SBU Intro Physics Labs, PHY 121 Simple Harmonic Motion Lab

Here, the spring constant $k$ comes from the usual definition by Hooke's law, in terms of the force of the spring at some displacement from the equilibrium position $x$¹²:

$$F=-k(x-x_0)$$

If we let $M$ denote the mass of the object oscillating, this accurately describes an ideal situation in which springs are massless. There is actually a correction² when we consider the case of springs whose mass is not negligible. If we let $M_G$ be the mass of the things oscillating (the glider, potentially plus more) and $M_S$ be the mass of the springs, then the effective mass M (which we plug into the above formula) is defined by:

$$M=M_G+M_S/3$$

We will also make use of the potential energy stored in a spring:

$$U=\frac{1}{2}k(x-x_0)^2$$

Note that the total kinetic energy also uses the same effective mass $M$ in place of the mass of the glider.

Procedure

Part I: Measuring the Spring Constant

In this part, we will measure the the displacement for various amounts of force, and use the slope to determine the spring constant.

First, disconnect the springs so the cart can move freely,¹ and (as always) level your air track.³

While you have everything disconnected, record (at the top of Part II) the mass of your glider and the mass of the two springs (combined)².

Then, reconnect the two springs, each from one end of the cart to the corresponding end of the track. Then, loop a string (tied to the cart) through the pulley on the other end (ensuring it doesn't get tangled with your spring).

Hang a 20g mass from the spring, and record the position with this mass.³ Take the uncertainty in hanging mass as 0.1g. Then increase the mass to 40g, and repeat (in increments of 20g) up to 100g.⁴

Part II: Measuring the Oscillation Frequency

Now, remove the string (or wrap it around the cart, so that it's out of the way). We won't need it for the rest of this experiment.

Open "SimpleHarmonic_Period" and set up the photogate, as usual.⁴ Place the photogate over the rest position of the mass (although the exact position isn't too important).

Begin with the glider alone. Record the additional mass as 0g. Pull the mass back, press "Start Collection," and have the mass oscillate back and forth (with an amplitude big enough to have a few swings back and forth, but small otherwise - don't stretch the springs too much). Then press "Stop Collection."

If the period seems roughly consistent between oscillations, record that period. (Neglect uncertainty in period.) Otherwise, run another trial.

Now: add a 20g mass to the top of the cart, record this as the "Additional Mass," and measure the new period.⁵ Repeat for three more 20g masses.⁶

Air Track for Parts II & III — Air Track, springs and glider in Parts II & III

Part III: Observing Conservation of Energy

Take all the masses off your cart. Measure the width of the flag on the cart, and record that on your sheet.

Now, open the LoggerPro file "SimpleHarmonic_Velocity". Double-check that the photogate is still connected and is centered over the middle of the cart's rest position.

Drag the cart some distance (~30cm) from its equilibrium position, and measure this displacement.⁷ Press "Start Collection," let go, and record the difference in time between when the photogate becomes "blocked" and when it becomes "unblocked" (in order to extract maximum velocity).

Analysis

Part I: Measuring the Spring Constant

For each hanging mass, calculate the tension. Propagate uncertainty.

Make a plot of tension vs. displacement (with appropriate error bars). Record the spring constant you measure from this plot (in the Part II section).

Part II: Measuring the Oscillation Frequency

Calculate the effective mass of the base glider-plus-springs system, and propagate uncertainty. Then, add on the additional mass for each data point to get the total effective mass, and propagate uncertainty.⁸

Calculate period squared for each period. Then, make a plot of period squared (no error bars) vs. total effective mass.

From the slope, calculate the spring constant (and propagate uncertainties). Compare your results to Part I (answer the question in the data table).

Part III: Observing Conservation of Energy

Calculate $v_0$ using your flag width and time difference.

Calculate maximal kinetic energy (using the effective mass of your system), and propagate uncertainty. (Note KE is maximal when PE=0, at the center point of oscillation.)

