Part 3: Mass versus Natural Frequency

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13 Part 3: Mass versus Natural Frequency

In this section we will measure the oscillation of a spring with a mass $m$ attached to it. We take into account the mass of the spring by expressing $m_t$ in Equation (4) as a sum of two parts. Most parts of the spring vibrate with smaller velocities than the mass at the end. Since this is difficult to measure explicitly, we will use the variable $m_0$ , the effective spring mass, which may not be equal to the actual mass of the spring. The total mass then becomes $m+m_0$ , and

(7) $\begin{equation*} f = \frac{1}{2\pi}\sqrt{\frac{k}{m+m_0}} \end{equation*}$

We can instead write the frequency, $f$ , in terms of the period ( $f=1/T$ ), and the equation above becomes:

(8) $\begin{equation*} m = -m_0 + \frac{kT^2}{4\pi^2} \end{equation*}$

PROCEDURE

1. Use the masses provided in the Table in Part 3 of the lab report. Those masses will give about a factor of 2 change in frequency.

2. For each mass (remember the mass of the pan!), attach it to the end of the spring, let it fall, and count the oscillations while you measure the time with a stopwatch. Note: that one does not need to release the weights from $x_0$ as done in Part 2. Also, remember to remove the meter sticks for clearance.

a. Measure the time for 50 oscillations with a stopwatch.

b. Record all of your measurements in your table in the Results section for Part 3.

3. Record the data from your table in the Excel sheet titled “PART 3” which will generate a plot of mass $m~(kg)$ versus $T^2$ .

4. Fit a straight line to your data in Excel, and interpret the slope of the straight line. To do this:

a. Click on a data point on your graph. This should highlight all of your data points.

b. Right click, which will open a window, and select ‘Add Trendline.’ From the pop-up window, select ‘Linear’ as well as ‘Display Equation on Chart’ and ‘Display R-squared value.’ The $R^2$ value is called the ‘co-efficient of determination’ and it tells you how well your data is fitted by a straight line. If your $R^2$ value is close to 1, this means your data is well-fit by a straight line.

Also interpret the significance of the intercepts of the line with the two axes of the plot.

Answer the following questions:

Question 7.

Are your results consistent with your determination of $k$ in Part 1? Calculate the % difference between the two values. Calculate the % difference between the two values.

Question 8.

Comment on the value of $m_0$ relative to the mass of the spring.

*** Helpful Practical Hint *** Use of small amplitudes and carefully centered weights may help prevent coupling to a pendulum mode where the springs start to sway back and forth (this will affect your ability to count the oscillations). Centre the weights by staggering the slots in the weights so that they aren’t all on top of each other.

13 Part 3: Mass versus Natural Frequency

PROCEDURE

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