1. In this presentation you will:
Learn about the pendulum and its properties.
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2. In this presentation you will learn about the pendulum and its cycle, period, frequency and amplitude. You will learn which factors affect its behaviour and how can they be calculated.
You will find out about the representation of the motion of a pendulum on a graph. You will learn about the concept of ‘simple harmonic motion’ and frequency.
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3. The Pendulum
A pendulum consists of a mass (bob) suspended from a fixed support on a string so that it swings back and forth in a regular (periodic) way under the influence of gravity.
The period of the pendulum is the time that it takes to complete a one whole oscillation or cycle.
cycle
The number of oscillations per unit of time (normally per second) is called the frequency.
The maximum distance that the bob swings from its central equilibrium position is the amplitude.
amplitude
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4. Pendulum motion in a graph
As the pendulum swings, it describes a movement that can be represented along a time axis as in this graph.
As you can see, period, frequency and amplitude are represented in this graph too.
The path of the motion varies harmonically and sinusoidally. That is, like a sine (or cosine) curve.
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5. Question 1
What three important factors related to the oscillation of a pendulum can be represented in a graph?
A) harmonic motion, displacement and restoring force.
B) time, displacement and the sine of the angle.
C) acceleration, periodic motion and oscillation.
D) period, frequency and amplitude.

6. Simple Harmonic Motion
The pendulum is a type of an oscillator which illustrates the concept of simple harmonic motion (SHM).
That is, the periodic motion that occurs when there is a restoring force opposite to and proportional to the displacement.
Motion
In a SHM pendulum, the acceleration of the bob is proportional to the displacement from the centre of motion, which is considered to be its central position.
Restoring Force
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7. Frequency The period (T ) of a pendulum swing is measured in seconds.
Since frequency (f ) is the number of oscillations (cycles or vibrations) in a second, we obtain that:
This unit of frequency is given the name ‘hertz’ (Hz) so 1 Hz = 1s-1.
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8. Frequency
For example, assume that a pendulum takes 3 seconds to complete a one whole cycle.
This means that in one second, it would make 1/3 of a cycle. So its frequency will be:
In many oscillating applications, frequencies are so high that need to be represented in higher multiple conventions like kilohertz (kHz = 103 Hz), megahertz (MHz = 106 Hz) or gigahertz (GHz = 109 Hz).
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9. Question 2 What is frequency? A) The time taken to complete a cycle.
B) The number of oscillations in a second.
C) The number of seconds in a cycle.
D) The maximum distance that the bob swings from its central equilibrium position.

10. Pendulum Properties
A pendulum has some factors affecting its frequency:
Length of string (l)
The frequency of the pendulum is dependent on the length (l) of the string. The shorter the string, the higher the frequency.
Amplitude
bob
The frequency is independent of the amplitude provided the initial angle is not large.
The frequency is independent of the mass of the bob. Since the acceleration of gravity of a falling object is independent of the mass of the object, a pendulum with a heavy bob will swing at the same rate as one with a lighter one.
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11. Question 3 How should a pendulum be modified to alter its frequency?
A) The length of the string should be modified.
B) The mass of the bob should be modified.
C) The amplitude of the pendulum should be modified.
D) None of these will modify the frequency of the pendulum.

12. Period Equation
You will not always have to time the pendulum swing to know its period. It can be calculated using:
Length of string (l)
where T is the period in seconds (s), l is the length of the pendulum in metres (m) and g is gravitational acceleration (9.8 m/s2).
g 9.8 m/s2
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13. Question 4
Would the period of a pendulum differ if it was on the surface of the moon?
A) It would decrease since gravity is lower on the Moon.
B) It would increase since gravity is lower on the Moon.
C) It would increase since gravity is higher on the Moon.
D) It would not differ since gravity does not affect period.