Calculate maximal potential energy (using your measured spring constant from Part I), and propagate uncertainty. (Note PE is maximal when KE=0, at the initial point of release.)

Answer the question about agreement with uncertainty.

Questions

Your TA will ask you to answer some of the following questions (they will tell you which ones):

Experimental questions:

We accounted for the mass of the springs in this experiment because we could, but the first thing to ask should have been: do we need to? We can make an estimate of the impact by assuming that the relative change in our spring constant is approximately the same as the relative size of the mass of the springs to the mass of the system: $\frac{\delta k}{k}\simeq\frac{M_S}{M_G}$. Using this estimation technique, approximate $\delta k$, the change in our result due to using effective mass, and compare it to your uncertainty in the spring constant, $\sigma_k$ (for part II). How important is this correction compared to other (random) errors in our experiment?
We level the air track in this lab because it's good lab procedure in general, but in fact in this lab a not-level (but still straight) air track shouldn't change any of our results. Explain why this is true (for each part).³

Theoretical questions:

In Part I you measured the effective spring constant $k_\text{eff}$ of the two springs acting together. If the two springs had $k_1$ and $k_2$ individually, how would they combine to get $k_\text{eff}$?

For further thought:

The $T^2$ relation seems to have been drawn out of nowhere, but there's actually a straightforward way to "prove" that the exponent is 2 from data alone, using logarithms. Basic log rules as applied to equation \eqref{T} give $\ln(T)=\frac{1}{2}\ln(M)+\ln\left(\frac{2\pi}{\sqrt{k}}\right)$ (remember that logs turn multiplication into addition and powers into multiplication). Make a plot of $\ln(T)$ vs. $\ln(M)$ (don't worry about error bars or units) using your data: what should the slope be? Is it what you expect? What does the spring constant $k$ change about the plot we made?

References and Tools

Guide to Uncertainty Propagation & Error Analysis (Quick Reference

PHY121/122 Plotting Tool

Photogate Reference Document

Air Track Reference Document

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For a review of simple harmonic motion, see KJF Ch. 14.

For more detail on where the factor of 1/3 comes from, see here.

See Air Track Setup Instructions.

See the Photogate Setup Instructions.

Check that it does, in fact, glide freely when released, and that there's not a significant amount of friction.

Use Table with Glider Masses and 9.7±0.1 grams each for the springs.

Ideally, you want to measure this precisely, so you want a good fixed point to measure from. Therefore, for this, the end of the cart makes more sense as a reference point than the middle of the cart (since the end of the cart is an easy identifier that's right above the ruler on the air track).

Depending on the orientation of your air track, it may make sense to record all your positions as negative - do this if you wish. (If the positions are decreasing as you hang more mass, making them all negative will give you a positive slope.)

There are various ways to do this, but the simplest is to hang it off the side (rather than put it on top, etc.). Just make sure it doesn't block your photogate.

In the interest of not having friction (as the masses exert torque on the cart, making it drag on the air track), make sure you balance out the locations of the masses as much as possible (i.e., don't put them all on the same side).

As with your position measurements earlier, it is best to measure the displacement using a position you can measure accurately, like one of the endpoints of the cart.

We neglect uncertainty in additional mass, so this uncertainty "propagation" is very easy.

Note that $x$ is the position of an object and the equilibrium position is $x_0$. So, the displacement from equilibrium is given by $x - x_0$.

In fact, the above formula for $T$ holds whenever you can express the force law in this way. Almost any force law can be approximated near a stable equilibrium point by Hooke's law, so almost every period can be approximately calculated (if you can figure out $k$) for small oscillations. An example of this procedure is the formula for the period of a pendulum.

Hint: there are two ways to show this mathematically, taking into account the "extra" effects of gravity. One is to show that the force still takes the term $F=k(x-x_0)$ for some $k$ and $x_0$ (and for each, you should either show that they don't change or explain why they don't matter). The second is to do the equivalent result for potential energy - show that $U=\frac{1}{2}k(x-x_0)^2$ still (again, potentially-different $k$ and $x_0$, explain why differences that occur don't matter).