1 | P a g e

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.

2 | P a g e

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 − 𝑥𝑜2), 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)

Force

Constant

(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 Force vs. Displacement graph for Table 1A:

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 + 5.2927

2)

%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

Forc

e

Displacement

GRAPH 1A: FORCE VS.

DISPLACEMENT

5 | P a g e

Table 1B. Determining the Force Constant of

8N/m Spring.

TRIAL Mass

(kg)

Force

(N)

X

(m)

Force

Constant

(N/m)

1 0.02 0.196 0.027 7.2593

2 0.03 0.294 0.04 7.3500

3 0.04 0.392 0.053 7.3962

4 0.05 0.49 0.065 7.5385

Average

Slope of the Line

Percent Difference

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 + 7.3860

2)

%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 − 𝑥𝑜2), 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

Forc

e

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 final

displacement

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%

Table

1B 0.0156 J 0.0154 1.31%

Sample Computation for Table 2:

Trial: Table 1A

Work:

𝑊 =1

2𝑘(𝑥𝑓

2 − 𝑥𝑜2)

𝑊 =1

2(5.2927𝑁

𝑚)(0.0922 − 02)

𝑊 = 0.0224 𝐽

Area under F vs. x graph:

Equation of the graph: y = 5.3542x - 0.0037

𝐴𝑟𝑒𝑎 = ∫ (5.3542𝑥 − 0.0037)𝑑𝑥0.092

0

𝐴𝑟𝑒𝑎 = 0.0222

Percent Difference:

% 𝑑𝑖𝑓𝑓 =|𝐸𝑉1 − 𝐸𝑉2|

(𝐸𝑉1 + 𝐸𝑉2

2)

% 𝑑𝑖𝑓𝑓 = |0.0224 − 0.0222|

(0.0224 + 0.0222

2)

%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).

https://en.wikipedia.org/wiki/Hooke%27s_law

8 | P a g e