Physics 120 Lab 5: Period of a Conical Pendulum
Union College Winter 2015
Physics 120 Lab 5: Period of a Conical Pendulum
The textbook has a short discussion of the conical pendulum, which is a mass hanging on a string and moving in a circle. The string, which is attached to a fixed point at the top, sweeps out a cone.
Part I: Empirical Determination of Period Equation
Use the equipment in the room and your best experimental skills to determine, empirically, the period of this motion. (If there is other equipment that you think could improve your experiment which is not provided, ask your instructor. There may something helpful that can be obtained in short order.)
Your goal is to determine what parameters the period depends on, and the functional relationship. For example, the period of oscillation of a mass on a spring depends on the mass to the ½ power, i.e.
,
and the spring constant to the -1/2 power, i.e.
,
where C just represents a constant, and is independent of the amplitude of the oscillation. When we put these together, and then fit the data, we could discover that the period equation
.
(Although, as you learned in lab 4, there is also a dependence on the mass of the spring, which affects this equation, and so the data would not, actually, fit this equation perfectly.)
Some helpful techniques:
1. You need to ensure that the mass moves in a circle, and not an ellipse. This is not possible to accomplish with 100% precision. Try to devise some clever techniques to make it go in a circle, and to examine it as it moves to ensure that it is, indeed, traveling in a circle.
2. Remember to isolate each parameter as best you can. That is, vary one parameter at a time, keeping all other parameters constant. And use only that data set to determine whether this parameter affects the period. Then, choose another parameter, and isolate it, getting a different data set. You may find, however, that some parameters are affected by others. You need, then, to determine which parameters are the independent variables. You may find a little experimentation will help you recognize how one parameter affects another.
3. If you have determined that the period depends on some parameter, let’s call it a, but it isn’t obvious what the functional form is, plot the log of the period on the y-axis and the log of a on the x-axis. This should make a straight-line and the slope of that line gives you the information you need. Consider a general relationship between T and a given by
,
where p is the power of a. The goal, then, is to determine the value of p. Now, we take the logarithm of both sides. This gives
.
By the rules of logarithms, we can rearrange this to be
.
So, if we plot log(T) on the y-axis and log(a) on the x-axis, we get a straight-line, i.e. y=mx+b, where the slope, m is equal to p, the power of a in the equation for T.
Part II: Theoretical Derivation
Use the principles discussed in Chapter 5 to model the motion of the conical pendulum and derive an expression for its period. Start with the fact that the mass is moving in a circle at constant speed. Does your derived equation fit your experimental results?
Part III: Making Use of the Conical Pendulum.
According to the text, Isaac Newton used the conical pendulum to get an accurate measure of the acceleration of gravity. Now that you have developed into an expert conical pendulum experimentalist, let’s test out this claim. Run a series of trials and solve for g using the equation given in the text (which should be derivable from your equation for T in Part II). In each trial, determine (either by estimating or calculating from a number of measurements) your uncertainties in the measured values. Be sure to record these uncertainties in your data table.
Conduct at least 5 trials, calculating g for each run. Calculate the average g and standard error.
Record your final best value of g, with its uncertainty. Assess and comment on how precise this measurement turned out to be. And, on how accurate it was.