Ø To investigate the spring constants of springs in series and parallel arrangements.
Ø To investigate the Spring Constant of two different springs.
INTRODUCTION
In this laboratory exercise we were conduct an experiment to determine the spring constant of a spring and then investigate the effect of putting the spring in parallel and in series with another, identical springs.
THEORY
The restoring force, F, of a stretched spring is proportional to its extension, x, if the deformation is not too great – it has not gone beyond its elastic limit. This relationship for elastic behaviour is known as Hooke's law and is described by:
Where k is the constant of proportionality called the spring constant. The spring’s restoring force acts in the opposite direction to its extension, denoted by the negative sign.
For a system like the one shown in figure 1 below, the spring's extension, x, is dependent upon the spring constant, k, and the weight of the mass, mg, which hangs on the spring. If the system of forces is in equilibrium then the sum of the forces down (the weight) is equal and opposite to the sum of the forces acting upward (the restoring force of the spring):
Figure 1: Illustration of Hooke’s Law. As additional weights are added, there is a linear increase in the length of the spring as long as the elastic limit has not been reached
SPRINGS IN SERIES AND IN PARALLEL
a. SPRINGS IN SERIES
Figure 2: Springs in series
If we load the first spring with a weight W, we see that it extends by x. Now we attach a second identical spring in series, and put on the weight W. The same force acts through each spring, so each spring stretches by x. Therefore the total stretch is 2x. Since k = F/x, the spring constant with two springs in series ks, is halves. It is given by the formula:
Where ka is the spring constant of one of the springs and kb the spring constant of the other.
b. SPRINGS IN SERIES
Figure 3: Springs in parallel
If the springs are arranged in parallel and a weight W is attached it extends x/2, half the extension that it would be for only one spring, since the stretching weight is shared between them. k = W ÷ 0.5x = 2k. The new spring constant kp is twice that of an individual spring. For a parallel arrangement the formula is:
Where ka is the spring constant of one of the springs and kb the spring constant of the other.
APPARATUS AND MATERIALS
- Springs
- Rulers (1m and 30cm)
- Clamp stand, boss and clamp
- Slotted Masses.
METHODOLOGY/PROCEDURE
1. The clamp, stand and boss; were set up. We made sure that the stand is balanced and not likely to tip over.
2. The spring was suspended from the clamp and carefully the un-stretched length of the spring was measured.
3. We attached the mass hanger to the end of the spring, and measured the new length of the spring, we calculated and recorded the extension in table of results. The extension was obtained by finding the difference between the un-stretched spring and the spring with mass on it.
4. We repeated steps 1-3 twice in order to obtain a set of data which will be averaged to obtain spring extension hence increasing the accuracy of the experiment.
5. We repeated steps 1-4 while increasing masses on the mass hanger stepwise.
6. The final length of the spring when the masses have been removed was measured and recorded.
7. We repeated the above procedure, for two springs in series.
8. We repeated the above procedure, for two springs in parallel.
ARRANGEMENT OF THE APPARATUS
Figure 4: Apparatus arrangement
POTENTIAL HEALTH AND SAFETY ISSUES
Before we carried out the experiment we identify the possible accident that we may encounter while carrying out the experiment as:
1. Being fallen on by mass weights.
2. Being injured by the spring.
As a safety measure we put on safety boots and dust coat. We made sure that we handled the equipment carefully to avoid injuries.
