Simple Harmonic Motion

Procedure

Attach one of the two springs to the metal rod. The tapered spring should be attached with the narrow end up. (Why?) Attach the mass hanger to the bottom of the spring and load it with some mass. You must use enough mass to achieve smooth oscillation, but you must NOT exceed the elastic limit of the spring! For the tapered spring, the limit is about 500 grams plus the hanger; for the straight spring, the limit is about 1000 grams plus the hanger.
The amplitude of oscillation only needs to be about 5 cm.
Record the attached mass (remember the 50 gram hanger) and the time for one complete oscillation (up and down). It is very difficult to time one oscillation -- how can we improve precision?
Remember to record error estimates with all of your measurements.
Use at least eight different masses spread over the allowable range (between smooth oscillation and the maximum values given above). If you choose masses too close together, your best fit line will be imprecise.
Change springs and repeat the experiment.

Write the equation relating mass (m), the spring constant (k), and the period (T) for an ideal massless Hooke's law spring loaded with a mass undergoing simple harmonic motion.
The world of the Physics laboratory is not ideal -- real springs have their own mass which oscillates with the load. In the equation you have written above, replace the mass (m) with the sum of the load mass (m_load) and the effective mass of the real spring (m_eff).
Which variables in this last equation are easy to measure in lab?
What combination of these variables would you plot to produce a graph that is a straight line?
What variable which is not easy to measure directly in lab can be derived from the slope of the best-fit straight line graph? Determine it. No error propagation is required. (Mathematica may be useful.)
What variable which is not easy to measure directly in lab can be derived from the y-intercept of the best-fit straight line graph? Determine it. No error propagation is required. (Mathematica may be useful.)
Why should the narrow end of the tapered spring be up?
Identify at least two sources of statistical error.
Identify at least two sources of systematic error.