A simple pendulum consists of a mass m hanging at the end of a string
of length L. The period of a pendulum or any oscillatory motion is the
time required for one complete cycle, that is, the time to go back and
forth once. If the amplitude of motion of the swinging pendulum is small,
then the pendulum behaves approximately as a simple harmonic oscillator,
and the period T of the pendulum is given approximately by
where g is the acceleration of gravity. This expression for T becomes exact in the limit of zero amplitude motion and is less and less accurate as the amplitude of the motion becomes larger. From this expression, we can use measurements of T and L to compute g.
Let us review simple harmonic motion to see where eq'n (1) comes from. Recall that simple harmonic motion occurs whenever there is a restoring force which is proportional the displacement from equilibrium. The simplest example of simple harmonic motion is that of a mass m on a spring which obeys Hooke's Law,
where x is the displacement from equilibrium and k is a constant called the spring constant. From Newton's Second Law, , we can write , or
This is a second-order linear differential equation -
"second-order" meaning that the equation involves a second derivative,
and "linear" meaning that it contains no non-linear x-terms such as x2
or sin(x). The solution to equation (3) is
(4) , where .
The constant A is the amplitude of the motion; the position x oscillates between -A and +A. The constant f is called the phase constant. A and f depend on the initial conditions, that is, the position and velocity at time t=0. The period T is related to w by
Now consider the simple pendulum — a mass m hanging on a string of length L. We can describe the displacement from equilibrium of the mass m either with the distance along a curved x-axis or with angle q of the string from the vertical. The distance x is related to the angle q by , where q is in radians.
There are two forces on the mass m: the force of gravity, mg, and the tension in the string, Fstring. The tension Fstring has no component along our curved x-axis, while gravity has a component along the x-axis equal to . The minus sign indicates that the direction (sign) of the force is toward the origin, always opposite to the direction (sign) of x and q .
If q is very small, then sin q » q , and we have, approximately,
and, using F = ma, we have
In this special case of small x or q , the restoring force is proportional to the displacement x, and we have a simple harmonic oscillator whose equation of motion is just like eq'n (3) except we have replaced the constant k/m with another constant g/L. The solution to equation (7) is exactly like the solution to eq’n(3) except we replace k/m with g/L. So, by looking at eq’n (4), we have, for a simple pendulum
Solving for T gives us eq’n (1). We emphasize that this expression is true in the limit of small amplitude motion (small q , small x). Note that, according to eq’n (1), the period is independent of both the mass m and the amplitude A.
If we do not assume that q is small so we cannot make the simplifying assumption , then the restoring force is
and the equation of motion becomes
In this case, we have a non-linear differential equation, which
is quite difficult to solve. In this more complicated case, the period
T still does not depend on m (since m does not appear in the equation of
motion, it cannot appear in the solution); however, the period T does depend
on the amplitude of motion. We will use the symbol q
o for the amplitude, or maximum value, of q
. The exact solution for the period T can be written as a infinite power
series in q o, and the first three
terms of the solution are
(The terms beyond the first three are quite small and can almost always be ignored.) In this expression, the angle q o must be in radians, not degrees. Note that this expression reduces to eq'n (1) in the limit of small angle q o.
In this lab, we will use a photogate timer to make very precise measurements of the period T of a simple pendulum as function of the length L and the amplitude q o. A photogate consists of an infra-red diode, called the source diode, which emits an invisible infra-red light beam. This beam is detected by another diode, the detector diode. When the mass m passes between the source and the detector, the infra-red light beam is interrupted, producing an electronic signal that is used to start or stop an electronic timer. In "period mode", the timer starts when the mass first passes through the gate, and does not stop until the mass passes through the gate a third time, as shown below. Thus the timer measures one complete period of the pendulum. Our timers are quite precise and read to 10-4s = 0.1 msec. Pressing the "reset" button sets the timer to zero and readies the timer for another measurement.
When making measurements, begin by positioning the photogate so
that the mass is directly between the diodes when it hangs straight down.
Then, after pulling the mass to the side, release it carefully so that
it swings through the photogate without a collision. Allow the mass to
swing back and forth a few times before making measurements to allow for
any wobble to settle out. After the mass is set swinging, several measurements
of T can be quickly made by simply pressing the reset button while the
mass is still swinging. You should not stop the mass and restart it after
each measurement, since this will introduce unnecessary wobble. Always
keep an eye on the pendulum while it is swinging so that it does not drift
into a collision with the delicate and expensive photogate.
In our pendula, the string is suspended over a support rod that turns freely on its axis. You will make measurements with 5 or more strings of varying length from about 40 cm to 120 cm. The length L of the pendulum is the distance from the center of the mass m to the pivot point, which the center of the top axle. To measure L, first measure the distance l from the top of the mass to the pivot point; then measure the height h of the mass. L = l + h/2. You should only measure l with the mass suspended, since the weight of the mass stretches the string somewhat.
Part 1. Dependence of T on L.
