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# Simple Harmonic Motion of a mass-spring system.

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Introduction

Immanuel Lutheran College

S.6 Physics (AL) 2003-2004

Experiment Report 1

Name:  Lam Kong Lan             Class:  6B              Class No.:  7

Title:  Simple Harmonic Motion of a mass-spring system

Objective:

1. To investigate the motion of a spring-mass system undergoing oscillation and to verify the relationship between the period of oscillation of a mass hanging from a spring and the mass.
2. To find out the force constant and effective mass of the spring.

Apparatus:

• Light spring
• Stop-watch
• Horizontal bar
• Balance
• Retort stand and clamp
• Slotted mass with hanger 2 ×100g and 5 × 20g

Theory:

By Hooke’s Law, for a mass m hanging from a spring, at the equilibrium position, the extension e of the spring is given by mg = ke where k is the force constant of the spring.

Let x be the displacement of spring from the equilibrium position, then we have an expression of the net force acting on the mass as  Fnet = -k( e + x ) + mg = -kx.

Here, the negative sign means that it is a restoring force and the direction of Fnet is always opposite to x.

Moreover, according to Newton’s second law, the equation of motion: Fnet = ma

∴  Fnet = ma = -kx,    then a = - (k/m)x = -w2x

∴ = As the mass m

Middle

Results:

(a)  Tabular form

 Mass Time for 20 oscillations (20T) Period of oscillation ( Period )2 m/kg t1/s t2/s mean t/s T/s T2/s2 0.02 4.83 4.89 4.86 0.243 0.059 0.04 5.73 5.91 5.82 0.291 0.084 0.06 6.49 6.41 6.45 0.322 0.104 0.08 7.55 7.24 7.40 0.370 0.137 0.10 8.50 8.38 8.44 0.422 0.178 0.12 9.33 9.29 9.31 0.465 0.216 0.14 9.95 10.72 10.34 0.517 0.267 0.16 10.86 11.08 10.97 0.549 0.301 0.18 11.78 11.67 11.73 0.586 0.343 0.20 11.19 13.34 12.26 0.613 0.376 0.22 12.86 12.81 12.84 0.642 0.412

(b)  Graphical form Slope of the graph = =1.82,   k = = 21.7Nm-1

Conclusion

1. Random errors arised because of reaction times of the timekeeper, which is about 0.02s for a normal person.
2. Reading errors arised when weighing the mass of the spring.          The maximum possible error will be ± 0.01g.
3. Errors arised when counting the number of oscillation of the mass.

(i)  When the mass used is very small, just about 20g, T is very short that the oscillation occurs at a very fast rate, for about 20 oscillations within a few seconds.  It is very difficult to observe one oscillation by the human eyes.  As a result, errors exist.

(ii)  When the mass used is very great, about 240g, T is long enough for observation.  However, as the value of the effective mass is very small, the large value obtained for T2and m will result in a great deviation of the value of the effective mass.

1. Modification of the experiment:

Use a more accurate balance or an electronic balance instead.

Reference:

1. Demontration of the experiment ( a video ) http://bednorzmuller87.phys.cmu.edu/demonstrations/oscillationsandwaves/periodicmotion/demo212.html

2.  Details of the experiment

http://physics.nku.edu/GeneralLab/211%20Simple%20Harmonic%20Spring.html

3.  Theory is based on Raymond W.N. Chan’s Physics Beyond 2000 ( 2nd Ed.) P. 77-78

1. Tao, Lee & Mak’s A-Level Practical Physics (2nd Ed.) P.35, 36

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