RESULTS
TABLE OF RESULTS FOR A SINGLE SPRING
MASS(m)-kg
FORCE(F)-N
EXTENSION(e) 1-cm
EXTENSION(e) 2-cm
EXTENSION(e) 3-cm
AVERAGE EXTENSION(e)- m*10-2
0.10
1.00
6.80
6.90
7.00
6.90
0.20
2.00
14.70
14.80
14.60
14.70
0.30
3.00
21.30
21.10
21.40
21.27
0.40
4.00
29.00
28.90
28.70
28.87
0.50
5.00
36.50
36.60
36.70
36.60
0.60
6.00
43.60
43.70
43.30
43.53
0.70
7.00
50.80
50.80
50.70
50.76
0.80
8.00
57.80
57.70
57.90
57.80
0.90
9.00
64.70
64.60
64.70
64.60
1.00
10.00
72.00
72.10
72.30
72.13
We converted the mass to force using the formula:
The average extension was found by applying the following formula
TABLE OF RESULTS FOR TWO SPRINGS IN SERIES
MASS(m)-kg
FORCE(F)-N
EXTENSION(e) 1-cm
EXTENSION(e) 2-cm
EXTENSION(e) 3-cm
AVERAGE EXTENSION(e)-m*10-2
0.10
1.00
3.80
3.70
3.60
3.70
0.20
2.00
7.70
7.50
7.60
7.60
0.30
3.00
11.40
11.50
11.60
11.50
0.40
4.00
15.40
15.30
15.60
15.43
0.50
5.00
19.20
19.30
19.40
19.30
0.60
6.00
22.90
22.80
22.70
22.80
0.70
7.00
23.80
24.00
23.90
23.90
0.80
8.00
30.30
30.20
30.40
30.30
0.90
9.00
34.20
34.30
34.20
34.23
1.00
10.00
38.50
38.30
38.40
38.40
We converted the mass to force using the formula:
The average extension was found by applying the following formula
TABLE OF RESULTS FOR TWO SPRINGS IN PARALLEL
MASS(m)-kg
FORCE(F)-N
EXTENSION(e) 1-cm
EXTENSION(e) 2-cm
EXTENSION(e) 3-cm
AVERAGE EXTENSION(e)-m*10-2
0.10
1.00
1.10
1.30
1.20
1.20
0.20
2.00
3.60
3.40
3.50
3.50
0.30
3.00
4.90
4.80
5.00
4.90
0.40
4.00
6.70
6.50
6.60
6.60
0.50
5.00
8.30
8.40
8.20
8.30
0.60
6.00
10.30
10.40
10.60
10.43
0.70
7.00
12.10
12.20
12.30
12.20
0.80
8.00
14.10
13.90
14.30
14.10
0.90
9.00
15.80
16.10
15.90
15.93
1.00
10.00
17.50
17.60
17.70
17.60
We converted the mass to force using the formula:
The average extension was found by applying the following formula
PLOTS
I. SINGLE SPRING
Figure 5: Plot of spring extension against force for a single spring
From the plot the spring constant (reciprocal of the gradient of the curve) can be calculated as follows:
Picking the two points on the plot:
II. TWO SPRINGS IN SERIES
Figure 6: plot of spring extension for two springs in series
From the plot the spring constant (reciprocal of the gradient of the curve) can be calculated as follows:
Picking the two points on the plot:
The spring constant for a single spring can be calculated using equation 3:
III. TWO SPRINGS IN PARALLEL
Figure 7: plot of spring extension against force
From the plot the spring constant (reciprocal of the gradient of the curve) can be calculated as follows:
Picking the two points on the plot:
The spring constant for a single spring can be calculated using equation 4:
DISCUSSION AND CONCLUSION
DISCUSSION
From the three plots it can be concluded that Hooke’s law holds to be true that is the restoring force, F, of a stretched spring is proportional to its extension, x, if the deformation is not too great – it has not gone beyond its elastic limit. From the calculated value it be observed that the spring constant used in the experiment is approximately 13.6252 N/M. From the plots it can be observed that some of the points fell outside the curve since there was an error in obtaining this data. This error was caused by faulty measuring equipment and parallax in reading data from the measuring equipment. However the equipment used in the experiment were accurate since we were able to obtain the best results as depicted in the plots.
The accuracy of the results can be improved by iterating the process of obtaining the data so as to increase the accuracy. Faulty equipment should be replaced by better ones.
CONCLUSION
In conclusion, the objectives of the laboratory exercise were all met. We were able to determine the spring constant of the two similar springs as 13.6252 N/M. We were also able to determine the spring constant for springs in series and parallel. Therefore the lab exercise was successful.
REFERENCE
Bueche, F. J., Introduction to Physics for Scientists and Engineers, Third Edition, McGraw-Hill, N.Y. (1980).
Wilchinsky, Z., "Theoretical Treatment of Hooke's Law," Am. J. Phys. 7, 134 (1939).
Sears, F. W., Zermansky, M. and Young, H. D., University Physics, 5th Edition, Addison-Wesley, N. Y. (1981), as cited in Yost, S. A., “The effect of spring mass on the oscillation frequency”, http://homework.phys.utk.edu/courses/spring2002/phys221/spring.pdf.
Peckham, D., “Hooke’s Law: St. Lawrence University Physics Dept. Phys 307 Laboratory Instructions”, Canton, N. Y. (1983), [it.stlawu.edu/~physics/InsertMadeupURLhere.html]
