Module 2.xhtml1.12005/04/28 12:03:18 GMT-52005/04/28 17:03:40 GMT-5PaulPadleypadley@rice.eduPaulPadleypadley@rice.educompound pendulumharmonic motionsimple pendulumThe Simple Pendulum
Shown
is a simple pendulum which has a mass
m
that is displaced by an angle
θ.
There is tension
(T⃗)
in the string which acts from the mass to the anchor point. The weight of the
mass is
mg⃗
and the tension in the string is
T=mgcosθ.
There is a tangential restoring force
=−mgsinθ.
If we approximate that
θ
is small (we have to make this approximation or else we can not solve the
problem analytically) then
sinθ≈θ
and
x=lθ.
(note that
sinθ
is only approximately equal to
xl
because
x
is the distance along the
x
axis) so that we can write:
F=ma=mx¨=−mgsinθ≈−mgθ≈−mgxlor
x¨+glx=0(Note
that We should immediately recongnize that this is the equation for simple
harmonic motion (SHM) with
ω=gl.
We could take another approach and use angular momentum to
solve the problem. Recall that:
L=Iω=Iθ˙I=ml2.Also
recall that the torque is the time derivative of the angular momentum so that:
τ⃗=r⃗×F⃗=ⅆL⃗ⅆt−lmgθ=Iθ¨θ¨+glθ=0Again
we would recognize that this is simple harmonic motion with
ω=gl.The Compound Pendulum
The compound pendulum is another interesting example of a pendulum that
undergoes simple harmonic motion. For an extended body then one uses the
center of mass and the moment of inertia. Use the center of mass, the moment
of inertia and the Torque (angular force)
τ⃗=r⃗×F⃗τ=r×FIθ¨=−lmgsinθ≈−lmgθθ¨+lmgIθ=0So
again we get SHM now with
ω2=lmgIOne
sees that this formalism can be applied to the simple pendulum (ignore the
string and one can consider the ball a point mass). The moment of inertia is
ml2.
So we get
ω2=lmgml2=glwhich
is just what we got before for the simple pendulum. We could write the
equation of motion for a simple pendulum
as:θ=Aei(ωt+φ0)
where
φ0
is determined by initial conditions.
A discussion of the Pendulum and Simple Harmonic Oscillator can be found at
http://monet.physik.unibas.ch/~elmer/pendulum/index.html