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Pendula
(or Pendulums, in the vernacular)
What is a pendulum?
Pendulum: Consists of a massive object called a bob
suspended by a string.
What is it good for?
•
Keeping time
•
Galileo (what other options did he have?)
•
Clocks
•
Calculating acceleration due to gravity
•
Other, less obvious things
•
Verifying the rotation of the earth
•
Understanding Energy: Potential and Kinetic
•
Study of simple harmonic motion
Velocity, acceleration & the Pendulum
Conservation of Energy & The Pendulum
•
Definitions
–
–
–
Potential energy describes the possibility of converting stored
energy into motion
Kinetic energy describes the energy of motion
Since (mass)energy can be neither created nor destroyed,
potential and kinetic energy can be exchanged interchangeable
This height indicates the
distance the pendulum
bob “falls” under the
influence of gravity. It
and gravity determine
how much potential
there is for motion
We are back at the
same height!
And at ½ of a cycle.
Kinetic energy is a
measure of how fast
the bob is moving.
Conservation of Energy & The Pendulum
We are back at the
same height!
And at ½ of a cycle.
Kinetic and Potential Energy
Kinetic Energy = ½ m v2
Potential energy = mg D h
With pendula, we are usually interested in the
period
What things might affect the period?
- mass
- angle
- length
- gravity
- wind resistance
What do you think the form of the equation might be?
Pendulum: Consists of a massive object called a
bob suspended by a string
Pendula go through periodic motion as follows:
Where:
l
T  2
g
T = period, or the time to go through one cycle
l = length of pendulum string
g = acceleration of gravity
Note:
1. This formula is true for only small angles of θ.
2. What would you expect to find in this equation and is not
there?
The motion of a pendulum is independent of the
mass
•
•
As long as the mass is big enough so that air resistance
can be ignored
As long as the mass of the string holding the pendulum is
very small compared to the mass of the bob
l
4 l
2
2 l
T  2
 T  4
 g  2
g
g
T
2
Where:
T = period
l = length of pendulum string
g = acceleration of gravity
As we increase l, what happens?
As we increase g, what happens?
l
2
2 l
T  2
 T  4
g
g
As we increase the angle, what happens?
As we increase the mass, what happens?
l
2
2 l
T  2
 T  4
g
g
Examples:
1) What is the period of a pendulum on Earth
with a length of 1m?
2) What is the period of a pendulum on Jupiter
with a length of 1m? (ag = 3g)
3) What is the period of a pendulum on the
moon with a length of 2m? (ag = 1/6g)
4) What is the length of a pendulum (on Earth)
with a period of 10 sec?
5) What is the length of a pendulum (on Earth)
with a length of 0.1 sec?
Examples(2):
1) What would the graph of period vs. length
look like?
2) What would the graph of period2 versus
length look like?
3) What would the graph of period2 versus g
look like?