Pendulum Experiment: The Graphical and Analytical Methods?

Question

I was wondering if you'd be able to help me with this. As of now, I only have a very, very basic understanding of physics. I took it in middle school and now I am being forced to take it again in high school.

This question relates to an assignment I was given in class that I do not understand. I do not expect you to solve this problem for me, but I would like to know if the formulas that I am using are correct, and if I am indeed going in the right direction, because, quite frankly, I can't understand half of the things my teacher says.

4π^2(L/T^2)= g

This is the equation we are dealing with. We're supposed to find what g is. L and T are variables we found by performing an experiment where we swung a pendulum. There are four separate numbers representing L (four lengths of the pendulum) and four numbers representing T that go along with each of the four Ls.

He told us to make two scatter plots, one with L along the y axis and T along the x axis, and one with L along the y axis and T^2 along the x axis. He then told us to draw lines to make them into graphs. He told me that the graph that went (T^2, L) should be a linear function and that the graph with the points (T, L) should be some kind of curved line. I don't know if this is true though, because he didn't seem to be paying attention when I asked him this...

Any way, to solve it via the analytical method, I plugged in an L and a corresponding T. I got an answer for g that was 9.04m/s^2. That doesn't sound right, but it doesn't sound horribly wrong, either.

To solve it via the graphical method, he told us we needed to find the slope of the line of the linear equation with the points (T^2, L). But thing is, I'm not sure where the slope comes into play when solving this problem. Does it replace the T^2 of 4π^2(L/T^2)= g or does it replace the whole L/T^2? Where does this variable go? I'm not sure exactly...I tried putting it in in the place of L/T^2, leaving me with 4π^2(slope)= g. The answer I got for putting it in was 11.05m/s^2. This is very different from the 9.04m/s^2 I got earlier, but it is the closest number that I was able to find...I adjusted the slope of me line multiple times using multiple methods but this was generally the result.

When doing this experiment, is it reasonable to find this kind of the difference in answers between the analytical and graphical methods? What do you think I am doing wrong?

The points used to find the line of best fit for L v. T^2

(0,0)
(0.4, 1.322^2)
(0.5, 1.453^2)
(0.6, 1.567^2)
(0.7, 1.635^2)

He did not tell us to use any particular angle when swinging the pendulum.




(i) The Analytical Method

g = 4 π2 (L / T2)
g = 4 π2 (.4 / 1.3222) Plug in a T from the T graph.
g = 4 π2 (.4 / 1.3222)
g = 9.04m/s2

(ii) The Graphical Method

Find the slope M.


Points on the line of T^2 - (1.3222, .40), (0, 0)

(This is an edit. I replaced the second point that I used initially (1.4532, .50) so that I would be sure that the line ran through zero...)

m = (y2 - y1) / (x2 - x1)
m = (0 -.40) / (0 -1.322^2)
m = -.40 / -1.322^2
m = .30

Plug M into the equation.

g = 4π^2/m
g = 4π^2 (.30)
g = 11.84m/s^2

Answer 1

OK, you start with the basic equation 4π^2(L/T^2) = g.
The L vs. T^2 graph should be linear because its slope L/T^2 = g/(4π^2) = constant. When extended it should pass through (0,0). This slope should replace L/T^2 in the basic equation, which you did. I think it's reasonable to get this kind of error, considering possible measurement errors in L and T. Note that since g ~ 1/T^2, a +5% error in T ≈ a -10% error in g. Another error source is the fact that the basic equation is not exact, due to the pendulum's acceleration being proportional to the sine of the angle, not to the angle itself which the equation assumes. (See the ref. if you want more details on this.) The larger the amplitude of the pendulum swing, the larger the actual value of T is relative to the value predicted by the equation. The error (Tact-Tpred) / Tpred is less than 2% up to 32 deg, but reaches 5% at 50 deg and 18% at 90 deg. So a measurement-based g value less than 9.8 could be reasonable. Can you remember roughly what the swing angle was for these measurements? As for the 11 m/s^2 value from slope, I'd be interested if you were to add the data points to your question. Does the extended graph pass through (0,0) as it should?
EDIT (having looked at yr added data):
First, the slope method as you applied it is just a repetition of the analytical method for point #1, since one of the two points is the origin. Second, tut-tut, you made an arithmetic error; 0.40/1.322^2 = 0.229 not 0.30. So you actually end up with the same g as analytical. For the heck of it, I computed the g values for each point and averaged them:
#1 = 9.04
#2 = 9.35
#3 = 9.65
#4 = 10.34
Ave. = 9.59
About 2% error for the average, not too shabby. But the spread is rather large, about 15%, and I think we'd have to blame that on a combination of measurement error and the amplitude effect.