Simple Harmonic Motion in a Spring-Mass System

Note: In this project, you will measure the motion of a simple harmonic oscillator made from a spring and some weights. Optionally, you can also measure its motion using a smartphone and a sensor app as described in the Using a Sensor App to Measure Period section.

1. Do your background research so that you are knowledgeable about the terms, concepts, and questions above. Be sure to record your data in your lab notebook as you go along.
2. Measure the mass of one of your weights, using the scale. If your scale does not measure small weights, you can weigh all five of your weights and divide by five. Then measure the mass of the spring.
3. Perform the following steps to collect your data:
   1. Hold one end of the spring in your hand and let it bounce gently down and then back up.
   2. Count the number of cycles the spring makes in 60 sec with no weight hanging from it.
   3. Hang one weight from the spring (using a fine wire or string, if needed).
   4. Count the number of cycles the spring goes through in 60 sec with the weight attached.
   5. Perform at least three trials for each weight.
   6. Repeat steps c–e for a series of different weights.
4. Keep track of your results in a data table like Table 1.

5. Make another table like Table 2 to convert your raw data into numbers that can be used to determine the spring constant and spring's effective mass.

6. Make a graph with "Added mass," m, in kilograms, on the y-axis, and T²/4π² in sec², on the x-axis. Use kilograms rather than grams so that the value of k is in units of N/m, which is equivalent to kg/sec². Usually you are instructed to graph the independent variable (mass in this case) on the x-axis and the measured parameter (T²/4π²) on the y-axis. You should ignore this rule in this project since graphing m on the y-axis will let you read m_e from the y-intercept.

   This graph of m vs. T²/4π² has the same terms found in Equation 4:

   Equation 4:

   $$m = k\frac{T^2}{4\pi^2} - m_e$$

   Let's look at the equation. It has a form similar to the equation of a straight line: y = ax + b, where a is the slope and b is the y-intercept. In fact, Equation 4 is an equation for a straight line, with slope equal to k, the spring constant, and y-intercept equal to the negative value of m_e. In other words, there is a linear relationship between m and T²/4π², so a graph of m vs. T²/4π² (Figure 2) will be a straight line with slope k and y-intercept -m_e. The reason you calculated T²/4π² was to be able to read the values of k and m_e from the graph of m vs. T²/4π².

   
   Figure 2. Graph of m vs T²/4π²

   How do you determine the slope of the line you have drawn? The slope is measured as change in m, divided by the change in T²/4π², over the same range.

   Equation 5:

   $$slope = k = \frac{\Delta m}{\Delta T^2 / 4 \pi^2}$$

   Δm/Δ T²/4π² can be read as "delta m over delta T²/4π²", with the Greek letter for delta, Δ, indicating "change of." Make another table like Table 3 to find the slope. Pick points that are near the ends of the graph, rather than adjacent points.

Once you have determined the value of the spring constant, k, from the slope of the line, you are ready to determine the effective mass of the spring. To do this, extend the straight line until it intersects the vertical y-axis. The line will intersect the y-axis at - m_e (negative m_e). Based on theoretical considerations, the absolute value of m_e should be around one-third of the mass of the spring.

Optional: Using a Sensor App to Measure Period

When your weights oscillate up and down, their motion is periodic, as described in the introduction. When you use a stopwatch to measure the phone's period, you do this based on its position (for example, how long it takes the phone to start at the top, go down to the bottom, and return to exactly where it started at the top). If you could trace the phone's vertical position on a graph over time, it would be sinusoidal. You can determine the period of a sine wave by measuring the horizontal distance between two peaks, or any other identical points between two periods (Figure 3).

Figure 3. The period of a sine wave.

Since the phone's position is periodic, its velocity (change in position over time) and acceleration (change in velocity over time) are also periodic. This means that if you can measure and graph them over time, you will also see a sine wave, and can determine the period from that sine wave. This is convenient because we can measure acceleration with a device called an accelerometer. Accelerometers are built in to most smartphones and video game controllers to allow motion control. Sensor apps such as phyphox allow you to record data using the phone's accelerometer. Based on this information, think about how you could do this experiment using a phone equipped with a sensor app, instead of a stopwatch, to measure the spring's period of oscillation:

1. Figure out a way to hang your phone from a spring (Figure 4). Materials like rubber bands, paper clips, tape, and string could all be useful. Try to make sure your phone stays vertical, without wobbling or tipping side-to-side.
2. The phone adds a lot of weight to the spring. You will need to include the weight of the phone in your calculations. Make sure you have a spring that is stiff enough to support the weight of your phone, in addition to the weights you use for the experiment. If you hold one end of the spring in your hand then release the phone, the phone should not hit the ground. If a mini Slinky is not stiff enough to support the weight of your phone, get a stiffer spring from a hardware store.

Figure 4. Phone attached to spring using a rubber band and paper clip.

3. Open your sensor app. If you use phyphox, select the accelerometer with g function. Make sure you choose the accelerometer graph (X, Y, or Z) that takes measurements in the same direction the phone is moving (do not take side-to-side measurements if the phone is moving up and down, or your readings will not be correct).
4. Measure the phone's oscillations using the sensor app.
   1. Hold one end of the spring in one hand.
   2. Confirm that you have selected the accelerometer that records the direction you would like to measure, based on your phone's orientation.
   3. When you are ready, press the play button in phyphox to start a recording.
   4. Make sure the phone is vertical and the spring is in its neutral position.
   5. Drop the phone and allow the spring to oscillate until it comes to a stop.
   6. Press the pause button to stop recording and save your data.
5. View the resulting graph after recording your data. Measure the time (x-axis) value between two adjacent peaks in the graph. In the phyphox app, use the 'pick data' tool to select the highest points of two adjacent peaks. Once you have picked a data point the app will display its x- and y-values. Find the difference between these two times to calculate the period. Figure 5 shows an example graph.

Example graph shows the acceleration recorded from a smartphone that is attached to a spring. The graph is a relatively constant wave pattern with time on the x-axis and acceleration on the y-axis. The peak of the acceleration is marked at the third peak with a value of 19.5 meters per second squared.

Figure 5. Example acceleration graph. The x-axis of this graph shows time in seconds (s) and the y-axis shows acceleration in meters per second squared (m/s²). The data cursor shows a peak in the graph at 4.1 seconds.

5. Does recording data in this manner give you any additional insight that you could not get just by using a stopwatch? For example, does the oscillation period change over time, or remain constant? What about the amplitude (height) of the oscillation?
6. Optional: Have a friend use a stopwatch to measure the period while you take measurements with the sensor app. Which method do you think is more accurate? Does your answer change if you take measurements over a certain number of periods (e.g. 10) or a certain amount of time (e.g. one minute)?