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Grigoriy Kurbatov

Physics Project

GrigoriyKurbatov

09.03.14

PhysicsProject

LoadedSpring Oscillator and Hooke’sLaw

Introduction:

Hooke’slaw or the law of elasticity was first discovered by Robert Hooke, anEnglish scientist in 1660 (Horibe, 2011). Hooke’s law posits thatfor considerably minimal deformations of matter, the deformity sizeor the displacement size is direct proportional to the load ordeforming force (Horibe, 2011). With these conditions in mind, theobject goes back to its initial size and shape upon elimination ofthe weight of an object. According to the law of elasticity, theelastic condition of solids can be elucidated by the fact that minordisplacements of ions, atoms, or molecules from normal positions isproportional to the load that brings about the displacement (Horibe,2011). The force of deformation may be used in solids by compressing,stretching, bending, squeezing, or twisting. Therefore, a metal wireshows elastic condition based on the law of elasticity since theminimal increase in length during stretching because of the forceused doubles every time the force is also doubled. In Mathematics,the law of elasticity or Hooke’s law postulates that the force Fused is equivalent to a constant (k) multiplied to the change inlength or displacement (x). Therefore, F = kx (Horibe, 2011). In thisequation, the value of kisnot merely reliant upon the type of elastic substance being takeninto account but also on its shape and dimension. At considerablymaximum values of force applied, the deformation of substance that iselastic is bigger compared to the expected value on the grounds ofthe law of elasticity, notwithstanding the fact that the substanceremains elastic and goes back to its initial size and shape followingthe deletion of force.

Hooke’slaw defines the elastic characteristics of substances only in theextent to which the displacement and the force are proportional(Horibe, 2011). At times, Hooke’s law is also defined by theequation F = -kx. Generally, in scrutinizing this expression, F doesnot longer apply to the force used and instead signifies theoppositely and equal directed reinstating force that leads to elasticsubstances going back to their initial dimensions and size. Hooke’slaw is likewise expressed in relation to stress and strain. Frommeasurements on certain bodies, the concepts of stress and strainallow the perception of Hooke’s law in general forms. Stress isrelated to the force producing a deformation. Strain is related tothe amount of the deformation. If stretching forced Fareused to the ends of a long elastic rod of square cross section lengthl,therod tends to grow longer and thinner and assumes the elongated shape.The longitudinal stress is described as the force per unitcross-sectional area. In a uniform rod, the stress is equal in allcross-sections. The longitudinal strain is the alteration in lengthper unit length.

Theory:

Theway of returning a system to equilibrium position, where the netforce acting on it is zero and the point of spring upward restoringforce equals to gravitational force of the mass, is calledoscillation. Oscillation system gives off energy. A system that isthrown off-kilter has more energy than a system in its equilibriumposition. As an example, if spring is stretched-out, it starts towave as it is released, because of lack of equilibrium position. Thelaw of conservation of energy states that energy is given to anothersystem when there is work acted upon it (Patterson, 1948, p. 151).This means that there is mere transference of energy from a singlebody to another body. In this type of transfer, no energy is formedor destroyed.According to the law of energy conservation, stretched-out springrequires to lose some energy to get to position of equilibrium.

Harmonicmotion is the movement of an oscillating body.

Theperiod of oscillation, T, of a spring is the amount of time it takesfor a spring to complete a round-trip.

Therestoring force of a spring is directly proportional to displacement,where kis the spring constant.

Springconstant, k, is depended on the material of spring and the dimensionsof the sample.

Aim:

The aimof experiments is to find out the value of spring constant. In thefirst experiment, by oscillating of spring and spending times tooscillate 20 times in variable masses. In the second experiment, bythe changing displacement of spring where the force increases byadding new mass to spring,

Diagram:

Apparatus:

Retortstand, boss and clamp, and G-clamp to fix stand to bench

Steelsprings

Masshungers with slotted masses, 100g

Timer

Meterrule

Shortlength of stiff wire to combine springs in parallel

Methods:

Inthe first experiment, apparatus were set up on edge of the table tostretch springs below the table to prevent hitting the mass on thetable and was fixed by G-clamp to bench. The boss was set up retortstand after the clump was screwed to it. The meter ruler was placednext to the clamp.

Afterthe settings, equilibrium position of mass hunger with 100 grams masson it was found and the hunger was pulled down. The time of 25oscillations were measured and filed in the table results. Therefore,100 grams were added after every 25 oscillations and recordings,until the mass became 700 grams. To get more accurate results it wasrepeated 3 times and then, the period, T, for each mass calculated.

Afterthe first part of experiment with one spring series, the secondspring was added in series and the procedure, which was describedabove was repeated. After all recordings and calculations, the graphwas drawn to find out a spring constant.

