The document discusses the principles of a simple pendulum and its motion, relating it to simple harmonic motion with energy transformations between kinetic and potential energy. It outlines an experiment to measure the period of the pendulum and derive gravitational acceleration by analyzing the relationship between the length of the pendulum and the square of its period. Key factors affecting the motion and measurements are detailed, including the required materials and methodology for conducting the experiment.

1. SIMPLE PENDULUM
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By Aditya Abeysinghe

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3. A simple pendulum when given a small displacement obeys
simple harmonic motion.
Thus, the energy conversion is as follows:
1. Maximum kinetic energy at the base
2. Maximum potential energy at the amplitude
3. Conservation of mechanical energy at any point in its
motion
Now let’s examine as to which factors affect the motion of a
simple pendulum.
PRINCIPLE BEHIND A SIMPLE
PENDULUM

4. It should be noted that
the tension is balanced
by the component of
weight mg cosθ.
The other component
of the weight mg sinθ is
used for the acceleration
of the object.
By newton’s second law of motion,
F = ma
F = -mg sinθ
Since θ is small, sinθ ≈ θ rad. Therefore, F = -mgθ
θ
T
mg
l
x

5. However, S = rθ and hence x = lθ. Therefore, θ = x /l
Thus, F = -mgx/l
By newton’s second law,
F = ma
Therefore, ma = -mgx/l and a = - (g/l) x
This is in the form a = -ω2x.
Therefore, the object obeys simple harmonic motion.
And ω = g/l.
However, T = 2π/ω
Finally, we get T = 2π √ (l / g)

6. Moreover, we can write the equation as follows:
T = 2π √ (l / g)
By squaring both sides,
T2 = (4π2 / g) l
This is in the form y = mx
Where the gradient of the graph = 4π2 / g
By performing the experiment and plotting a
graph, we can easily find the gravitational
acceleration using the gradient.

7. Required materials:
• Simple pendulum
• Stop watch
• Meter ruler
• Stand with a fixed pointer
• A weightless string
THE EXPERIMENT

8. 1. Place the apparatus and keep the length of the
pendulum to be 2m.
2. Place the pointer close to the lowest point of the string
3. Displace the string either clockwise or
counterclockwise/anticlockwise and then release the
string
4. At the first instance when the string passes the
pointer the stop watch is activated.
5. Take the readings to 50 complete revolutions.
6. Repeat the above experiement if the difference
between the time periods of revolutions is greater
than 0.5s
7. Finally, decrease the length of the string by 0.25m
gradually and draw the graph between l and T2
METHOD

9. 1. Use a string since when the string is in
motion, the increase the length of the string
2. To prevent changing the length of the string, the
string should be swayed perpendicular to the
stopper, to where the string is connected above.
3. The length of the string, l, should be measured
from the center of gravity of the weight to the
stopper above.
4. By keeping the pointer at the lowest point of the
string’s path of motion, the time can be
accurately measured since the string sways at its
highest speed at the lowest point
IMPORTANT POINTS