Idea Transcript
The Simple Harmonic Pendulum Joel Ballard Spring 2012
Contents 1 Introduction 1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 1
2 Assumptions
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3 Variables
3
4 Equations
4
5 The Small Angle Approximation
6
6 Conclusion
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Abstract
Modeling the motion of the simple harmonic pendulum from Newton’s second law, then comparing this with the small angle approximation model using MATLAB.
1 1.1
Introduction History “Pendulum” from the Latin “pendulus”meaning “hanging”
Galileo Galilei began experimenting with pendulums in 1602. Galilei first became interested as a university student when Galilei was watching a lamp swinging in a cathedral in Pisa, Italy. Galilei discovered that the period almost entirely upon length. Galilei theorized that a clock could be made using a pendulum. Christiaan Huygens was the first person to use this idea when Huygens constructed a clock using a pendulum in 1665. For the day it was very accurate, only losing one minute per day. Huygens later improved this to a loss of ten seconds per day.
1.2
Applications
Pendulums have many applications and were utilized often before the digital age. They are used in clocks and metronomes due to the regularity of their period, in wrecking balls and playground swings, due to their simple way of building up and keeping energy. They are even found in various scientific instruments, from seismographs to early torpedo guidance systems, due to their sensitivity to disturbance. A predecessor to the seismograph was based on an inverted pendulum, Chang Heng’s Dragon Jar invented at around 123 A.D..
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Assumptions
All models are full of assumptions. Some of these assumptions are very accurate, such as the pendulum is unaffected by the day of the week. Some of these assumptions are less accurate but we are still going to make them, friction does not effect the system. Here is a list of some of the more notable assumptions of this model of a pendulum.
• Friction from both air resistance and the system is negligible.
• The pendulum swings in a perfect plane.
• The arm of the pendulum cannot bend or stretch/compress.
• The arm is massless.
• Gravity is a constant 9.8 meter/second2 .
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3
Variables
m = mass at the swinging end of the pendulum (kilograms)
g = acceleration due to gravity (meter/second2 )
L = length from the swivel point to the center of mass (meters)
θ = angle between the string position to the string position at rest (radians)
t = time (seconds)
T = period of the pendulum (time for one complete cycle) (seconds)
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4
Equations
We will now derive the simple harmonic motion equation of a pendulum from Newton’s second Law. F = ma Acceleration due to gravity will be a function of θ. At θ = π2 −→ |a| = g and at θ = 0 −→ a = 0, considering the relation of acceleration and θ we arrive at a = −g sin θ
(1)
arc length (arcL ) of the pendulum can be thought of as the “position” of the system arcL = Lθ acceleration of the system will now be a=L
d2 θ dt2
plug in equation (1) to get the simple harmonic motion of a pendulum shown in equation (2) d2 θ g + sin θ = 0 dt2 L
(2)
Now we will solve equation (2) to get T (period) reduce the second order differential equation to a first order " # 2 dθ d2 θ dθ g d 1 dθ g ∗ + ∗ sin θ = 0 −→ − cos θ = 0 dt dt2 dt L dt 2 dt L integrate to get the differential equation
dθ dt
2 −
2g cos θ = C L
with the initial conditions of θ0 (0) = 0 and θ(0) = θ0 we can solve for C C=−
2g cos θ0 L
put our C into the equation
dθ dt
2 =
2g (cos θ − cos θ0 ) L 4
Now take the square root of both sides while ignoring the negative because we are solving for time and either time will be the same distance from our t0 . It is just a matter of forwards or backwards in time. On the left hand of our equation lies the rate of change of the angle with respect to time, but we are going to solve for the period, so we need the time with respect to the angle, because of this we are going to inverse the entire equation and integrate from 0 to θ0 . We will now multiply the whole thing by four to get the period. The change in time to get from 0 to θ0 is only one forth of the entire cycle of the pendulum. This gives us our new equation of s Z θ0 1 L 1 √ dθ (3) T =4 g 2 0 cos θ − cos θ0 This is where MATLAB can prove to be a great asset because equation (3) cannot be solved in terms of elementary functions.
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The Small Angle Approximation
The small angle approximation states that θ ≈ sin(θ) at small angles. Using this we adjust the equation g g d2 θ d2 θ = − ≈− θ sin θ −→ dt2 L dt2 L and it is almost the same and far easier to solve. d2 θ g ≈− θ dt2 L λ2 +
(4)
g =0 L
p λ = ± Lg i in the form of a ± bi using the complex roots case for solving second-order equations y(t) = eat [A1 cos(bt) + A2 sin(bt)] We can solve for A1 and A2 which are arbitrary constants using the initial conditions θ(0) = θ0 and θ0 = 0. We get A1 = θ0 and A2 = 0. Thus we come to the solution r g t (5) θ(t) ≈ θ0 cos L Using the above equation we create the period equation by setting θ = 0. We can solve for t, and this gives us one fourth of the total period. Now we multiply by four and get s L T ≈ 2π (6) g
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Which means at “small angles” less than 10 ◦ or so, the period of the pendulum is completely dependent on the length of the arm and gravity, because θ ≈ sin θ at “small angles” as demonstrated in the graph below.
Now using odesolve on MATLAB and breaking tem of two first order equations
and breaking
d2 θ dt2
+
g Lθ
x01
=
x02
=
x2 −g sin(x1 ) L
d2 θ dt2
+
g L
sin θ = 0 into a sys-
(7) (8)
= 0 into x01 x02
= x2 −g = x1 L
(9) (10)
we get the two graphs below. Showing the difference between the simple harmonic model and the small angle approximation model.
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Conclusion
Pendulums are not too difficult to predict if a person only needs a few significant figures, or is willing to stay at fairly small angles. So next time a person is stuck on a desert island and needs a way to measure short amounts of time or keep perfect rhythm for their island song. A pendulum is easy to make and with a little bit of math, easy to understand, one could even use the swaying of their hammock, assuming a fairly uniform driving force. If tropical paradise is too old fashioned, and one’s dreams are far bigger than this small planet can offer, then gravity could become the variable instead of length. One could become the MacGyver of space travel and use a pendulum to calculate the gravity (and mass) of a newly discovered world.
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References [1] Differential Equations with Boundary Value Problems Polking, Boggess, Arnold [2] The Simple Pendulum www.acs.psu.edu/drussell/Demos/ [3] Pendulum (mathematics) www.wikipedia.org [4] Pendulum www.wikipedia.org [5] Mathematical Swingers: The Simple Pendulum as a Log Application www.http://my.execpc.com [6] The History of the Pendulum www.bukisa.com [7] Pendulum Shifts to Active ETFs www.etftrends.com [8] hammock-on-the-beach www.rajatandena.files.wordpress.com
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