Spring Constant Lab Manual PDF

Preview

Summary

Spring Constant Lab Manual - PHYS 1007....

Description

PHYS 1007 Fall 2021

3. Spring Constant Experiment Introduction In this experiment you will be introduced to the process of linearizing an equation that you will use to determine the spring constant for a given spring. You will use two different methods to collect your data: measuring extension as a function of the applied load (static method) and measuring the frequency of oscillation of a mass suspended from the spring (dynamic method).

Theory

Part 1: Static Method (𝒌𝒔 )

The ability of an object to regain its original shape after having been deformed by a force is called elasticity. All solid substances have this property. However, after certain point in deformation (called the elastic limit) the object remains permanently deformed. If the deformation is kept within the elastic limit, the spring’s extension is proportional to the applied force. This is known as Hooke's law and is expressed as 󰇍𝑭󰇍 = 𝑘𝒙 󰇍

where 𝐹 is the applied force measured in Newtons (𝑁 =

𝑘𝑔𝑚 𝑠2

(3.1)

), 𝑥 is the spring’s extension length in meters, 𝑁

and 𝑘 is a proportionality constant called the spring constant and expressed in 𝑚.

In the first part of the experiment you will use Hooke's law to determine the spring constant, 𝑘𝑠 , of a spring. The basis of this experiment is illustrated in Figure 1. The force applied on the spring is the gravitational force 𝐹𝑔 = 𝑀𝑔. 𝐹𝑔 = 𝑘𝑠 𝒙

(3.2)

Figure 1: Extension of a spring, 𝒙, due to an applied force, 𝑭𝒈 .

1

PHYS 1007 Fall 2021

According to Equation (3.2), there is a linear relationship between the force and the extension of the spring and we can thus describe the relationship between the variables using the equation of a line 𝒚 = 𝑚𝒙 + 𝑏.

By applying different forces on the spring and measuring the corresponding extensions, a graph of 𝐹𝑔 vs 𝑥 would show a straight line of slope, 𝑚 = 𝑘𝑠 and intercept 𝑏 = 0. The static method spring constant, 𝑘𝑠 , can thus be determined from the slope, 𝑚, of the 𝐹𝑔 vs 𝑥 graph as follows 𝑘𝑠 = 𝑚

(3.3)

𝜎𝑘𝑠 = 𝜎𝑚

(3.4)

And the uncertainty on 𝑘𝑠 is equal to uncertainty on the slope, 𝜎𝑚 :

Figure 2: Linear relationship between the force and extension of the spring.

Once the we know the value of the spring constant, (i.e. the spring is calibrated) we can then apply Equation (3.2) to find an unknown parameter, such as the acceleration due to gravity on a planet other than Earth.

Part 2: Dynamic Method (𝒌𝒅 )

If a mass suspended from a spring is pulled down and then released, the system will undergo simple harmonic motion (SHM). The position, 𝑥 , of the mass becomes a function of time, 𝑥(𝑡), and it can be shown that the motion of the mass will have the following form, 𝑥(𝑡) = 𝐴 sin(𝜔𝑡 + 𝜑)

where 𝐴 is the amplitude of the SHM, 𝜔 (omega) is its angular frequency in

(3.5) 𝑟𝑎𝑑 𝑠

and 𝜑 is the phase. The

phase is not important in this experiment since it depends on the moment you measure the system, which is completely arbitrary. Similarly, you could have used cos(𝜔𝑡 + 𝜑).

The angular frequency of oscillation, 𝜔, is related to the dynamic method spring constant, 𝑘𝑑 , and to the mass oscillating on the spring, 𝑀 , by 𝜔= √

𝑘𝑑 𝑀

(3.6)

2

PHYS 1007 Fall 2021

The period, 𝑇 , of small oscillation of the mass thus depends on the elasticity of the spring, measurable as the spring constant, 𝑘𝑑 , and on the mass itself.

𝐴

𝑇=

1 𝑓 𝑓=

𝜔 2𝜋

Figure 3: Simple harmonic motion experienced by a mass on a spring.

