.002.

[Although f can be computed, the computation is rather complicated and depends on the precise shape of the spring.]

In this experiment, you will first determine the spring constant k of a
spring, by hanging various weights from the spring and measuring the extension.
Then, you will measure the period of oscillation of a mass m hanging from the
spring, and will compare this measured period with period predicted by eq’n (5) [with m_{eff}
used instead of m]. Finally, you will repeat your measurement of the period
with two other masses and check that the period is proportional to .

**Procedure**

**Part 1: Measurement of the spring constant**

Begin by weighing the spring (you will need the weight in part 2). You might want to check the reliability of the digital balances by weighing the spring on two different balances. Use the digital balances with 0.1g resolution. The spring used in this lab has a tapered coil which serves to reduce the interference of the windings with each other and make the spring behave more like a perfect Hooke’s spring. Always hang the spring with the larger windings downward.

With the empty mass tray hanging from the spring, measure the position of an edge of the tray. This will be the zero position, which you will subtract from all subsequent positions. There is a mirror by the meter stick scale so you can avoid parallax when you measure positions. Now add the slotted weights to the tray, one at a time, and measure the positions of the same edge as each mass is added. Add masses in 50 gram increments to a total of 500 grams: D m = 50 g, 100 g, 150 g, …. Use the balance to check that the masses are accurate.

Plot the weight added (Dm*g) vs. Dx. ,
the position change from the zero position. Use the known value of g = 9.796
m/s^{2}. This graph should be a straight line with slope k.

To determine the slope, compute for each data point, and compute the mean, the standard deviation, and the standard deviation of the mean of your several k values.

[Side comment: You might think to determine the best value of k by measuring
the slope of (Dm*g) vs. Dx. using the "slope" function in Mathcad. This is not quite right since what we want is the
best fit line* which goes through the origin ,* while the slope function finds
the slope of the best fit line, which, in general, has a non-zero intercept.]

**Part 2: Measurement of the period**

Remove the weight tray
from the spring and load the spring with a 100 gram mass with a hook.
Carefully, set the mass oscillating with a small amplitude motion and use a
stop watch to time the interval for several complete oscillations. If the time
for N complete periods (N » 50 or more)
is T_{total} , then the period is . By measuring the time for
many periods, the uncertainty in T, due to your reaction time, is reduced by a
factor of N.

.

[DO __NOT__ measure one period 50 times; measure
the time for 50 periods, __once__.]

If you have time, repeat this measurement at least once, to check the reproducibility of this method.

Now compute the period using eq’n (5) with m_{eff} in place of m

(10) .

Also compute , the uncertainty in T_{calc}_{
}(see note on uncertainty calcs, below)_{. }Compare
the calculated and measured periods.

Repeat this procedure with m = 200g and m = 500g (but you need not repeat the calculation of - it’s too time-consuming.). Make a plot of , using your three data points. On the same graph, plot the line y = x. Finally, compute the quantity for each of your three data points. Do theory and experiment agree?

[Review: For a function like , involving multiplication, division, and powers, the fractional uncertainty is given by .]

**Questions:**

**1.** What are the units of the
spring constant k in the MKS system of units (MKS = __m__eter-__k__ilogram-__s__econds)?
What are the units in the cgs system
(centimeter-gram-second)? Hint: It is customary to write spring constants as
force/distance rather than mass ´ distance / seconds^{2}. The unit of force in
the cgs system is the dyne, which itself has units of
grams ´
cm / seconds^{2}.

**2**. A given spring stretches 20
cm when it is loaded with a 200 gram weight. What is the spring constant of the
spring?

**3.** Sketch the graph (Dm*g) vs. Dx. Indicate
the slope and intercept.

**4.** What happens to the period
of a mass-on-a-spring simple harmonic oscillator if the mass is tripled? What
happens to the period if the spring constant is reduced by a factor of 2? How
does the period depend on the amplitude of the oscillation?

**5. **True or False: for a simple
harmonic oscillator consisting of a mass on a spring, the period measured when
the mass is hanging vertically is different than the period measured when the
spring and mass are supported horizontally .

**6.** Show with a sketch what a
graph of T^{2} vs. looks like. What are the slope and intercept?

**7.** (Counts double). Assuming
that k, m_{eff}, d k, and d m_{eff}_{ }are known, show how to compute dT_{calc},where T_{calc} is
given in eq'n(10).

**8.** What condition or conditions are necessary for simple harmonic
motion to occur?

**9. **A mass m hanging from a spring with spring constant k is taken to
the Mars, where g is 1/3 of its value on Earth, and the period of the
oscillation is measured. Is the measured period on the
Mars the same as the period measured on the Earth? Explain your answer.