Objectives
The main aims of this experiment are to determine if the oscillation of a mass hung vertically from a spring are an example of simple harmonic motion, to measure the gravitational acceleration of the earth and to determine the accuracy of the experimental results. The value of k (the spring constant) for the spring being used will also be measured and the relationships between the spring constant, mass and period of will also be investigated.
Theory
If a mass is attached to the string, it will stretch and reach equilibrium. The relationship between the spring extension and force applied is described by Hooke’s Law:
F=-kx
where F is restoring force, k – spring constant, and x is for displacement. This relationship is linear, and the minus sign before the right part of the equation indicates that restoring force is exerted in the opposite direction to the displacement (Nave, 2016).
If the spring with a mass attached to it is pulled down and released it will move due to restoring elastic force due to k (Russell, 2016). The motion of the spring is simple harmonic motion with position of the mass changing in a sinusoidal fashion with respect to time. Natural frequency of the harmonic oscillator is described by the equation:
ω=mk.
The frequency itself can be expressed in terms of period, so the formula for the period of oscillation can be derived (Nave, 2016):
ω=T2π →T=2πmk
Results & Calculations
Observations for Part 1:
Using the experimental data, the graph of extension of spring versus the mass of the hanging weight was plotted to find the value of spring constant.
Figure 1. Graph of Extension vs. Mass
The slope of the graph equals 1.64 m/kg. The formula for the slope of this line is g/k, so the spring constant can be found using the following formula:
k=gslope=9.81 m/s21.64 m/kg=5.98 N/m
Actual spring constant for the spring we used in this experimented is reported to be 6 N/m, so the percent error can be found:
%Error=Observed Value-Actual ValueActual Value×100%=5.98-66×100%=0.33%
Observations for Part 2:
Using the experimental data, the graph of squared period of oscillation versus mass of the hanging weight was plotted to find the value of spring constant.
Figure 2. Graph of Squared Period vs. Mass
The slope of the graph comprised 6.94 m/kg. The formula for the slope of this line is 4π2k, so the spring constant can be found using the following formula:
k=4π2slope=39.446.94 m/N=5.68 N/m
Actual spring constant for the spring we used in this experimented is 6 N/m, so the percent error will be:
%Error=Observed Value-Actual ValueActual Value×100%=5.68-66×100%=5.33%
Part3: Calculating the value of gravity:
The slope of the line equals 4.25. This value can be used to calculate the experimental value of gravity. The formula for the slope of this graph is 4π2g, so the value of gravity acceleration is found using the formula:
g=4π2slope=39.444.25 s2/m=9.28 m/s2
Theoretical value of the gravity acceleration is 9.81 m/s2, so the percent error of our experimental value will comprise:
%Error=Observed Value-Actual ValueActual Value×100%=9.28-9.819.81×100%=5.40%
Discussion
Two experimental values of spring constant were calculated, and the experimental value of gravity was obtained. These values comprised:
Part 1, k = 5.98 N/m.
Part 2, k = 5.68 N/m.
Part 3, g = 9.28 m/s2.
The percent errors of calculated experimental values compared to theoretical comprised 0.33%, 5.33%, and 5.40% respectively.
In the first experiment, the graph was linear due to the nature of the relationship with slight deviations from the straight line. The gradient of the line is the tangent of the angle between the line and x-axis. This value was calculated using Excel's function Trend Line, which builds a trend line and returns the equation of the approximation line. The experimental value is very close to the theoretical which is indicated by the small value of the percent error (only 0.33%).
In the second experiment, the graph was also linear. The gradient of the line was obtained with the help of Excel. Percent error in this experiment is significantly higher than in the previous one (5.33% vs. 0.33%). However, its value is still relatively small and almost falls within 5% which indicates that the experiment was carried out without significant errors and with a high level of precision.
In Part 3 of the experiment, the results of two previous experiments were combined to find experimentally the value of gravity. The values of spring extension from the first part and squared period of oscillation from the second part were used to plot a graph. The value of gravity was found from the slope of the approximation line. The percent error was the highest of three experiments and comprised 5.40%. This value, however, is still relatively low and suggests that the experiment was conducted without significant errors.
The experiment can be improved by increasing the number of experiments which will alleviate random and experimental errors. Also, it can be improved by the use of precision instruments.
Precautions
This experiment is relatively safe and poses little possible danger for the experimenters. The basic precaution is to avoid pulling the spring down with weight hanging on it because accidental release may cause the weights to fly from the hanger and hit someone in the face.
Conclusion
In this experiment, the value of spring constant was found experimentally using the method of measuring spring extension and period of harmonic oscillation of the spring. The values found comprised 5.98 N/m and 5.68 N/m respectively. The percent errors of the experimental values were relatively low (0.33% and 5.33% respectively), which shows that no significant mistakes were made over the course of the experiments. In part 3 of the lab, the experimental value of gravity was found using the data from the first and second experiments. The found value of g comprised 9.28 m/s2 with the percent error of 5.40%. Objectives of the lab were fully accomplished.
References
Nave, R. (2016). Hooke's Law. [online] Hyperphysics.phy-astr.gsu.edu. Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/permot2.html [Accessed 31 Jan. 2016].
Nave, R. (2016). Simple Harmonic Motion. [online] Hyperphysics.phy-astr.gsu.edu. Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html [Accessed 31 Jan. 2016].
Russell, D. (2016). The Simple Harmonic Oscillator. [online] Acs.psu.edu. Available at: http://www.acs.psu.edu/drussell/Demos/SHO/mass.html [Accessed 31 Jan. 2016].