We can, thefore, express the relationship between the period, the mass and the spring constant by combiing informaton from Figure 3 with Equation (3.6): 2𝜋 𝑘𝑑 = √ 𝑀 𝑇

(3.7)

In the second part of the experiment you will measure the period of oscillation of different masses suspended from the spring. Once again, we would like to take the advantage of a graphical analysis. This time, however, the equation we are working with is not already in the form of a straight line ( 𝒚 = 𝑚𝒙 + 𝑏). We will, therefore, need to reorganize the variables in Equation (3.7) so we can obtain a linearized relationship, i.e. a relationship that can be described by the equation of a line, 𝒚 = 𝑚𝒙 + 𝑏. We begin by identifying our independent variable, 𝑥 . This is the value we are going to be controlling. As mentioned above, we will be controlling the masses that we suspend from the spring. This means the mass term, 𝑀 , in Equation (3.7) is the independent variable in our case: 𝑀=𝑥

According to 𝒚 = 𝑚𝒙 + 𝑏, the definition of a linear relationship (first-order polynomial) is such that the 1 independent variable has a power of +1. The 𝑀 term in Equation (3.7), however, has a power of − 2 .

Thus, we know that we will need to complete a series of algebraic manipulations, so that our resultant equation has the appropriate power on the independent variable.

We them need to identify the dependent variable, 𝑦. This is the value that we are going to be measuring or otherwise determining as a result of applying different values of the independent variable, i.e. the values of 𝑦 are the result of the values of 𝑥 . We know that in our case we will be measuring the period of

3

PHYS 1007 Fall 2021

oscillation, 𝑇 . Therefore, we can conclude that the dependent variable is related to 𝑇 . The exact form of this relationship will become clear once the linearization process is complete.

We now know everything we need to proceed with linearizing Equation (3.7): we know that the term 𝑀 needs to have a power of +1 and that the term related to 𝑇 needs to be alone on the left-hand side of the equation. Following these conditions, we obtain: 𝑘𝑑 2𝜋 = √ 𝑀 𝑇

Squaring both sides 1

𝑀 −2

𝑀 −1

4𝜋 2 𝑘𝑑 = 𝑀 𝑇2

Inverting both sides 𝑀 −1

𝑀 +1

𝑀 𝑇2 = 2 4𝜋 𝑘𝑑

Isolating 𝑇related term 𝑇2

𝑇2 4𝜋2

𝑻𝟐 =

4𝜋2 𝑴 𝑘𝑑

𝒚 = 𝑚𝒙

+𝑏

Our resultant equation follows the linear form of 𝒚 = 𝑚𝒙 + 𝑏, with the condition that 𝑀 = 𝑥 . The dependent variable is the square of the period of oscillation: 𝑦 = 𝑇 2 and the y-intercept term is 𝑏 = 0. Therefore, if we plot 𝑇 2 vs 𝑀 we will obtain a straight line according to 𝑻𝟐 =

4𝜋2 𝑴 𝑘𝑑

(3.8)

In the 𝒚 = 𝑚𝒙 + 𝑏 expression, the slope is the term, 𝑚, that multiplies the independent variable. In the

Equation (3.8) above, the term that is multiplying the independent variable is that: 𝑚=

4𝜋2

𝑘𝑑

4𝜋 2 𝑘𝑑

. Therefore, we can say

(3.9)

From this expression we can obtain the value for the dynamic method spring constant. 𝑘𝑑 . The uncertainty on 𝑘𝑑 is then related to the uncertainty on the slope, 𝜎𝑚 via: 𝜎𝑘𝑑 =

Figure 4: Linear relationship between 𝑻𝟐 and M for a spring undergoing simple harmonic motion.

4𝜋 2 𝜎𝑚 𝑚2

(3.10)

Again, once 𝑘𝑑 is know, Equation (3.8) can be used to determine an unknown value, such as unknown mass.

4

PHYS 1007 Fall 2021

Procedure Part 1: Static Method (𝒌𝒔 )

For the static method, you will suspend different masses from the spring and measure resultant extension of the spring from its equilibrium position. 1.

Download and open the Spring Constant Data Analysis Excel spreadsheet from the LAB Brightspace page. You should use this file to enter the data you collect. Note that the file has two pages: one for Static and one for Dynamic Method.

2.

Follow the Spring Constant Simulation link provided on the LAB Brightspace page and select the Lab option.

3.

Select the following settings:

Spring Constant 1: Large

Displacement:

Damping: None

4.

Check that the Gravity is set to Earth, 9.8 m/s2.

5.

Click and drag the ruler next to the spring.

6.

Click and drag the orange cylinder to suspend it from the spring.

7.

Collect a set of mass and spring extension readings and record them in Columns A and B of the Static Method page in the data analysis file: ✓ Use mass values from 50g to about 300g in increments of approximately 20g. ✓ Using the ruler, measure the extension (the green displacement arrow) of the spring for each mass. You can approximate to about 0.5cm on the ruler in the simulation, so be very careful that you record the readings as accurately as possible. ✓ You may find that mass oscillates a bit on the spring, so use the Stop button

to

stabilize the system. 5

PHYS 1007 Fall 2021

8.

