REASONABLENESS OF RESULTS

A. SIMPLE PENDULUM EXPERIMENT

A simple pendulum is used to determine the acceleration due to gravity (small g) using the formula:

g = 4 PI^2* L/T^2

where L is the length of the pendulum and T is the period of the pendulum. Since there are only two variables (L and T) measured in this experiment, all of the error in the determination of g comes comes from the error in the measurements of L and T. The errors in the measurements of L and T depend on the equipment used. Generally, a meterstick is used for measuring L; thus it is reasonable to assume that the error in the measurement of L should not be greater than 2 mm ( 1mm on each end of the penulum). Similarly, the error in the measurement of T will depend on the stopwatch or other timing device used. Typically, a stopwatch accurate to 0.1 sec is used in the experiment, and to minimize the error in the measurement of T, time for 25 or so oscillations is measured and the average period calculated. Thus, error in the measurement of T will be about 0.02 sec. [Note: reaction time of the person measuring time should be included for accurate results.] In order to set reasonable error limits for the measured value of g, we proceed as follows:

g = 4 Pi^2 * L/ T^2 ................(1)

Take natural log on both sides of Eq. (1).

ln g = ln (4Pi^2) + ln L - 2 ln T ........(2)

Differentiating Eq.(2)

dg/g = 0 + dL/L - 2 dT/T ................(3)

In Eq.(3), dg, dL, and dT may be cosidered as errors in the measurement of g, L, and T respectively. It should be noted that measured values of L and T may be higher or lower than the actual values of L and T, i.e. dL and dT may be positive or negative. In order to get the maximum permissible error range in the measurement of g, the sign of dT in Eq. (3) is changed. Thus, both terms on the right side of Eq.(3) are numerically added. The result is:

dg/g = dL/L + 2 dT/T ..................(4)

The fractional errors on right side of Eq.(4) are multiplied by 100 to give the percent error range in the determination of g. This is the error range within which the result of your measurement should fall. For a 1 meter long pendulum with a 2.54 cm diameter ball, the percent error should be:

dg/g = (.01/100 +.01/2.54+2*.25/1.0)*100 .........(5)

NOTE: The numbers used are best estimates. Eq. (5) results in 1.8% error.

Next, calculate the percent error in your experimental value of g (by finding the percent difference between your value of g and the standard value of g for your location to be supplied by your instructor). If your value of g is within 1.8 % of the standard then your experiment has been successful. If your value of g is not within 1.8% of the standard, then in addition to normal (regular) measurement errors, you have some other sources of error. More on this later.

B. THE YOUNG'S MODULUS EXPERIMENT

In this experiment , the Young's Modulus (Y) for a metallic wire is determined by the formula:

Y=M*g*L/Pi*r^2*l......................(1)

Where M is the mass used to stretch the wire,

NOTE: Please see a physics text for a discussion of Young's Modulus experiment.

Take natural log of both sides of Eq. (1):

ln Y = ln M +ln g +ln L -ln Pi -2* ln r- ln l..............(2)

Differentiating Eq. (2):

dY/Y = dM/M+dg/g+dL/L+2dr/r+dl/l.................(3)

As before, we have made all terms positive on the right side of Eq. (3) and we look at dY, dM, dL, dr, and dl as errors in the measurements of Y, M, r, and l respectively. By assigning reasonable values (depending on the equipment used to measure ) to dM,Dl, dr, and dl and by using the measured values of M, L, r, and l, we ca n determine the permissible error limits for the measurement of Y.