The relationship between the length and the period of motion of a pendulum

Introduction: I chose to investigate this topic out of pure curiosity to see how the length of a pendulum affects its period of motion. A pendulum is a suspended point of mass, hung from a fixed point on an inextensible cord. When it is pulled and released from one side of its equilibrium, at x°, the pendulum swings back and forth on a vertical plane under the influence of gravity (La Né Powers, 2006). The motion is periodic and oscillatory; I am determining the oscillation or otherwise known as the period of motion (Resnick & Malliday, 1977, pp. 310-311). The period of motion is the amount of time taken to swing back and forth once, measured in seconds and symbolised by T (Kurtus, 2010). Galileo discovered pendulums and he found that the period of motion is proportional to the square root of the length - T∝√l (Morgan, 1995).

Due to the research carried out, I have discovered that the correct method of measuring the independent variable (length of the string) is from the fixed point it is hung from (fulcrum) to the center of the mass (Cory, 2004)(Encyclopedia Britannica, 2011). The formula F=-mg sin⁡θ shows that when a pendulum is displaced from its equilibrium, it is brought back to the center by restoring force ("Pendulum," 2008). Newton’s second law, F=Ma=(d^2 (Lθ))/(dt^2 ) , shows that the arc which the pendulum swings through is actually a segment of a circle – with the radius being the length of the pendulum. The combination of these formulae demonstrates that the mass of a pendulum is independent to its period of motion (Encyclopedia Britannica, 2011). I concluded from this that a specific weight for my pendulum is not necessary, although it must remain constant. As seen in the above equation, this restoring force is...

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...of motion (T), measured in seconds and milliseconds. Time is recorded for five periods and averaged (T=t/5). Repeated five times for each length and averaged.

Constant variables: the environmental conditions (enclosed indoor area), the weight of the pendulum, repeated the same amount of times for each length, released from 10°, and the pendulum is released with the same tension in the string each time

Equipment:

160cm of 8 strand braided nylon bricklayer’s line

17.07grams worth of 5/16” zinc plated mudguard washers

Scientific scales reading from 100-0.01grams

A stopwatch measuring to the milliseconds

Spring clamp with a hole in the handle

Blu-Tack

180° protractor

A capable assistant

Stool (if needed)

Procedure:

Clamp the spring clamp to an object over 160cm high without obstructions underneath and with the hole facing downwards.