Now switch the Gravity setting to Planet X.

9.

Choose a mass value (𝑀𝑋 ) for the orange cylinder and measure the resultant extension (𝑥𝑋 ). Use 1g

as uncertainty on mass and 0.5cm as uncertainty on extension. Record your values in the following table. Table 1: Static Method, Planet X

Mass of the Orange Cylinder, 𝑴𝑿

Extension of the spring, 𝒙𝑿

Value Uncertainty Units

6

PHYS 1007 Fall 2021

Part 2: Dynamic Method (𝒌𝒅 )

For the dynamic method, you will oscillate the suspended masses and record the period of oscillation. 10. Once you have fully finished collecting data for Part 1, reset the simulation by pressing

11. Select the following settings:

Spring Constant 1: Medium

Displacement Period Trace

Damping: None

Slow

12. Check that the Gravity is set to Earth, 9.8 m/s2. 13. Click and drag the ruler and the timer next to the spring.

14. Click and drag the orange mass to suspend it from the spring. 15. Collect a set of mass and period readings and record them in Columns A and B of the Dynamic Method page in the data analysis file: ✓ Use mass values from 50g to about 300g in increments of 20g. ✓ Using the ruler, ensure that for each mass, you have the same amplitude of oscillation (i.e. you pull down on the mass by the same amount each time). In the example below, I used an amplitude of 20cm for each trial. ✓ Use the period trace as a guide for when to start and stop timing the motion.

7

PHYS 1007 Fall 2021

16. Choose either the red or the blue unknown mass (𝑀? ) and measure its period of oscillation (𝑇? ). You can use 0.02s as uncertainty on the period. Record your value in the following table: Table 2: Dynamic Method, Unknown Mass

Period of Oscillation, 𝑻?

Unknown Mass, 𝑴?

Value Uncertainty

Red or Blue?

Units

17. Do not forget to include a screen captures of the simulations in your lab report.

Data Analysis

Part 1: Static Method (𝒌𝒔 )

18. Your Static Method data should be on the appropriate page in your data analysis file in rows 5-17 of Columns A and B. Use what you have learned in the previous lab about calculated columns to

8

PHYS 1007 Fall 2021

populate Columns C, D and E with the values of the mass in kilograms, extension in meters and the force of gravity. ✓ As part of your Calculation section, include 1 sample calculation of how you obtained the force of gravity. 19. We would now like to plot the straight line with 𝐹𝑔 on the y-axis and 𝑥 on the x-axis.

✓ For convenience, y-values should in the right column and the x-values in the left column as Excel automatically takes the left column as the one containing the x-values. You will notice that this has already been arranged for you in the layout of the data table in your analysis file.

✓ Please note that the data provided in the examples below is not for a different set of settings in the simulation. Your values, therefore, will be different those in the examples.

20. Left click and drag to highlight the data you want to plot.

9

PHYS 1007 Fall 2021

21. With the data highlighted, click INSERT/Charts /Scatter and select a scatter plot with only the data points.

Once you make your selection, you should have a graph with data points, but no line connecting them yet:

10

PHYS 1007 Fall 2021

22. Once you have your graph, it is a good idea to check that the x- and y-values are correct.

23. Right click on a data point on your graph to pull up the options. Select Add Trendline… A menu (Format Trendline) should appear on the right-hand-side. You need to select Linear. You also need to make the equation appearing on the graph by making sure you check-mark the boxes for Display Equation on chart and Display R-squared value on chart.

11

PHYS 1007 Fall 2021

24. At this point your graph should look something like this:

Note, that although it has a line of best fit now, along with the corresponding straight-line equation, it is far from complete. It is still needs to be formatted a little. It is up to you what to do with the Chart title. You can click on it and enter your own or you can delete it entirely. Please note that the short title there is not an acceptable substitute for a proper caption (see Section B of General Information) that is expected for each of your graphs and tables.

12

PHYS 1007 Fall 2021

You also need to make sure that the axes are properly labelled with quantity and units. To do this: left click somewhere on the graph, go to CHART TOOLS/Design/Add Chart Element/Axis Titles. This will allow you to label each axis.

When you are done, your graph should look something like this:

It is ideal for the data to occupy as much room as possible on the graph. In the case of the graph in the example, the x-axis scale can be adjusted to further zoom in on the data and its trendline.

13

PHYS 1007 Fall 2021

To adjust the scale on an axis, right click on the axis itself and select the Format Axis… option. There you can set the start and end point (Bounds) for the axis in question.

Here is what the graph above looks like with a more appropriate scale:

Note that the data is now occupying most of the area of the graph. Once you have verified that the scale on your graph is appropriate, you can copy/paste the graph into your report.

