Part 2: Stored Energy versus Displacement

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Part 2: Stored Energy versus Displacement

We wish to measure the energy stored in a spring with a displacement $x$ . The pan, with masses on it, is dropped from the no-load starting position determined in the previous section ( $x_0$ ). The lowest position of the pan will coincide with a velocity of zero (because the pan changes direction). The change in height is $h$ . Conservation of energy requires that the change in potential energy of the masses, $mgh$ , is stored as potential energy $U=\frac{1}{2}kh^2$ in the spring (see Equation (2)).

Equation (3) indicates a relationship between potential energy and $x^2$ . Since $h$ is equivalent to $x$ in our notation, and the work done is $mgh$ , we get

(5) $\begin{equation*} mgh = \frac{1}{2} kh^2 \textrm{ or } mh = \frac{kh^2}{2g} \end{equation*}$

so then, taking the logarithm of both sides:

(6) $\begin{equation*} log(mh) = 2log(h) + Constant \end{equation*}$

PROCEDURE

The change in height is determined with the aid of a paper slider, which you will need to attach to your ruler.

1. Put the pan (with no additional weight) on the spring, and let it come to rest. Move the slider up to the bottom of the pan.

2. Pull the pan up such that the bottom of the pan is at $x_0$ , which was measured in Part 1. Drop the pan such that it pushes the slider down the ruler. (Tip: you may want to drop the pan a few times to ensure the slider measures the true lowest position of the drop.) Record the mass of the pan and the new position of the slider after the fall under the Part 2 sheet in the Excel file. As you enter data, Excel will automatically compute the change in position, the potential energy, and their logarithms.

3. Add 10 g of weight to the pan and repeat step 2. Drop the mass from $x_0$ allowing the pan drop to a new lower position. Repeat this process but with 5 or 6 different masses (please do not exceed 120 g total mass on the spring), and enter the results in the table provided in the Results section.

a. Caution: To minimize damage to the pans and weights, be careful when you drop the pan each time, and make sure they don’t come off the spring and fall to the floor.

4. Determine the line of best fit through a trendline of the data. Find the slope of the line and the intercept. Re-scale the axes so that the data fits the chart nicely. Ensure you have descriptive titles and add your plot to the Results section.

5. Copy the data from the Excel file into your report.

Answer the following questions in the Lab 2 report, Discussion section:

Question 3.

Is the log-log plot consistent with the stored energy varying as $h^2$ ? What did you expect the slope of the line to be? How different is your experimental result from the theoretical value?

Question 4.

Determine the analytic equation for the constant in Equation (6). Show your work to the TA, and record your final result.

CHECKPOINT

(Show your analytic equation for the constant to the TA before moving on.)

Question 5.

Is what you obtain for the constant from this graph consistent with what we know about the spring from Part 1? Justify your reasoning and calculate the percent difference between the two values.

Hint: To calculate the percent difference between two values, A and B, use the following formula:

$\frac{\|A-B\|}{(A+B/2)}*100\%$

If the answer is $\le 10\%$ , then you may conclude that $A=B$ within 10% uncertainty.

Question 6.

What sort of systematic errors does the procedure have with it, and how can we reduce these errors within the lab?

Part 2: Stored Energy versus Displacement

PROCEDURE

CHECKPOINT

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