In this section, we’ll make a pendulum and discuss how to take data with it.
Building the apparatus:
To make a pendulum, all you need is a string and a weight – on a pendulum, the weight is called a plum-bob. Some ideas for your plum-bob: a hex nut, roll of tape, chess piece, pretty much anything with some weight to it, and with an easy spot for connecting it to a string. Tie the string to the weight, and you’ve made a pendulum. Now we need a way to hang our pendulum so that we can time its oscillations. You could use a doorknob, coat hanger, command hook, or the edge of a desk. If you don’t have somewhere like that, you can make a stand by using some heavy support like a spray bottle full of liquid or a stack of books, and then taping an “arm” – something like a pencil or ruler – to the support such that you could hang your pendulum from the arm. No matter how you choose to hang the pendulum, you must:
- Set you hanger up such that you can change the length of your pendulum.
- Be sure that the the pendulum swings freely, such that the string and plum-bob do not rub against any walls or your stand.
So, for this lab, the plan is to time the period of the pendulum for multiple lengths of the pendulum. We know they obey the formula: T=2π(L/g)p , where T is the period – the time it takes for the pendulum to do one full oscillation, L is the length of the pendulum, g is the local gravitational constant, and p is some number. This formula is true when the pendulum swings at small angles, but more on that in a moment. We’ll be solving for p, after using logarithms to simplify our formula into a linear one. So for the data taking, we will be measuring the period of the pendulum multiple times for a given length such that we can find the average and uncertainty on the period for that length. We find that it is hard to time just one oscillation – one period, and it is easier to measure the period by observing some number of oscillations, then dividing the time by the number of oscillations. This is the same method as stacking 10 nickels, measuring the height of the stack, then dividing the measurement by 10 to return the height of one nickel. So, for each length of the pendulum, you will preform a number of trials, where in each you will time some number of oscillations, divide by that number of oscillations, and record the period. Thus you will have (L,T) data points with uncertainties on L and T (how will you find uncertainty on L?) with which you will create (x,y) data points with uncertainties that obey a linear formula, and then we can fit a line.
About small angles, the formula we are using only works when θ ≤ 15°. Here, θ is the angle between the string when we let it go and the string when it is at rest (vertical). Additionally, every trial you must pull the pendulum back to the same angle. We are seeing how the period changes with a change in L, so we want every other factor to remain constant throughout our data taking procedure. It is helpful to mark the angle somehow to be sure the angle stays constant. To measure the angle to be sure it is under 15°, use a protractor. If you don’t have a protractor, just print a picture of one, as long as you don’t warp it – change the width of the picture disproportionally to the height – you’ll have a perfectly functional protractor. Of course you could just as well use trigonometry to draw and cut out a triangle with an angle of 15° and use that.
So in summary, the steps are as follows:
- Make a pendulum.
- Make a pendulum stand such that you can vary the length of the pendulum, and the pendulum swings freely.
- With the pendulum at some length L1, take a number of trials with a consistent θ ≤ 15°, where you time the pendulum for each trial to find the period.
- Measure L1 and its uncertainty (How?).
- Repeat steps 3 and 4 for a new length, but with the same angle you used for L1.
- Take the averages and standard deviations of your L and T data to get (L,T) data points with error bars.
- Figure out the linearization & solve for p – the rest of the lab.