Near each pendulum apparatus, there should be 5 strings with approximate lengths of 40, 60, 80, 100, and 120 cm. If your apparatus is missing some of these lengths, just cut some more string from the ball provided. Measure the period T for each of the five lengths. Keep the amplitude of motion very small, so that eq’n (1) is approximately correct. By pressing the reset button several times while the mass is swinging, observe the period for several trials. By looking at the numbers for many trials, estimate the average value of the period and its uncertainty. (Do not record all the trials and do not compute the average - that is too time-consuming. We want you to observe and estimate.)
Now we shall see whether your data are in accord with eq’n (1). Squaring both sides of eq’n (1) yields
So, if this equation is correct, then a graph of T2 vs. L should be a straight line with slope , while a graph of T vs. L should not be a straight line. [Side comment: a graph of "A vs. B" means A on the y-axis and B on the x-axis: Y vs. X, always. ]
Using Mathcad and your data, make plots of T2 vs. L and T vs. L. On the plot of , also plot the straight line y = m?x, where the slope m = . Use the known value of g = 9.796 m/s2.
Part 2. Precision determination of g.
Choose the longest string and re-measure the length L as described above. Since the longest string is longer than 1 meter, use a 2-meter stick. If you are working with a partner, it's a good idea for both of you to measure L independently and then compare answers. Estimate and record the uncertainty in L.
Now re-measure the period T. Begin by positioning the photogate carefully and set the mass swinging with a very small amplitude (a few degrees, just enough to pass completely through the photogate each swing). Observe several trials of the period T and estimate the uncertainty in T by considering the spread in your values of T.
Compute g from eq’n (1) using your measured values of L and T. Also compute d g, the uncertainty in g, which is given by
We don’t expect you to understand where this expression comes
from just yet. Regard it as a gift from on high, for now. Compare your
answer with gknown = 9.796 m/s2 by giving the percent
discrepancy between your value and the known value.
NOTE: In Mathcad, you cannot define an expression like ; you can only define a variable like . So, your Mathcad document should have a definition like:
And remember: all the variables used in the definition (g, L, T, have to be defined previously in the Mathcad document.
Eq’n (13) is the best estimate of the uncertainty due to random measurement errors only. It does not take account of any possible systematic errors.
Part 3. Dependence of T on amplitude.
Notice that the Mathcad Primer shows, step by step, all the analysis for this section.
Using the string which is nearest to 1 meter in length, measure the
period T as a function of the maximum angle q
o. Measure the angle q o
using the protractor which is attached to the pendulum rig. Measure T for
several angles, from about 5o up to about 70o or
so, at intervals of about 5o. (Avoid angles larger than 70o,
simply for safety’s sake.) Be extremely careful to avoid collisions
with the photogate. You need not take multiple trials at each angle; one
good measurement at each angle is enough. Convert your angles from degrees
to radians and then make a graph of T vs. q
o (in radians) . On the same graph, plot the eq’n (11)
using your measured value of L and the known value of g. Theory and experiment should agree pretty closely. Comment on any discrepancy.
1. What condition or conditions are required for simple harmonic motion to occur?
2. Demonstrate that the solution of eq’n (3) is given by eq’n (4). Do this simply by inserting eq’n (4) into eq’n(3) and checking that an equality is obtained. As an example, consider the differential equation , where C is a constant. The solution of this equation is , where A and B are constants. To check that this is indeed the solution, first do the derivatives that appear in the equation: and . Then plug into the equation to check for an equality:
3. (a) What is the definition of the slope of a line? (b) Sketch a graph of T2 vs. L for a simple pendulum with a very small amplitude of motion [no numbers on the graph!, just a qualitative sketch] (c) what is the slope of the line in part (b) ?
4. Indicate how you will do the calculations in Mathcad for Part 1, described on the bottom of page 1.5. Just indicate the procedure with pen and paper; don’t actually prepare a Mathcad document.
For instance, suppose you were doing a lab in which the goal is to measure g by timing the fall of an object. The lab instructions tell you to measure the time t for an object to fall a distance d, for 5 different distances. According to theory, d and t are related by . You are told to use your data to plot d vs. t2 and on the same graph plot the line y = mx, where m = g/2. There are several ways you could do this. One way is the following...
5. What is the definition of angular measure in radians? Using the definition, show that there are 2p radians in 360o. [Hint: the statement "p radians = 180o" is a conversion factor, not a definition. Look up the definition in your physics book.]
6. What is the fractional increase in the period T of a simple pendulum when the maximum angle q o is increased from an extremely small angle (much less than 1o) to (a) 3o, (b) 60o ?
[Hint: think about whether to use degrees or radians in your calculations.]
[Note: If T increases from 1.213 sec to 1.229 sec, the fractional increase is
. The percentage
increase is 1.3%.]
7. Sketch a graph of T vs. amplitude q
o for a simple pendulum. (No numbers on this graph! Just make
a clear, qualitative sketch showing what the graph looks like.)
8. In an experiment with a simple pendulum,
the following measurements are recorded: .
(a) Based on these measurements, what is the value of g and what is the value of d g, the uncertainty in g?
(b) What would be the value of d g if we ignored the small uncertainty in T? [Remember to show all your work and to give final answers with the appropriate number of significant figures.] [Hint: in your calculations, should L be in cm or m?]