Inthe second experiment, apparatus were set up as in the firstexperiment. Also, there was used a stiff wire to hang springs inparallel. There were 3 parts of the experiment: one spring, twosprings in series and two parallel springs. The equilibrium positionof springs with mass of 100 grams on it was measured. After eachadding of 100 grams to mass hanger, displacement was calculated. Thegraph was plotted. Spring constant was calculated by Hooke’s lawand the graph was drawn

Results:

Thefirst experiment:

1 Spring

Mass(kg)

1 Reading(sec)

2 Reading(sec)

3 Reading(sec)

Average(sec)

Period(sec)

T2

0.1

9.90

9.58

9.72

9.73

0.39

0,15

0.2

13.90

13.87

13.91

13.89

0.56

0,31

0.3

17.72

17.61

17.72

17.74

0.71

0,50

0.4

20.0

19.98

20.02

20.0

0.80

0,69

0.5

22.28

22.10

20.83

21.79

0.87

0,75

0.6

24.41

24.31

23.94

24.22

0.97

0,99

0.7

25.85

25.98

24.05

25.96

1.09

1,08

Calculationsof the first experiment:

Gradient= T2= 1,10 =1,630 s2/kg

m 0,675

k=4π2= 24.2 Nm-1

1,63

2 Springs in Series

Mass(kg)

1 Reading(sec)

2 Reading(sec)

Average(sec)

T

T2

0.1

14.70

14.79

14.75

0,59

0,35

0.2

20.18

20.09

20.14

0,81

0,66

0.3

24.91

24.94

24.93

1,00

1

0.4

28.48

28.19

28.34

1,13

1,28

0.5

31.70

31.59

31.65

1,27

1,62

0.6

34.41

34.13

34.27

1,37

1,88

0.7

37.08

36.65

36.87

1,47

2,16

Calculations:

Gradient= T2= 1.2 = 3,2 s2/kg

m 0.375

k=4π2= 12,33 Nm-1

3,2

Thesecond experiment:

One spring

Mass(kg)

1 Reading of extension(cm)

2 Reading of extension(cm)

0,1

3,5

3,5

0,2

7,7

7,8

0,3

11,6

11,6

0,4

15,8

15,6

0,5

16,8

16,8

0,6

23,6

23,6

0,7

29,5

29,6

Two springs in series

Mass(kg)

1 Reading of extention(cm)

2 Reading of externtion(cm)

0,1

7,5

8,2

0,2

15,7

16,5

0,3

23,8

24,7

0,4

32,3

32,8

0,5

40,2

40,8

0,6

48,7

48,9

0,7

56,2

57,1

Two parallel springs

Mass(kg)

1 Reading of extension(cm)

2 Reading of extension(cm)

0,1

0

0

0,2

2,1

2,1

0,3

4,1

4,2

0,4

6,2

6,2

0,5

8,3

8,3

0,6

10,4

10,4

0,7

12,5

12,5

Calculations:

Discussion:

Duringthe first experiment, mass and oscillation were known and used tofind the spring constant. To calculate the period, oscillations arerequired to use them in the formula, where period is time divided bynumber of oscillations. The time, which needs to do 25 oscillationswith each mass (from 0, 1 kg to 0, 7 kg) was measured while 25oscillations, 3 times. The average time was counted and filed intable. The spring constant was calculated as24.2N/m. Inthis experiment, the graph shows an ascending line going to theright. This means that the heavier the object, the longer the time ittakes for oscillation to happen.

Whilethe second experiment, Hooke’s law was used to receive springconstant. The spring constant is … N/m. Theresults of the experiment show that as the mass increases, the forcealso increases. Hence, it can be said that the mass of an object isdirectly proportional to the force applied.

Errorsin the first experiment are 17.95%, because of some aspects. Loadedspring with small masses oscillates faster and counts of oscillationbecome much difficult. The experiment was probably not entirelysuitable the main problem been trying to get the system to oscillateas vertically as possible. If the system oscillates at just 10degrees off the vertical then only 98% of the amplitude actually actson the vertical component of the motion.

Errorsin the second experiment are 1.23%. The springs and mass hunger werenot moving, so the experiment did not have potential for human errors

Conclusion:

Tosum up, while these experiments, it was proved that an easy massattached to a resistant spring will oscillate back forth quickly,whereas a heavy mass attached to an easily stretched spring willoscillate back and forth very slowly. The aim of this experiment wasachieved. The spring constants for each case were found with minorerrors.

References:

Becherrawy,T. 2012, Mechanicaland Electromagnetic Vibrations and Waves,ISTE, WILLEY, London.

Horibe,S. 2011.&nbspRobertHooke, Hooke`s Law &amp the Watch Spring.[online] Available at:http://www1.umn.edu/ships/modules/phys/hooke/hooke.htm[Accessed: 30 Mar 2014].

Patterson,L. D. 1948. Robert Hooke and the conservation of energy.&nbspIsis,pp. 151–156.

Wilchinsky,Z., &quotTheoretical Treatment of Hooke`s Law,&quot Am. J. Phys. 7,134 (1939)

Grigoriy Kurbatov 8