PHY11L A4 E205

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E205: HOOKE’S LAW FRISNEDI, Nadine T.

OBJECTIVE

The purpose of this experiment is to study the elastic properties of the spring. In physics, Elasticity is the ability of solid materials to return to its original shape, length or size when the deforming forces are removed.

Another purpose of this experiment is to determine the force constant of the spring. The force constant is the factor or the characteristic of the spring. I can be simply defined as the stiffness of the spring. The experiment can help the students understand how the displacement or elongation and the deforming force of the spring is important in determining its force constant. The third objective of this experiment is to investigate the relationship between deforming force and amount the spring stretches. Hooke’s Law is a principle in physics that states that the deforming force needed to compress or extend a spring is directly proportional to the elongation of the body. The experiment will show the students the proof of this relationship.

The last objective is to determine the total work done on the spring when it is being stretch. The experiment will also show how the Force constant is needed in determining the Work done on the spring.

Through the experiment, the students will be able to gain more knowledge and appreciation about the concepts of elasticity and Hooke’s Law with the use of spring. At the end of the experiment, it is expected for students to learn how to determine the Force constant of a spring and to compute for the Work done on the spring. The students will not just learn how to compute for the Force constant

of the spring using the given formulas but also by getting the slope of the line based from the graph that will be made using the gathered data. The experiment will help the students be able to understand the applications of the given laboratory formulas in solving problems involving Physics and will surely be helpful in studying other concepts about it. Another thing about this experiment is that it is very easy to conduct and it is not time consuming, thus students will enjoy doing it.

MATERIALS AND METHODS

Before the experiment was actually performed, the students were given guidelines first. The group came up with a strategy for conducting this experiment which is to make sure that the procedures were being followed correctly while making sure that the equipment used are handled carefully especially the springs since they are sensitive and really important in order to perform the experiment properly. It was also instructed to us that the amount weighs to be used should be placed cautiously and not letting the weights hang on it for a long time so that the spring won’t be stretched so much.

The materials for these experiment are the following: a set of Hooke’s Law Apparatus which is composed of support rod, support arm, notch, clamp and transparent scale plate and stretch indicator, a 4N/m spring, an 8N/m spring, a mass hanger and a set of weights. The springs and the Hooke’s Law Apparatus were the main equipment that was used in performing the experiment.

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Figure 1. The materials and equipment used in the experiment.

The equipment was assembled first. The 4N/m spring was hanged to the notch on the support arm of the support rod. After that, the stretch indicator was connected to the bottom of the spring. The clamp was adjusted so that the indicator reading was exactly at zero. The top of the stretch indicator was used as the base for measuring. Lastly, the mass hanger was connected to the bottom of the stretch indicator.

Figure 2. Setting up the Hooke’s Law apparatus.

Figure 3. Making sure that the stretch indicator is aligned to zero.

For the second part of the experiment, it is focused in the determination of the force constant of the spring. The spring that was used on this part has an ideal force constant of 4N/m. This part requires a lot of equipment care. The students were instructed not to stretch the spring too much. It is simply because the spring is small and sensitive. The elasticity of the spring may change or become permanently damaged. A 20g mass was placed on the mass hanger. The displacement done by the stretched spring that was indicated by the stretch indicator was recorded and a photo was taken to have an accurate data. The weight of pan must not be included in the computations.

Figure 4. A 20g weight was placed on the hanger connected to the 4N/m spring.

After that the force constant of the spring was computed using the using the equation of Hooke’s Law which is 𝐹 = 𝑘𝑥.The procedures were repeated for another three trials however the added weights were increased by 10 grams per trial.

Figure 5. Getting the displacement of the 4N/m spring for the third trial.

3 | P a g e After doing all the four trials, the average value of

the force constant was computed. Using MS Excel, the force vs displacement graph was plotted. Also, with the use of the slope function of MS Excel, the slope of the line of the graph was computed. Another way is when the line graph has been plotted, choose from the settings that will show the equation of the graph that look’s like: y=mx+b, where m is the slope. After that, the percent difference was calculated with the slope as the first value and the average force constant as the second value. All the data was put to Table1A. All of the procedures done for the first spring were repeated using a different spring, which is the 8N/m spring. All of the data gathered for the second spring were put to Table 1B.

