When Approximations Fail

Prelab:

1. Using the following equations,

and the answer is 3.50 +/- 0.01 rad/sec.

Note:

a. Significant digits: 3 (since the length has 3 significant digits);

b. Uncertainty: Don't forget the absolute value sign.

c. Format: please use "±" sign while reporting a physical quantity.

(or 3.501±0.001 rad s^-1).

Compare and ∆ω with σ_ω (uncertainty of ω), check whether the observed value is within the 68% range of the theoretical value.

Lab:

Procedure #1:

1. What does it mean experimentally for the small angle approximation to be valid?

-ANS: The small angle approximation will be valid until the observed angular requency is statistically distinguishable (in the predicted range) from predicted angular frequency.

2. What will be observed if the approximation is invalid?

ANS: The observed angular frequency is statistically distinguishable from the predicted value (outside the 68% confidence interval).

Procedure #2:

Based on the model, what is the predicted angular frequency (or quivalently the period) for your pendulum?

-ANS:

Calculate your angular frequency using omega=sqrt(g/l), where l is the length of your pendulum, which is comprised of three parts -- the radius of the disk, the length of the string, and the radius of the ball.

*Since there will be 3 significant digits in your length, the angular frequency should also have 3 significant digits.* (Although the length measured by a caliper has more decimal digits than a ruler, when you add them together, the number of decimal digits would be dominated by that of the ruler, then there will only be 3 significant digits.)

Caution:

The number of significant digits for the uncertainty should be the same as that of the center value (the predicted angular frequency).

Data & Plot:

1. You should report the length (by adding the three components measured using ruler and caliper) of your pendulum with its uncertainty in the report.

2. Decimal digits should be the same (between the value you measured and its uncertainty).

3. The data and plot should contain the errors. If the error is too small to be discernible in the plot, you should write it down in your figure legend.

4. Do not fit the data -- there's no linear relationship between the amplitude and the frequency.

5. The unit for the amplitude is Radian, not Meter.

Procedure #4:

When approximation fails? You should,

1. Roughly determine that from your figure, or from your experiment.

2. If your measured data is not consistent with the predicted value even for small angles (actually it happened in one group during the lab, and it turned out to be caused by a wrong measurement of the length of the pendulum), try to explain it: it might probably because of the incorrect measurement of the length of the pendulum, which should be composed of three parts: radius of the disk, length of the string, radius of the ball.

Important uncertainty sources:

1. The length of the pendulum is not measured correctly.

2. The motion of the pendulum is not in a plane, but in a roughly circular orbit.

3. More cycles can yield a better fitting result in the angular frequency, but the amplitude will decrease with time, thus influence the fitting result.