Pendulum
A mass, called a bob, suspended from a fixed point so that it can
swing in an arc determined by its momentum and the force of gravity.
The length of a pendulum is the distance from the point of
suspension to the center of gravity of the bob. Chance observation of
a swinging church lamp led Galileo to find that a pendulum made
every swing in the same time, independent of the size of the arc. He
used this discovery in measuring time in his astronomical studies. His
experiments showed that the longer the pendulum, the longer is the
time of its swing.
If we assume the angle q is small, for then we can approximate sin θ
with θ (expressed in radian measure). (As an example, if θ = 5.00° =
0.0873 rad, then sin θ = 0.0872, a difference of only about 0.1%.)
With that approximation and some rearranging, we then have
∂ 2θ mgL
+
θ =0
2
∂t
I
Physical Pendulum,
Small amplitude
UPendulum = mgh = mgL (1-cosθ)
For smaller displacements, the movement of
an ideal pendulum can be described
mathematically as simple harmonic motion
(like the mass-spring), as the change in
potential energy at the bottom of a circular arc
is nearly proportional to the square of the
displacement. Real pendulums do not have
infinitesimal displacements, so their behaviour
is actually of a non-linear kind.
The Physical Pendulum
A "physical" pendulum has
extended size and is a
generalization of the bob pendulum.
An example would be a bar rotating
around a fixed axle. A simple
pendulum can be treated as a
special case of a physical pendulum
with moment of inertia I. ( I = ∑ miri2)
Period of a physical pendulum
(Note: l is now the length from the
suspension point to the center of
mass CM instead of L)
Example:
Simple Pendulum: I = mL2
Leg: I = 1/3 mL2
Simple Pendulum
All the mass of a simple
pendulum is concentrated in the
mass m of the particle-like bob,
which is at radius L from the
pivot point. Thus, we can
substitute I = mL2 for the
rotational inertia of the
pendulum.
L
T = 2π
g
for small amplitudes!!
Exercise 4: A clock has a pendulum that performs one full
swing every 1.0 sec. The object at the end of the string
weights 10.0 N. What is the length of the pendulum?
L
T = 2π
g
Solving for L:
(
)
9.8 m/s 2 (1.0 s )
gT 2
L=
=
= 0.25 m
2
2
4π
4π
2
Pivot
Length: L
Mass: M
θ
ICM= 1/12 ML2
CM
Parallel-Axis Theorem
IPivot= 1/12 ML2 + M (½ L)2 = 1/3 ML2
mg
The period is
1
ML2
I
2L
3
T = 2π
= 2π
= 2π
1
gMl
3
g
gM ( L)
2
Ring
Disc
r
r
CM
IPivot= Mr2 + Mr2 = 2Mr2
CM
IPivot= ½ Mr2 + Mr2 = 3/2 Mr2
I
2 Mr 2
T = 2π
= 2π
gMl
gMr
I
= 2π
T = 2π
gMl
2r
T = 2π
g
3r
T = 2π
2g
3
Mr 2
2
gMr
Natural Frequency for different species. To calculate the
moment of inertia we assume the a leg can be treated as rod
of length L
2L
T = 2π
3g
Human: Length of leg L = 1m
Dachshund L=0.2m
T = 2π
Tdachshund
2 1m
= 1.6s
2
3 10m / s
= 0.7 s
If we assume that our legs swings with an max angle of 10degrees or 0.174 rad
v(Human) = s/t = 0.174rad * 1m / T/4 = 0.4 m/s ≈ 1mi/h
The Foucault pendulum (pronounced "foo-KOH"), or Foucault's
pendulum, named after the French physicist Léon Foucault, was
conceived as an experiment to demonstrate the rotation of the Earth.
It is a tall pendulum free to oscillate in any vertical plane. The first
public exhibition of a Foucault pendulum took place in February 1851
in the Meridian Room of the Paris Observatory. A few weeks later,
Foucault made his most famous pendulum when he suspended a 28kg bob with a 67-metre wire from the dome of the Panthéon in Paris.
In 1851 it was well known that the earth rotated: observational
evidence included earth's measured polar flattening and equatorial
bulge. However, Foucault's pendulum was the first dynamical proof of
the rotation in an easy-to-see experiment, and it created a sensation
in both the learned and everyday worlds.
http://www.youtube.com/watch?v=jtkr70fHF08
Foucault's Pendulum in
the Panthéon, Paris.
Damped Oscillations
We know that in reality, a spring won't oscillate for ever. Frictional forces will
diminish the amplitude of oscillation until eventually the system is at rest.
A mass in air oscillates many times before it comes to rest. A mass in a liquid like
molasses is hardly to oscillate at all.
When dissipative forces such as friction are not negligible, the amplitude of
oscillations will decrease with time. The oscillations are damped.
G
G
F = −bv
To incorporate friction, we can just say that there is a frictional force that's
proportional to the velocity of the mass.
b: damping constant
This is a pretty good approximation for a body moving at a low velocity in air, or in
a liquid.
What we expect is that the amplitude of oscillation decays with time. It is
described with an exponential decay of the amplitude with time, instead of the
amplitude being constant.
Equation of motion is now
The solution is
d 2x
dx
m 2 = − kx − b
dt
dt
2
d x b dx k
+
+ x=0
2
dt2
m dt m
d x
dx
2
+γ
+ ω0 x = 0
2
dt
dt
(http://www.abdn.ac.uk/physics/vpl/pendulum/damped.html)
A
with
Mechanical energy decreases with time
A = A0 e
− bt / 2 m
critically damped: The damping force is such that the system returns to equilibrium
as quickly as possible and stops at that point. ( ζ=1)
overdamped. The damping force is greater than the minimum needed to prevent
oscillations. The system returns to equilibrium without oscillating, but it takes longer
to do so than a critically damped system. . ( ζ>1)
underdamped: It oscillates about the equilibrium point, with ever diminishing
amplitude. ( ζ<1)
Damping factor
zeta
http://lectureonline.cl.msu.edu/%7Emmp/applist/damped/d.htm
http://www.abdn.ac.uk/physics/vpl/pendulum/applet/applet.html
A spring with a mass of 6 kg has damping constant 33 kg/s and
spring constant 234 N/m. Find the damping constant that would
produce critical damping.
1.
2.
3.
4.
=1
N
kg
b = 1* 4 ⋅ 6kg ⋅ 234 = 12 39
m
s
Driven Oscillations and Resonance
d 2x
dx
m 2 = − kx − b + F0 cos(ωDt )
dt
dt
steady state x(t ) = AD cos(ωDt + φ )
F /m
AD =
(ω02 − ω 2 ) 2 + ωγ 2
When the driving frequency is close to natural frequency,
its amplitude of motion can be quite large.
Resonance curve
A plot of amplitude A versus driving frequency
Small damping
The amplitude can become very large for frequencies
close to natural frequency
Large damping
The amplitude has low, broad peak near the natural
frequencies.
Applet resonance