14. Frequency Equation
In a similar way, since f = 1/T, we can work out the pendulum frequency using:
Length of string (l)
where f is the frequency in hertz (Hz), g is acceleration due to gravity (9.8 m/s2) and l is the length of the pendulum in metres (m).
g 9.8 m/s2
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15. Question 5
Would the frequency of a pendulum differ if it was on the surface of the moon?
A) It would decrease since gravity is lower on the Moon.
B) It would increase since gravity is lower on the Moon.
C) It would increase since gravity is higher on the Moon.
D) It would not differ since gravity does not affect the frequency.

16. Length Equation
The length of the string is the variable that modifies the period or frequency of a pendulum (variation of mass or amplitude will not affect those). Length can be calculated using:
Length of string (l)
where l is the length of the string in metres (m), g is gravity (9.8 m/s2) and f is the frequency in hertz (Hz).
g 9.8 m/s2
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17. Question 6
What happens if the string length of a pendulum is increased?
A) Both period and frequency of the pendulum increase.
B) Both period and frequency of the pendulum decrease.
C) The period of the pendulum increases and the frequency decreases.
D) The period of the pendulum decreases and the frequency increases.

18. Calculation of Gravity Acceleration
The acceleration of free fall (gravity) is approximately 9.8 m/s2.
More accurate experiments show that g varies over the earth surface so that in different locations, slightly different gravity readings are obtained. Look at the table to see some examples.
Location
Gravity
in m/s2
London
9.812
Calcutta
9.788
Tokyo
9.798
Sydney
9.797
North Pole
9.832
This is due to peculiarities in the earth’s morphology and the earth’s crust thickness and density, distance to the earth centre and latitude.
Looking at the period equation, we observe that gravity g in your area can be calculated with a pendulum applying the equation:
where l is the length of the pendulum in metres (m) and T is the period in seconds (s).
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19. Continuous Pendulum Motion
Newtown first law of motion states that, any moving body will continue its motion forever until a force acts on it to stop it.
This could lead us to think that, a pendulum swinging will keep its motion forever, as long as nothing (or nobody) stops it.
Why does a pendulum reduce its oscillation gradually until eventually stops it?
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20. Friction in a Pendulum
The answer is friction. The friction of the bob with the air it swings through, is a force working in opposite direction to the motion.
Even in a vacuum, there would be friction between the string and the fixing point. If necessary, friction could be reduced to very low levels, but there will always exist a friction factor that eventually, will stop the oscillation.
Motion
Friction Force
If there was no friction of any kind, the pendulum would swing back and forth forever because of the law of conservation of energy.
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21. Question 7
Why will a pendulum gradually reduce its oscillation until it eventually stops?
A) The reduction of oscillation in a pendulum is due to friction.
B) The reduction of oscillation in a pendulum is due to the gravity force.
C) The pendulum does not reduce its oscillation as stated by the law of conservation of energy.
D) The pendulum does not reduce its oscillation as stated by Newton's first law.

22. Pendulum Clock
The main application of the pendulum as we have seen, is to use its regular motion to control the mechanism of a clock.
Every cycle of the pendulum moves a gear one notch so the hands of the clock move around.
An important factor that affects a clock’s timing is heat.
As heat normally expands metals, it stretches the pendulum making it longer, therefore varying its period.
This is the reason why clock pendulums are adjustable in length.
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23. Foucault Pendulum
Another interesting application of the pendulum is the Foucault Pendulum. This pendulum is used to demonstrate the Earth’s rotation.
In 1848, Jean Foucault observed that when a large pendulum swings over a long period of time, the pendulum appears to rotate its oscillation during the day.
What is really happening is that the pendulum is actually swinging in the same direction but the Earth has rotated under the pendulum.
A Foucault pendulum located on the pole, it would complete a full rotation in 24 h (23h 56m), the time that the Earth needs to make a full rotation on its axis.
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24. Question 8 What did the Foucault pendulum demonstrate?
A) Regular motion
B) Time flow
C) Earth's gravity
D) Earth's rotation

25. Summary After completing this presentation you should be able to:
Show knowledge about the pendulum and its properties.
Show understanding of the concepts of ‘simple harmonic motion’ and frequency.
Apply the equation to calculate the frequency of a pendulum.
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