PH 1140 Lab 3: Mass-Spring Oscillator III

Overview

In this third experiment with the mass-spring system, you will be repeating a few now-familiar steps, but mostly you will be learning something new about handling uncertainties, about curve fitting, and about the relative phases among the four measured quantities; Fx(t), x(t), vx(t), and ax(t). Again, there are two parts to this experiment: Part 1 involving mass and spring constant measurements, the theoretical prediction of the angular frequency of a mass-spring oscillator, complete with uncertainty; and Part 2 involving the actual measurements of Fx(t), x(t), vx(t), and ax(t) during 10 seconds worth of oscillations so that you can determine experimentally the angular frequency of oscillation, and other system parameters such as relative phases and the relationships among amplitudes.

So far this must sound too familiar—even redundant. Today we will consider a topic ignored in our work so far: Is our assumption that the spring is massless correct? Can we make a correction to perhaps improve the comparison of the values for the angular frequency determined from our force constant measurements and those found by a curve fit to sine wave created when the spring oscillates? Does this effort provide a better comparison between Theory and Experiment?

Please make use of the Data Sheet and the Report found below to help you clarify how to perform this experiment.

Procedure: Part 1

Today you will continue working with mass-spring system. This time all of your work will be done with a single spring so that you can more easily focus on other aspects of the behavior of the mass-spring oscillator.

Your station should be prepared with all appropriate Vernier sensors and a container of masses. Please be certain to have two springs one rather skinny and the other relatively fatter.

First measure the mass of the spring, the mass hanger, and the slotted masses on one of the mass balances available in your lab room. After you measure the mass of the spring (about 6g, fat spring, ~4g, skinny one), measure the mass of the hanger-plus-50-g-slotted-mass combination, and then add additional masses in 100-g increments, measuring the total mass each time, until you reach a total of 600g. The total mass at each measurement should be within half of an integer number of 100g units—if it is not, ask your lab instructor for a properly-sized replacement mass. Record the spring mass on your data sheet for future reference. (Later you can use half a gram as the slotted mass uncertainty, unless you measured deviations from stated values consistently and substantially less than half a gram in magnitude.) Notice that we are attempting to be more precise today with our measurments.

Now complete the necessary measurements to determine the force constant, k, as you have done before.

Procedure: Part 2

Now that you have measured masses and determined the spring constant in order to predict the angular frequency of subsequent oscillation (complete with uncertainty), the next step is to put a mass/spring system into oscillation so that you can compare the measured angular frequency of oscillation with the theoretically-predicted value, as well as to make measurements of the relative phases.

Open the Logger Pro File for Part 2. Reduce the mass hanging from the spring to 300g (total mass 303 ± 3g, counting the spring mass). With the mass hanger motionless, zero the force sensors, and then set the system into oscillation by pulling the mass hanger down about 1 to 1.5 cm from equilibrium and releasing it. When you are satisfied, begin collecting data (which automatically ends after 10 seconds), and, if necessary, autoscale the data.

All four graphs are sinusoidal in nature, so we're going to employ a different fitting routine than the usual linear fit in order to quickly extract system parameters from these graphs. This Logger Pro file is automatically set up to fit each curve with a sinusoid, and then provide a data box superimposed on each graphical region listing the respective amplitudes (A), angular frequency (B), and the phase (C), (and the vertical zero-offset (D) which you will never use in this experiment).

First, look at the parameters shown in each data box labeled B. B is the angular frequency of oscillator. All 4 boxes should have essentially the same ω value, all with incredibly tiny uncertainties. Record that best-fit ω value on your Data Sheet for later comparison with theory.

Now click on the Data tool bar button, and "Clear All Data". Change the total mass hanging from the spring to 400g, damp out its motion at equilibrium, zero the Sensors, put the system in oscillation with a 1 to 1.5 cm amplitude as before and collect 10 seconds of Fx(t), x(t), vx(t), and ax(t) data. As before, the data display on the Logger Pro template will include the sinusoidal fitting parameters "A", "B", and "C". Record all three values on your Data Sheet for subsequent comparisons with theory.

Questions to Consider When Writing Your Report

  1. Write down the spring constant for the single-spring in industry-standard form (that is, properly rounded off and with units).
  2. In the previous experiment, it was mentioned that for most purposes the angular frequency is a more useful parameter of the oscillating system than the period of oscillation itself. For the ideal mass-spring system, the angular frequency is derived to be ω = √(k/m). Calculate the theoretically-predicted ω for this single-spring system for a couple of intermediate mass values as follows. Because the spring has about 6g of mass (in contrast to the idealized mass-spring system where the spring is MASSLESS!), we suggest that you include the possible effect of spring mass by adding half of the spring mass (3g!) to the mass of the hanger-plus-slotted-masses, and then taking the uncertainty of the mass to be ± 3g. That should span ALL possibilities from the spring mass having no effect to having a 100% effect. Both extremes seem unlikely, but no matter what the real situation is, the proper value to use must be somewhere in between those two extremes. So now as Problem 2, determine the two values of ω from the above equation for the given mass values of 303 and 403g, and the k determined by measurement. This was the fat Spring. Do similar measurements for the skinny spring. Please also do this for the skinny spring as well, once again using half of the measured mass of the spring in addition to the mass on the hangar.
  3. We determine the UNCERTAINTY of any parameter such as ω by expressing its differential, Δω, in terms of the uncertainties (differentials) of k and m. The general form for the uncertainty of a function f(x,y) in terms of the uncertainties of x and y is

    General formula for error propagation

    In many cases, such as this one, we get compact, easy-to-remember expressions by dividing Δω by ω itself (obtaining thereby something called the FRACTIONAL UNCERTAINTY). From the relationship ω = √(k/m), we obtain

    Fractional uncertainty of omega

    Note that absolute value signs are used to make sure that deviations from a central values of k and m do not inadvertently diminish one another. Also, please be aware that this gives us a worst-case uncertainty, because the individual uncertainties are simply added together in absolute value. Determine the uncertainty Δω for each ω calculated in Problem 2, and then record ω ± Δω in industry-standard form (after keeping extra digits throughout the intermediate calculations in order to avoid round-off error).
  4. Compare the "B" quantities with the predicted ω for m = 303g from Problem 2, and report the comparison here in terms of fractional deviation.
  5. Now compare amplitude relationships. In theory, ωAx = Av, where ω is given by the "B" parameter (the subscript on the A's indicates which amplitude, or "A" value, to use). Calculate the fractional deviation from theoretical prediction as |Av – ωAx|/Av. Do the same thing for ω2Ax = Aa, and AF = mAa. Generally, the fractional deviations will be under 2% and in one case may be as large as 5% (do you know why that is?).
  6. Now examine the relatives phases (the "C" parameters, which are reported in units of RADIANS) between the three pairs of parameters: position and velocity, velocity and acceleration, and position and acceleration. First, determine the absolute difference |Cx – Cv| and convert that radian measure to an angle measured in a fraction of π. Then do the same for |Cv – Ca| and |Cx – Ca|, recording results as the answer to this Problem. Also as part of the answer to this problem, find the pair of graphs (now including Fx(t) as well) that are basically IN phase with one another. (Relative phase is probably the single most difficult concept for students in PH 1140 to comprehend. We hope that this visual display of 4 graphs associated with the oscillations of a mass-spring system, combined with the above calculations, help in demystifying the concept of relative phase.)

Files and Logger Pro Templates

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