Figure 6. Using the 8N/m spring, the stretch indicator was made sure to be at zero.

Figure 7. Getting the displacement of the 8N/m spring for the first trial.

Figure 8. A 50g weight was placed on the hanger connected to the 8N/m spring.

For the last part of the experiment, the students focused in the determination of the Work done on the spring. The data from Table 1A and Table 1B were used for this part of the experiment. The work done in stretching the spring was computed using the equation,𝑊 =1

2𝑘(𝑥𝑓 2_{− 𝑥}

𝑜2), where k is the average force constant, 𝑥𝑓 being the displacement of trial 4 in the first part of the experiment and 𝑥𝑜= 0. The area under the graph of force vs displacement was determined by getting the linear regression model of the graph using the stat mode function in the calculator. MS Excel can also be used in determining the equation of the line. After plotting the line graph, choose the setting that will show the equation of the graph. After getting this equation, integration was applied with the limits from 0 to the amount of displacement from trial 4. Shortly after, the total work done and the area under the graph of force vs. displacement were compared by getting the percent difference.

OBSERVATIONS AND RESULTS

In Table 1A and Table 1B, the mass was already given. The force was computed by getting the product of the mass and the acceleration due to gravity on Earth, the displacement was determined by getting the reading on the scale of the indicator.

4 | P a g e The force constant was computed by using

Hooke’s Law which is: 𝐹 = 𝑘𝑥, where F is the deforming force, x is the displacement and k is the force constant. The force was divided by the displacement and its quotient is the force constant. After that the average force constant was determined. The force vs. Displacement graph was plotted in Excel and the slop was computed using the function in Excel. The percent difference was determined by using the slope of the line and the average force constant.

Table 1A. Determining the Force Constant of 4N/m Spring.

TRIAL Mass _(kg) Force _(N) X (m) Constant Force (N/m) 1 0.02 0.196 0.037 5.2973 2 0.03 0.294 0.056 5.25 3 0.04 0.392 0.074 5.2973 4 0.05 0.49 0.092 5.3261 Average Slope of the Line Percent Difference

5.2927 N/m

5.35

1.16%

The graph shows the relationship between force and displacement using the 4N/m spring. This shows that force and displacement have directly proportional relationship. The slope in the graph is positive since it is increasing. The slope of the graph was labelled as the accepted value of the force constant and was then used in computing for the percent difference.

Sample Computations: Trial 1 Force: 𝐹𝑜𝑟𝑐𝑒 = 𝑀𝑎𝑠𝑠 𝑎𝑑𝑑𝑒𝑑 ×9.8𝑚 𝑠2 𝐹𝑜𝑟𝑐𝑒 = 0.02𝑘𝑔 ×9.8𝑚 𝑠2 𝐹𝑜𝑟𝑐𝑒 = 0.196𝑁 Force Constant: 𝐹 = 𝑘𝑥 𝑘 =𝐹 𝑥 𝑘 =0.196𝑁 0.037𝑚 𝑘 = 5.2973 𝑁/𝑚 Average Force Constant:

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐹𝑜𝑟𝑐𝑒 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝑘1+ 𝑘2+ 𝑘3+ 𝑘4 4 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐹𝑜𝑟𝑐𝑒 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 =5.2973 + 5.25 + 5.2973 + 5.3261 4 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐹𝑜𝑟𝑐𝑒 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 5.2927 𝑁/𝑚 Percent Difference:

Slope of the Line = 5.35

% 𝑑𝑖𝑓𝑓 = |𝐸𝑉1− 𝐸𝑉2| (𝐸𝑉1+ 𝐸𝑉₂ 2) % 𝑑𝑖𝑓𝑓 = |5.35 − 5.2927| (5.35 + 5.2927₂ ) %𝑑𝑖𝑓𝑓 = 1.16% y = 5.3542x - 0.0037 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.02 0.04 0.06 0.08 0.1 Fo rce Displacement GRAPH 1A: FORCE VS.

