INTRODUCTIONHooke's Law states that the restoring force of a spring is directly proportional to a small displacement. In equation form, we writeF = -kxwherexis the size of the displacement. The proportionality constantkis specific for each spring. If it is not stretched to the point where it becomes permanently deformed, the behavior of a properly wound coiled spring, when subjected to a stretching force, can to follow Hooke's Law. 1.[Bueche, p. 95] (Note that Hooke's Law applies more generally too many more systems than just ordinary springs.) To see whether an ordinary screen door spring behaves similarly, one such spring was suspended by one end from a horizontal support and masses were hung from its other end to stretch it. The resulting data were used to construct a graph of load as a function of elongation, from which it was possible to obtain the spring constant of the spring. In addition, for one value of load the spring was given a small additional stretch and released, thereby setting the system into vertical oscillation. Assuming this motion to be simple harmonic, its period also yields a spring constant, thereby providing an additional check.

OBJECTIVETo verify the equation and determine the relationship of frequency and amplitude and verify Hookes law and to find the combined spring rate of the two springs used on the spring-mass system. Next, verify the mass/frequency relationship of a vibrating spring-mass system and to determine approximate values for gravitational acceleration and the effective weight of the spring.THEORYThe swinging pendulum of an old fashion clock, or quartz crystal in a modern watch, the motion of the piston in a car engine, a beating heart, the alternating current of electricity, the vibration of atoms and molecules about their equilibrium position as well as sound, radio, and light waves - all undergo a periodic or oscillatory motion, which repeats itself regularly in time.The study of the property of a periodic motion will be performed with a mass m suspended on a spring Fig.1. There are two forces act on the mass: the force of gravity and the restoring force of the spring. 2.In a position of equilibrium the net force on the mass equals zero, so these two forces are equal and aim at opposite direction. The spring exerts the force in the opposite direction of elongation or contraction. The force F is called a restoring force and is described by Hookes Law.F = - kx , (1) where k is the spring constant, a characteristic of the spring, and x is the change of the spring elongation or contraction.

If a body, which obeys Hookes Law, is displaced from equilibrium and released, the body will undergo simple harmonic motion (SHM). Many systems, such as water waves, sound waves and ac circuits, exhibit this type of motion.A particularly easy example to study is a mass on a spring. This system will undergo simple harmonic motion (SHM) with a period, T, given by Equation (2).

A particularly easy example to study is a mass on a spring. This system will undergo simple harmonic motion (SHM) with a period, T, given by Equation (2).

(2)where k is the constant from Hooke's Law and m is the combined mass, defined by Equation (3).

M = mass of object & pan on spring + mass of spring (3)

Apparatus

Figure 2: Cussons Vibrator Drive Unit

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Figure 3: P 1906 Linear Vibration Apparatus

PROCEDURE

1) Three tests were conducted using linear vibration apparatus. They were spring rate test and spring-mass system test.2) In Spring Rate Test, the vibrator (Figure2) was set up and the height of the suspension bar was adjusted until the lower restraining spring was extended by 40mm without load units on the carrier. 3) The rule was aligned at zero on some convenient point on the carrier then six load units were added progressively. 4) The results were tabulated and then plotted in a graph of change in extension against change in load. The stiffness of the spring system was determined from the slope of the graph.5) In spring-mass system test, the apparatus was set up as (Figure 3) for the spring-rate test. The vibration generator was connected to the drive unit and the keeper rod was removed. 6) The system was vibrated over the frequency range 4-20Hz and the natural frequencies of vibration for a full range of mass loads were established. The graph of against mass loads was plotted from the results.

Safety and Precaution1. The electrical supply voltage, frequency and configuration is ensured that are compatible with the equipment which requires a single/ three phase supply and a reliable protective earth or ground connection.2. The equipment is isolated from the electrical supply when it is not in use.3. The equipment is always isolated from the electrical supply before removing the front or rear panels to prevent expose of any part that in operation, carries a potential of 30V dc. Or 50 V ac.

Results:

Spring Rate TestTable 2: Spring extensions under different load unitsLoad Units, WExtensions, E (mm)

13.0

25.0

38.0

410.0

513.0

615.5

Figure 1: Graph of extensions against load unitsSpring-Mass System TestTable 3: Frequency and period at resonant point under different load unitsLoad Units, WFrequency, F (Hz)Period, TpTp

016.50.0610.004

18.50.1170.014

26.50.1540.024

35.40.1850.034

44.70.2130.045

54.30.2310.054

64.00.2500.062

Figure 2: Graph of Tp2 against load units

Analysis:From the graph (figure 3),Gravitational acceleration, g = = = 9.87 m/s-2Effective weight of the spring: + carrier = OD + = 0.4w = 1.193g

Discussions:From Figure 1, the extension of the spring increases linearly with the increase of load units. This verified the Hookes law where the stress is proportional to strain. Errors may exist in this experiment due to parallax error. So, random error exist can be reduced by conducting the experiment few more time to obtain more consistent & accurate results and also by eliminating uncontrolled variable or properly shielding or grounding the measuring system.In Figure. 2, the period at resonant point increase linearly with the increase of load units. The calculated gravitational acceleration, g and effective weight of the spring are 9.87 m/s-2 and 1.193g respectively. The calculated values deviate from the actual value because the system resonates over a very narrow frequency range and the resonant point is hard to identify.

Conclusion:Amplitude of a vibrating system decreases with increasing frequency and spring obeys Hookes law. It was also found that frequency of a vibrating system decreases with increasing load where gravitational acceleration and effective weight of spring can be obtained by plotting mass /frequency relationship.

References

1) Figiola, R. S., Frederick Bueche (2011). Theory and Design for Mechanical Measurements: Fifth Edition. New York, MA: John Wiley & Sons.

2) R.Keith Mobley., (1999). Vibration Fundamentals. Wildwood Avenue, MA: Butterworth - Heinemann, Inc., pg 193.

3) Hall, J. G., Allanson, J. E., Gripp, K. W., & Slavotinek, A. M., (2007). Handbook of Physical Measurements: Second Edition. New York, MA: Oxford University Press.

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