Simple Pendulum

1
Simple Pendulum
Q
Rcm
q
L cos q
Equation of Motion
mg
Simple Pendulum
2
Compound Pendulum
Q
Center of mass
L
q
Rcm cos q
Equation of Motion
mg
Compound Pendulum
3
Pendulum
Simple Pendulum
Compound Pendulum
Approximation
4
Different Centers
Center of mass Center of gravity
Center of gyration
(Should be called radius of gyration)
Center of oscillation
Parallel Axis Theorem
Center of percussion -- Later
5
Compound PendulumEffective Length
G
leff
h
b
h
for us
a
6
Exact SolutionEnergy Method
For when the angles are not small
k is the largest angle for an energy E
7
Energy Method (continued)
Elliptical Integral
8
Approximate Solution
To find period, set
9
Exact Solution (by definition)
See Mathematica notebook
Incomplete Elliptical Integral of the first kind
(EllipticFf,a in Mathematica)
Period
Exact
Approx.
Complete Elliptical Integral of the first kind
(EllipticKa in Mathematica
10
Inverse Function
Jacobi Amplitude function
Jacobi Sine function
kp-.1
k.01
k3p/4
kp/2