Transcript Physics 141 Mechanics Yongli Gao Lecture 4 Motion in 3-D

Physics 141
Mechanics
Lecture 21
Oscillation
Yongli Gao
• You may not know it, but every atom/molecule in
your body is oscillating.
• For any system, there's at least one state that the
system is of the lowest potential energy. This is a
point of stable equilibrium, or the bottom of the
valley in a potential vs. position curve.
• If the system is of a small displacement from the
point, it'll experience a restoring force, pointing to
the bottom of the potential curve. The force
accelerates the system so it'll swing across the
equilibrium point to the other side, and the restoring
force will reverse as well. Thus, it'll oscillate
around the point of equilibrium.
Period and Frequency
• The period T of an oscillation is the time taken for
the oscillating system to repeat itself, or, to complete
one oscillation. For example, the time for a
swinging pendulum starting from one extreme point
to the come to the same point. Same position,
velocity, and acceleration. T is in second.
• The frequency f of an oscillation is the number of
complete oscillations per unit time. Clearly
1
f 
T
• The unit for frequency is hertz. 1 hertz = 1/s
•
•
•
•
•
•
Simple Harmonic Oscillator
The simplest oscillation is a particle of mass m
attached to a massless spring of spring constant k on
a horizontal frictionless plane.
From Hooke's law, F=-kx
d2x
From Newton's 2nd law, F  ma  m 2  kx
dt
The solution is the simple harmonic oscillation
x(t)  Acos(wt  f )
(SHO)
where A is the amplitude of the oscillation,
k
is the angular frequency,
w
m
wt+f the phase, and f the phase constant. 2
T
The period of an SHO is
w
Any periodic motion can be expressed as the sum of
SHO's of different frequencies.
Energy of an SHO
• In an SHO, the kinetic energy K and potential
energy U convert to each other back and forth, but
the total energy E=U+K is a constant.
x(t)  Acos(wt  f )
1 2 1 dx 2 1
2
K  mv  m   m Aw sin(wt  f )
2
2 dt  2
1
1 2 2
2 2
2
 mA w sin (wt  f )  kA sin (wt  f )
2
2
1 2 1 2
U  kx  kA cos2 (wt  f )
2
2
1
sin 2   cos2   1  U  K  kA2
2
General Small Oscillations
• Any small oscillation about an equilibrium position
can be approximated as SHO. Suppose the the
potential energy is U(x) and the equilibrium position
is x0. At equilibrium the force is zero, U
F(x0 )  
|x  x 0  0
• Taylor expansion
x
2
U
1 U
2
U(x)  U(x0 ) 
|x 0 (x  x0 ) 
|
(x

x
)
 ...
0
2 x0
x
2 x
1  2U
1 2
2
 U(x0 ) 
| (x  x0 )  a  k
2 x0
2 x
2
U
 F( )  
 k

k
1  2U
w 

2 |x 0
m
m x
Circular Reference
• SHO motion can be viewed as the
x-component of a uniform circular
motion.
y
r
wt
f
x
r(t)  Acos(wt  f )i  Asin(wt  f )j
Simple Pendulum
• A simple pendulum is formed by hanging a particle
of mass m to a pivotal point O by a massless string
of length l. About O,
I0  ml 2
 0  mgl sin   mgl
d 2  0 mgl
d 2
g
I0   0    2  
 2  
2
dt
I0
ml
dt
l
w 
g
2
l
,,T 
 2
l
w
g
• This has been used in the past centuries for clocks.
Physical Pendulum
• A physical pendulum is formed by allowing a rigid
body fixed to a pivotal point O to oscillate
frictionlessly. About O,
I0  ICM  mlCM 2
 0  mglCM sin   mglCM
d 2  0
d 2
mglCM
I 0   0    2   2  

2
dt
I0
dt
ICM  mlCM
w 
mglCM
2
I CM  mlCM 2
,,T 
 2
2
ICM  mlCM
w
mglCM
• This is true for real pendulums.
Damped Oscillation
• Real objects may experience friction or viscosity as
they oscillate. The motion is damped oscillation.
• Viscous force
d2x
k
b dx
Fd  bv  2   x 
dt
m
m dt
 x(t)  A0e  t / 2 cos(w' t  f )
b
k
b2
  ,, w' 

m
m 4m2
• The amplitude is damped, and the energy dissipates
as
E  E0 e t
Forced Oscillation
• You can also drive an object to oscillation by
applying a periodic force, like walking on a hanging
bridge.
d2x
k
F0
F  F0 cosw Ft  2   x  cosw F t
dt
m
m
 x(t)  xm cos(w Ft)
k
F0
  xmw F cos w F t   xm cosw F t  cosw F t
m
m
F0
 xm 
m(k / m  w F 2 )
• The amplitude depends on both w=√k/m and wF. If
the driving frequency wF is the same as the natural
frequency w, the amplitude reaches the maximum
and we have resonance.
2