Posted: May 12th, 2022

Determination of acceleration Due to Gravity

Determination of acceleration Due to Gravity

Results and Graphs
Table 1: Average T2 vs Mass (kg)
Average T2 Mass (kg)
0.67 0.1
0.84 0.15
0.96 0.2
1.04 0.25
1.27 0.3
Table 2: Distance (m) vs Mass (kg)
Distance (m) Mass Kg
0 0.05
0.026 0.1
0.048 0.15
0.061 0.2
0.081 0.25
0.096 0.3

Fig 1: T squared vs mass

Fig 2: Distance vs mass
Question #1: Simple harmonic motion is a type of motion in which a restoring force which is applied is proportional to the displacement and acts in the opposite direction. The motion is similar to the oscillatory motion where there is maximum displacement the frequency among other factors that determines motion of oscillatory bodies.
Question #2:
Gradient from the second graph=(0.3-0.15)/(1.3-0.8)=0.15/0.5=0.3 kg/s^2
gradient=(0.3-0.1)/(0.1-0.02)=0.12 kg/m
k=(4π^2)/0.12=32.9868 N/Kg,
Intercept=0.05 kg
Acceleration due to gravity =0.3×32.9868=9.896 m/s2±0.086 . The difference in the error is not of a bigger margin which implies that minimal errors were made during the experiment.
Question #3: The graphs are both straight line graph but do not pass through the origin. Both graph shows positive correlation. The graph intercepts at the mass of 0.05 and hence due to experimental errors, the graph could not pass through the origin. The graph is a straight line which is a clear indication that it is proportional.
Question #4: The errors experienced were the error due to the timing of counting the oscillations and error due to friction in the spring. The other type of error is the weight due to the spring which effects on how the oscillations are determined during a given period of time.
Both errors are of the random type. This can be attributed to the precision of the instrument. They are also considered random because they are easy to be determined and hence corrected.
The magnitude of the error can be calculated from using the standard deviation. The values were within range of 0.12±0.08 sd. The value was chosen because it was within the range of values obtained during the experiment. The standard deviation samples close values that are calculated from the experiment and hence determining the magnitude and the range of the error.
Question #4b:
The accuracy can be improved by the use of a stiffer spring with a known spring constant and reduction of the energy loss in the spring. The error could also be reduced by proper timing during measuring of the oscillations. In addition the accuracy can be minimised by positioning the spring in a position that its weight does not affect the final results
Question #5: I would count a lesser number of oscillations because a stiffer spring has a high spring constant which results in the reduction of the movement of the spring. The spring constant is a property of the spring, therefore energy loss in stiffer spring is high than in less stiff spring.
Question #6: The use of multiple parameters is important because they are used in the reduction of the possible errors during measurements and calculations. The parameters are treated separately and therefore there is distribution of error thereby increasing the accuracy of the results. The parameters are used in determination of the parameters being investigated
Question #7: Most of the objectives have been met during the experiment because most of the parameters that affect the acceleration due to gravity are well determined during the experiment. The value of the spring constant was also determined which was used in the calculation of the acceleration due to gravity. The other factors affecting the experiment such as external forces were investigated. Error determination is important and was used to demonstrate the extent of the errors and the factors leading to the errors. This is important because it minimises future error in similar situations.
Section B
Question #1: The oscillation period decreases due to the increase in the displacement, the spring acquires more energy hence increase in the maximum velocity. The maximum velocity is mainly as a result of increase in the displacement. Generally the process of the movement of the spring is as a result of change of energy from one form to another. From potential energy which is on the spring when it is being released, it is then converted to kinetic energy and the process repeats itself.
Question #2:
g=GM/R^2 =(〖6.67×10〗^(-11)×〖5.98×10〗^24)/〖6380〗^2
=〖39.8866×10〗^13/〖4.07044×10〗^7
=9.799 m/s^2
Source:Theory.uwinnipeg.ca
Question #3: The gradient can be calculated as follows:

gradient=(0.3-0.1)/(0.1-0.02)=0.12 kg/s^2
k=(4π^2)/0.12=32.9868 N/Kg,
k=(4π^2)/0.12=32.9868 N/Kg
The graph obtained is a straight line graph which can be summarised as the force applied on the spring is directly proportional to the extension unless the elastic limit is exceeded. It therefore obeys the Hooke’s Law. The spring constant of the spring indicates the stiffness of the spring.
b) Gradient=(0.30-0.15)/(0.096-0.048)=0.15/0.048=0.3125 kg/s2
(0.12-0.3125)/0.12×100=-1.604%
Question #4:
Gradient from the second graph
=(0.3-0.15)/(1.3-0.8)=0.15/0.5=0.3 kg/m
We use the following points:
(0.026, 0.1), (0.096, 0.3)
Gradient=(0.3-0.1)/(0.096-0.026)=0.2/0.7=0.2857 kg/m
% gradient=(0.3-0.2857)/0.3×100
=4.7667%
Acceleration due to gravity =0.3×32.9868=9.896 m/s2±0.086
Question #5: The error calculated is calculated as follows
〖4.7667〗^2×〖-1.604〗^2= 22.72%+2.573%
=25.29%
The results indicate the propagation of errors during the experiment.

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