# 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:

- 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.
- 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

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

- Modification of the experiment:

Use a more accurate balance or an electronic balance instead.

Reference:

- 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

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

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