14

PHYS 1007 Fall 2021

25. You should note that the equation of the straight line that you have applied to your data does not automatically generate the uncertainty values for the slope and the intercept. This is something that needs to be done separately. You can use the LINEST function for this. Start by left clicking and highlighting 4 cells designated for you in the file (B20, B21, C20, C21):

Then type: =linest(known_y’s,known_x’s,true,true). For the known_y’s left-click and drag to select all the y-values, for the known_x’s left-click and drag to select all the x-values. Therefore, according to the cell numbers in your data analysis file you would enter =linest(E5:E17,D5:D17,true,true). Once you have done that, press Ctrl+Shift+Enter.

15

PHYS 1007 Fall 2021

When you are done entering the formula your 4 cells should be filled as follows: 𝑆𝑙𝑜𝑝𝑒 𝜎𝑠𝑙𝑜𝑝𝑒

𝑌 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝜎𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡

16

PHYS 1007 Fall 2021

You should now have all the information you require to fill the Linear Fit Summary Table (Table 3). Every time you create a linear graph in this course you need to include such a table underneath that graph in the Observations section. Table 3: Linear fit summary for Static Method

𝒎

Slope

𝝈𝒎

𝒃

Y-intercept

𝝈𝒃

Value Units 26. State the 𝑘𝑠 ± 𝜎𝑘𝑠 value you found and explain how it relates to the information in the Linear Fit

Summary Table. You have now calibrated the spring on the Large Spring Constant setting in your simulation. This means that you can now use the value of the spring constant that you found to find the acceleration due to gravity (𝑔𝑋 ) on Planet X.

27. Adapt Equation (3.2) to allow you to solve for 𝑔𝑋 .

✓ Recall from the previous lab that when you are isolating for the quantity you wish to solve for (𝑔𝑋 in this case) you need to manipulate your equation such that the unknown quantity

is alone on the left-hand-side of the equal sign and all the other quantities (𝑀𝑋 , 𝑥𝑋 , etc.) are

on the right-hand-side of the equal sign. ✓ Remember to show your work.

28. Calculate the value for 𝑔𝑋 based on the equation you have created in the previous step. 29. Use what you learned in the previous lab to create the uncertainty propagation equation that will allow you solve for 𝜎𝑔𝑋 .

✓ Remember to review Example 2 on page 6 of Section C of General Information.

✓ Show your work. 30. Calculate the value for 𝜎𝑔𝑋 based on the equation you have created in the previous step.

17

PHYS 1007 Fall 2021

31. Perform a t-test to compare the expected y-intercept predicted by Equation (3.1) with the y-intercept from the 𝐹𝑔 vs 𝑥 graph. Assume that the predicted y-intercept value has an uncertainty of zero.

Part 2: Dynamic Method (𝒌𝒅 )

32. Repeat the process of creating a linear fit for data on the Dynamic Method page of your analysis file. ✓ Refer to steps 18-25 above. ✓ Remember to include the graph you create and the Linear Fit Summary Table in your Observations. 33. Determine the spring constant for dynamic method, 𝑘𝑑 ± 𝜎𝑘𝑑 , using the slope of the 𝑇 2 vs 𝑀 graph.

✓ First you will need to re-arrange Equation (3.9) to isolate for 𝑘𝑑 . Show your work. 34. You will notice that the units you are getting on your 𝑘𝑑 ± 𝜎𝑘𝑑 value do not appear to be the expected SI units of

N

m

. You need to convert the 𝑘𝑑 units to the SI units. Remember to explicitly show this

step in your Calculations.

35. You have now calibrated the spring on the Medium Spring Constant setting in your simulation and can use it find the mass, 𝑀?, of the cylinder you chose in step 16.

✓ Re-arrange Equation (3.8) to isolate for mass and use it to calculate 𝑀? .

✓ Use Example 2 on page 6 of Section C of General Information to create the uncertainty

propagation equation for uncertainty on the unknown mass, 𝜎𝑀? . Show your work. Once

the equation is ready, use it to calculate 𝜎𝑀? .

Discussion As part of your discussion, you need to answer the following questions: 1. Which of the two methods (static, dynamic) would you consider more reliable? Explain your reasoning. 2. Explain how the simulated experiments you used can be performed in real life. How would you be able to obtain a value of oscillation period without the Period Trace option of the simulation? 3. So far we have discussed of a single spring. Now, consider a parallel and series combinnation of two springs (Figure 5). Which system will have a higher total spring constant? Explain.

18

PHYS 1007 Fall 2021

Figure 5: a) Two springs in parallel. b) Two springs in series

19...