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SHM: determining acceleration due to gravity

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Introduction

Student Name: WONG Wing Yan

Class and class no. : 6G2 (22)

Date of experiment: 25th March, 2009

Experiment 5 – SHM: determining acceleration due to gravity

Objective

In this experiment, we are going to study the simple harmonic motion of a simple pendulum. The acceleration due to gravity (g) can be estimated by the following set-up.

Apparatus

  • Retort stand and clamp
  • String 1.5m long
  • Protractor
  • Pendulum bob
  • G-clamp
  • Metre ruler
  • Stop watch

Setup


Theoryimage11.pngimage10.pngimage09.pngimage03.pngimage47.pngimage04.pngimage02.pngimage14.pngimage45.pngimage48.jpgimage02.pngimage13.pngimage46.pngimage49.gifimage34.pngimage07.pngimage44.pngimage08.pngimage12.pngimage05.pngimage23.pngimage06.pngimage00.pngimage01.png

A simple pendulum can perform a simple harmonic motion as shown in figure 2. The acceleration due to gravity can be determined if we know the period (T) of the SHM and the length of the string (image55.pngimage55.png).image15.png

image16.pngimage19.pngimage18.pngimage17.png

image20.pngimage21.pngimage22.png

image24.png

image26.pngimage25.pngimage27.png

image28.pngimage29.png

image30.pngimage31.png

image32.pngimage33.pngimage35.png

image37.pngimage36.png

        A light string with its upper end fixed and lower end attached to a pendulum bob mass m is shown. When the string is held to make and angle image65.pngimage65.png with the vertical, the bob displaces an arc length image71.pngimage71.png.

...read more.

Middle

 (e.g. < 10image52.pngimage52.png), we have sinimage51.pngimage53.pngimage53.png and therefore sinimage54.pngimage54.png. Hence, it becomes:

image56.png

As the bob continues to move, it performs a simple harmonic motion with angular velocity (image57.pngimage57.png) and it has an acceleration (image58.pngimage58.png). By comparing it with the equation, we have:

image59.png

However, the period can be shown as follows:

image60.png

image61.png

image62.png

Note that period of the simple harmonic motion is independent of the mass.

Procedure

  1. The apparatus was set up as shown in Figure 1.
  2. The bob was moved to a height so that the attached string was taut and an angle of 10image52.pngimage52.png was made with the vertical.
  3. The period (t1) for 20 complete oscillations was measured and it was recorded in the following table.
  4. Steps 2 and 3 were repeated for another measurement of period (t2).
  5. Steps 2 to 4 was repeated by shortening the length of the string by 5 cm each time for measuring further 7 sets of data.
  6. The graph of the square of the period (T2) against the length of the string (image55.pngimage55.png) was plotted.
  7. A best-fitted line was drawn on the graph and the slope was measured.
  8. The acceleration due to gravity (g) was calculated from the equation, its slope is given by:

image63.png

image64.png

Precautions

...read more.

Conclusion

Conclusion

        In this experiment, we try to study the simple harmonic motion of a simple pendulum. By looking for the value image82.pngimage82.png , we can estimate the acceleration due to gravity (g).

We compare the experimental value to theoretical value and find that the experimental one is a bit smaller than the theoretical value, which is due to the error of this experiment.

It is also clear to period is independent of the mass of the body. From the equation :

image83.png

we can know the relationship between T and image55.pngimage55.png, which is image84.pngimage84.png.

image38.png

image39.png

image41.pngimage40.png

image43.pngimage42.png

We can improve the experiment by measuring the angle carefully but not larger than 10image52.pngimage52.png. We can also reduce the surface area of the weight, then we can reduce the air resistance. The time for more periods of oscillations (like 40 periods) could be taken for minimizing the measurement error. A more precise result could be obtained.

Reference

  • New Way Physics for Advance Level – Book1(Mechanics) ; Manhattan Press (H.K.) LTD, page 263-265
  • http://en.wikipedia.org/wiki/Simple_harmonic_motion

...read more.

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