PHY225aHW

Homework assignment for PHY225a lab. Due date Sep 11,2007

100 points total

1. In class we learned that if z is a function of two variables x and y,

and we have measured both x and y and determined their mean values and standard deviations, then we can find the standard deviation in z using the formula below. Note that the derivatives with respect to x and y are evaluated using the mean values of x and y.

Here is a problem for you to try. In lab #7 you will measure the charge over mass ratio (e/m) of the electron. You will do so by shooting electrons from an electron gun through a uniform magnetic field, which is perpendicular to their velocity. The electrons will move in a circle. You will measure the magnetic field B, the accelerating voltage of the electron gun V (this will determine the velocity of the electrons) and the radius of the electron orbit r. You will find e/m from the formula:

a) ( 20 points) Write an expression for the standard deviation in the quantity e/m in terms of the measured quantities and their standard deviations.

^{Now, divide both sides by f to
get a relation
between the fractional errors.}

^b)^{(
20 points)
If you can measure each V, B and r with an accuracy of 1%, what would
be the
relative (%) error in your measurement of e/m ?}

2. You have a pendulum and have done several measurements to determine the length of the string and the period of the pendulum. You tabulated your data ( Table1) and your goal is to determine the acceleration due to gravity. The accepted value of g in Nashville is g = 9.79822 m/s^2 .

	Period [s]	Length[cm]
	2.026	102.8
	2.021	102.7
	2.024	102.7
	2.020	102.6
	2.025	102.8
	2.024	102.6
	2.022	102.9
	2.022	102.8
	2.024	102.5
	2.019	102.8
	2.023	102.7
	2.024	103.0
average	2.0228	102.74
STDEV	0.0021	0.14

Table 1. Measurements of the period and length of the pendulum.

(a) (15 points) Find the average values and the standard deviations for the period and the length .
I have used EXEL to give me the average and standard deviation. For now I keep more significant digits that the precision of the measurement. I'll report my final result for g with just as many significant digits as it makes sense ( after I determine the error).
(b) (30 points) Using the results from (a) determine the acceleration due to gravity. Make sure to report your result together with its statistical error.

I determined the error using the error propagation formula. After taking the derivatives, I found that:

Then, I plugged in the numbers above.

(c) (15 points) Is your result in agreement with the accepted value of g ?
Explain how you got to your conclusion. Are there any other sources of error (besides the statistical error) that you should take into consideration ?

This tells me that the probability that my result is consistent with the accepted value is much smaller than 5 %. Remember that +/- 2 standard deviations corresponds to a confidence interval of 95.4 %, i.e. - the probability to get outside this interval just by chance is less than 5% . I also notice that the accepted value is quoted with many significant digits ( this means that the error is just in the last digit), while my results is not so precise. Since I observe a large deviation from the accepted value, I must have made a systematic error that I have not taken into account yet. This can be due to a miscalibrated instrument or a procedure that gives offset to one or both measured quantities.