, the value of g can be found out using a simple equation. The variables in the experiment are ‘g’ which is acceleration due to gravity which is our measured variable; ‘T’ is time period which is the dependent variable which we be observing; ‘l’ is length of the entire pendulum out of which we will be controlling only the length of the string as the length of the hook and the mass bob will be the same; finally ∏ is the constant variable.
The relationship between the variables is such:
- Acceleration due to gravity ( g ) ∞ Length of pendulum ( l )
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Acceleration due to gravity ( g ) ∞ 1/ square of time period ( T2 )
Thus the value of ‘g’ can simply be calculated by putting in the values in the appropriately modified equation.
(A diagram showing the entire setup
of the apparatus.) (A diagram showing the oscillation of a
simple pendulum.)
To conduct this practical work the following apparatus was used:
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A strong table clamp- This piece of apparatus was used for the purpose of providing a base for the pendulum to suspend it from.
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Small wooden blocks- Two small wooden blocks were being used. This was due to the reason that the clamp could not directly hold the pendulum due to the thinness of the string. Thus the string was placed between the wooden pieces and the attached to the hook of the string.
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String- A string of negligible mass is taken which is about 150 to 250 cm long. This forms the main part of the pendulum
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Mass bob- A mass bob of unknown weight is taken. One precaution of taking the mass bob is that it must have a small hook so that the string can be attached to it.
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Metre rule- This is used to measure the length of the string that is the controlled variable.
- Stopwatch- This is used to measure the time taken by a preset number of complete oscillations that we are allowing the pendulum to make.
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Calculator- For processing and calculation a calculator is used to avoid errors.
- Tie the string to the mass bob of unknown weight and suspend it from the table stand as shown in the diagram. Make sure that the two small blocks of wood are in such a position that they clamp the pendulum and they are in turn clamped by the clamp of the stand.
- In order to get proper value that will be finally calculated out I have decided to take 5 readings of the oscillations of the pendulum using four different lengths of the pendulum (20cm to 90cm) so as to remove most possibilities of random errors.
- To begin with the length of the string of the first reading is determined so that the length of the mass bob may also be incorporated in it so as to get a convenient length of the pendulum.
- When the apparatus is set up as shown in the diagram described earlier, the pendulum is taken and set to a particular angle not more than 15 degrees from its mean position.
- the pendulum is then left and it sets into motion. The time period for 10 oscillations is seen and recorded with a stopwatch.
- To obtain the mean value of one oscillation of the pendulum of length 30cm, the total time taken for all oscillations is divided by 10 which is the number of oscillations which the pendulum is allowed to swing for.
- This procedure must be repeated four more times and then after the data has been collected for all five times, the average value of these will be taken as the final reading of the time period of the simple pendulum of length 20cm.
- Repeat the entire process of experimentation with all the different lengths of the pendulums and tabulate the data.
- After tabulation of data, it must be processed to obtain the appropriate results. The data collection and processing aspect is seen later in this report.
On taking readings with vernier calipers, the diameter of the mass bob was found out to be 2.55cm. The vernier calipers showed a positive zero error of 0.09cm. Thus the diameter was found out to be 2.55 – 0.09 = 2.46cm. The radius of the mass bob is thus 1.23cm. The length of the hook again measured with vernier calipers was found to be 1.25cm, which after adding for the zero error will be 1.25 – 0.09 = 1.16cm. Thus the length of the entire system of the mass bob and the hook is 2.39cm.
The lengths of the pendulums will thus be the length of the string added to the length of the mass bob and hook.
The lengths thus are:
- Length of string + Length of the mass bob and hook = 20cm + 2.39cm = 22.39cm.
- Length of string + Length of the mass bob and hook = 30cm + 2.39cm = 32.39cm
- Length of string + Length of the mass bob and hook = 40cm + 2.39cm = 42.39cm
- Length of string + Length of the mass bob and hook = 50cm + 2.39cm = 52.39cm
- Length of string + Length of the mass bob and hook = 60cm + 2.39cm = 62.39cm
- Length of string + Length of the mass bob and hook = 70cm + 2.39cm = 72.39cm
- Length of string + Length of the mass bob and hook = 80cm + 2.39cm = 82.39cm
- Length of string + Length of the mass bob and hook = 90cm + 2.39cm = 92.39cm
The reading for the time periods of the different lengths of the pendulum is:
- For pendulum of length 22.39cm:
- For pendulum of length 32.39cm:
- For pendulum of length 42.39cm:
- For pendulum of length 52.39cm:
- For pendulum of length 62.39cm:
- For pendulum of length 72.39cm:
- For pendulum of length 82.39cm:
- For pendulum of length 92.39cm:
To plot an appropriate graph the plot variables that have been selected by me are length and time period of the pendulums. According to the equation: L = t2. Readings of length of the pendulums are already stated. These are plot variables on their own and do not need to be accounted for errors as they zero error have already been taken care of. The plot variable that needs to be processed is thus t2.
As the value of only t has been taken, the square of time period of 1 oscillation will be the value of t2. On calculating the value of t2 was found out to be:
When we see the modified formula for the finding out the value of acceleration due to gravity, it is:
On applying partial differenciation, the formula we get is:
The uncertainty in the reading of the stopwatch was 0.1s. This is the value of delta t.
For the different values of time period, the uncertainties are:
- For t = 0.95s:
- For t = 1.15s
- For t = 1.32s:
- For t = 1.47s:
- For t = 1.62s:
- For t = 1.73s:
- For t = 1.83s:
- For t = 1.92s:
Thus the error bars are:
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s2
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s2
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s2
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s2
Now using these Values of L and T2 which are the final plot variables, the graph will be plotted, and the slope of this graph will give the value of acceleration due to gravity or ‘g’. The values of L will be plotted on the x-axis and the values of T2 will be plotted on the y-axis.
The slope of the Line of best fit was found out to be 24.44 units. As this was the slope pf the line of, the slope of the line will be = 0.41 units. The value of ‘g’ will thus be. (We have to multiply the equation by 100 as he values of ‘g’ calculated by the graph has to be in standard SI Units, and as we have taken all values of length as in cm, we must multiply by hundred to convert these values to meters). From this we get the value of g to be equal to 9.65 ms-2 which taken from the literature value of it is 0.16 ms-2 away from the mark. Hence the percentage error is 1.63%.
The methods used were according to me accurate and the results were also seen to have come out to be accurate. The conditions under which this experiment was performed were nothing out of the ordinary, as it was done in a standard room. The limitations and the weaknesses are confined to a few random errors and a systematic error which was found in the vernier calipers. To reduce the errors the following precautions were taken:
- Errors in the instrument must be considered beforehand.
- Readings and calculations must be checked.
- The pendulum must not swing in a circular motion but in a straight motion.
- Lengths of the string should be taken after taking in account the stretching of the string.
- Finally, the values must be read correct to the right decimal place and significant figure.
Citations-
- {http://www.saburchill.com/physics/practicals/006.html}. This was referred to for the literature value of ‘g’.
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. This source was referred to for the diagram.
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. this was referred to for the diagram showing the oscillation of a pendulum.
