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Introduction A simple pendulum consists of a mass called the pendulum bob suspended from a support by a thread. In a vertical plane, the back-and-forth motion of the pendulum bob from its highest displacement on one side to its highest displacement on the other side is called an oscillation. This motion is periodic. The time taken for a complete oscillation is called the period (T). The number of oscillations completed in a unit time is called the frequency (f). The frequency of the oscillations is the inverse of the period, f = 1/T. When a simple pendulum is displaced, a restoring force will pull the pendulum back towards its equilibrium point. This force changes its direction as the pendulum’s motion moves the pendulum past its equilibirium point so that the force remains pointed towards the equilibrium point. The motion of the pendulum will be a simple harmonic motion if the restoring force is opposite and directly proportional to the pendulum’s displacement from its equilibrium position[1]. The period can then be calculated using the equation[^2 ] where T is the period of oscillation, m is the mass of the pendulum, g is the acceleration of free fall. It can be shown that when a pendulum bob is released from a small angle, the period of oscillation is only depedent on the length of the pendulum and not the bob’s mass or the angle from which it is released. In such cases, the period of oscillation is given by the equation[^1 ]
which can also be written as where T is the period of oscillation, l is the length of the pendulum, g is the acceleration of free fall. This reseach attempts to verify the correlation between the period of oscillation T and the length of the pendulum l , hence verifying the proportional correlation between T^2 and l. Using the result obtained and the above equation, we may be able to obtain a value for g.
1. Research question: To what extent does the length of a pendulum affect the period of oscillation? 2. Independent variable: Length of a pendulum, l (m) Dependent variable: Period of oscillation, T (s) Constant variables: Acceleration of free fall (g), angle of oscillation (°), mass of bob (g), the force at which the pendulum is released 3. Methodology 3.1. Apparatus: Retort stand, split cord, thread, pendulum, stopwatch, metre rule Figure 1. Diagram illustrating the set-up of the apparatus
5. Data l (m) T 1 (s) T 2 (s) Tavg (s) T (s) T² (s²) 0.875 37.52 37.93 37.73 1.89 3. 0.750 34.49 34.68 34.59 1.73 2. 0.650 32.34 32.32 32.33 1.62 2. 0.550 29.75 29.47 29.61 1.48 2. 0.450 27.15 26.95 27.05 1.35 1. 0.350 23.80 23.85 23.83 1.19 1. 0.250 20.00 20.05 20.03 1.00 1. Table 1. Data collected for l , T 1 , T 2 , Tavg, T, and T² Justification of uncertainty 6. Graph Figure 2. Graph of T²/s² against l/cm
7. Conclusions The graph of T^2 against the length l of the pendulum is a straight line of best fit in the form of y = mx + b, indicating a linear relationship between the two values, which is in line with the theory. The shorter the length of the pendulum l , the smaller the value of the period T and the smaller the value of T^2. Though the line of best fit does not go through the origin (0,0), its x-intercept is still extremely close to the origin, which suggests the presence of slight errors in our experiment. Therefore, we can still conclude that our result does verify the relation The gradient of the graph is correspondent with the value 4π^2 / g in the equation. Hence, the value for the acceleration of free fall g can be calculated by dividing 4π^2 by the gradient of the line of best fit The calculated value of g, 9.76 m s-^2 , has a percentage error of 0.5%. This shows that the obtained result is consitent with the accepted value, 9.81 m s-^2 , to a large extent. 8. Evaluation of Plan and Procedures The experiment might have been affected by the below sources of errors: 1) The angle from which the pendulum is released differs from one time to another, 2) The time taken for 20 oscillatons is measured, which could have introduced the effect of friction that slows down the motion of the pendulum, 3) There is human reaction time when using the stopwatch, 4) The force with which the experimenter releases the pendulum is not consistent, causing some pendulums to be released with more momentum than others.