5 | P a g e Table 1B. Determining the Force Constant of

8N/m Spring.

TRIAL Mass _(kg) Force _{(N) (m)} X Constant Force (N/m) 1 0.02 0.196

0.027

7.2593

0.04

7.3500

0.053

7.3962

0.065

7.5385

7.3860 N/m

7.71

4.34%

The Force vs. Displacement graph for Table 1B:

The graph shows the relationship between force and displacement using the 8N/m spring. This shows that force and displacement have directly proportional relationship. The slope is also increasing and is bigger compared to the slope of the graph for Table 1A.

Sample Computations: Trial 1 𝐹𝑜𝑟𝑐𝑒 = 𝑀𝑎𝑠𝑠 𝑎𝑑𝑑𝑒𝑑 ×9.8𝑚 𝑠2 𝐹𝑜𝑟𝑐𝑒 = 0.02𝑘𝑔 ×9.8𝑚 𝑠2 𝐹𝑜𝑟𝑐𝑒 = 0.196𝑁 Force Constant: 𝐹 = 𝑘𝑥 𝑘 =𝐹 𝑥 𝑘 =0.196𝑁 0.027𝑚 𝑘 = 7.2593 𝑁/𝑚

Average Force Constant:

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐹𝑜𝑟𝑐𝑒 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝑘1+ 𝑘2+ 𝑘3+ 𝑘4 4 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐹𝑜𝑟𝑐𝑒 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 =7.2593 + 7.35 + 7.3962 + 7.5385 4 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐹𝑜𝑟𝑐𝑒 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 7.3860 𝑁/𝑚 Percent Difference:

Slope of the Line = 7.71

% 𝑑𝑖𝑓𝑓 = |𝐸𝑉1− 𝐸𝑉2| (𝐸𝑉1+ 𝐸𝑉₂ 2) % 𝑑𝑖𝑓𝑓 = |7.71 − 7.3860|

(7.71 + 7.3860₂ ) %𝑑𝑖𝑓𝑓 = 4.34%

In Table 2, we were to find the Work done on spring for this part. For this table, the preliminary data came from Table 1A and Table 1B. The final displacement was the displacement recorded for the fourth trial of both springs. The force constant to be used came from the average force constant that was computed for both springs. The work done in stretching the spring was computed using the equation,𝑊 =1

2𝑘(𝑥𝑓 2_{− 𝑥}

𝑜2), where k is the average force constant, 𝑥𝑓 being the displacement of trial 4 in the first part of the experiment and 𝑥𝑜= 0. y = 7.7137x - 0.0138 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.02 0.04 0.06 0.08 Fo rce Displacement GRAPH 1B: FORCE VS. DISPLACEMENT

6 | P a g e The area under the graph of force vs displacement

was determined by getting the linear regression model of the graph using the stat mode function in the calculator. MS Excel can also be used in determining the equation of the line. After plotting the line graph, choose the setting that will show the equation of the graph. After getting this equation, integration was applied with the limits from 0 to the amount of displacement from trial 4. Shortly after, the total work done and the area under the graph of force vs. displacement were compared by getting the percent difference. The percent difference was calculated with the total work done as the first value and the area under the graph of force vs. displacement as the second value.

Table 2. Determining the Work Done on the Spring

TRIAL _displacement final average force _constant

Table 1A

0.092m

5.2927 N/m

Table 1B

0.065m

7.3860 N/m

TRIAL work area under F _{vs. x graph difference} % Table

1A

0.0224 J

0.0222

0.89%

1B

0.0156 J

0.0154

1.31%

Sample Computation for Table 2: Trial: Table 1A Work: 𝑊 =1 2𝑘(𝑥𝑓 2_{− 𝑥} 𝑜2) 𝑊 =1 2( 5.2927𝑁 𝑚 )(0.092 2_{− 0}2₎ 𝑊 = 0.0224 𝐽 Area under F vs. x graph:

Equation of the graph: y = 5.3542x - 0.0037 𝐴𝑟𝑒𝑎 = ∫ (5.3542𝑥 − 0.0037)𝑑𝑥 0.092 0 𝐴𝑟𝑒𝑎 = 0.0222 Percent Difference: % 𝑑𝑖𝑓𝑓 = |𝐸𝑉1− 𝐸𝑉2| (𝐸𝑉1+ 𝐸𝑉₂ 2) % 𝑑𝑖𝑓𝑓 = |0.0224 − 0.0222| (0.0224 + 0.0222₂ ) %𝑑𝑖𝑓𝑓 = 0.89%

DISCUSSION & CONCLUSION

The elastic properties of the spring tells us that there is a limit to which something can stretch to. Staying within the limit, expectedly, it will return to its original size as the elasticity does not change. However, passing the limit, it will not turn back to its normal size.

For the determination of force constant of the spring, the formula that was derived from Hooke’s Law provided us a definition that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit. The data from Table 1A and 1B, shows us that we have determined the force constant of 4N/m and 8N/m spring. The concept of Hooke’s Law was used in this part. The force constant was computed by using the equation: 𝐹 = 𝑘𝑥, where F is the deforming force, x is the displacement and k is the force constant. The force was divided by the displacement and its quotient is the force constant.

Our data from both Table 1A and 1B shows us that as the deforming force increases, the distance the spring stretches also increases. This means that force is directly proportional to the displacement of the spring. In conclusion, I could say that the first three objectives were then achieved from the second part of the experiment.

7 | P a g e For the determination of work done on the spring,

the area of the graph (F vs. x) is nearly identical to the total work done. This is because the relationship gives the area, where the force F is plotted as a function of distance. In the more general case of a force which changes with distance, the work may still be calculated as the area under the curve.

In Table 2, we have determined the work done on the spring. According to our data, the 4N/m spring has a greater work done compared to the 8N/m spring. Since the 4N/m spring has a lower force constant, it is easier to stretch so if will have a greater displacement than the 8N/m spring. Force constant and displacement is also directly proportional to work so as the force constant and displacement increase, work also increase. The area of the graph of Table 1A is greater than the area in Table 1B. Since the total work done on the spring was determined, our group have fulfilled the last objective of the experiment.

The percent difference that we got in this experiment were very low. We have 1.16% for Table 1A and 4.34% for Table 2B. For Table 2 we got a percent difference of 0.89% and 1.31%. Since the experiment was easy to perform, there was a lesser chance of committing errors. The possible sources of error can be the inaccurate measurement of the displacement of the spring since we might tend to make assumptions when the stretch indicator is not horizontally aligned. In recommendation to the students who will be doing the same experiment, doing a few sub-trials can be helpful in verifying the measurements. They should also wait for the spring and the stretch indicator to stop moving and become when getting the measurements. Taking clear photos of the trials is also a good thing to do when in doubt of the measurements.

ACKNOWLEDGMENT & REFERENCE

I would like to thank my groupmates for being so cooperative, initiative and relaxed upon conducting the experiment. I appreciate all of their efforts since without their help, our experiment will have a great chance of failure. I also thank them for helping in making the Excel file that served as our data sheet become presentable and organized. I also thank our professor, Prof. Ricardo F. De Leon, Jr. for guiding all throughout the experiment. I thank him for instructing us on how we should set up the materials and equipment for our experiment. I also thank him for teaching us how to make a proper graph of the data we have and also for teaching us how to use the slope function in MS Excel. I also would like to acknowledge the lab assistants for reminding us how to handle the materials and equipment and telling us about the important things to remember such as the weights to be added. Lastly, I would like to thank my family for supporting me in my studies as I pursue my degree in Mapúa.

References:

General Physics 2 Laboratory Manual, Mapúa Institute of Technology, Manila: Department of Physics

Walker, J., Halliday, D., & Resnick, R. (2014). Principles of Physics. 10th Edition. 159-161. Williams, M. (February 2015).

http://www.universetoday.com/55027/hookeslaw /

Hooke’s Law (Retrieved August 2015).

https://answers.yahoo.com/question/index?qid= 20081119031341AAVPAHl

Hooke’s Law (Retrieved August 2015).