this lecture simple harmonic motion (shm) motion near an equilibrium position can be approximated by...
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THIS LECTURE
SIMPLE HARMONIC MOTION (SHM)
Motion near an equilibrium position can be approximated by SHM
Some examples of SHMs
An example of SHM: mass on spring k
x
kxxm From Newton’s and Hooke’s law we derivethe differential equation
)cos( tAxSolution:
mk /Angular frequency
Amplitude (A) and phase constant () are determined by the initial conditions
Period 2
T
Water molecules can vibrate in a number of ways with well-defined frequencies
O
H
Water molecules
Water molecules can vibrate in a number of ways with well-defined frequencies
O
H
Water molecules
An example of SHM: mass on springs
)cos( tAxSolution:
mkeff /Angular frequency Period 2
T
x
xkxkkxkxkxm eff )( 2121
From Newton’s and Hooke’s law we derivethe differential equation
21 kkkeff
xkxm eff
An example of SHM: mass on springs
)cos( tAxSolution:
mkeff /Angular frequency Period 2
T
x
xkxkxkxm eff 2211
From Newton’s and Hooke’s law we derivethe differential equation
21
21
kk
kkkeff
xxx 21
Period
21
2kk
mT
21
21 )(2
kk
kkmT
Which of these systems has the shortest T?
General motion
x
Motion near an equilibrium position can be approximated
by SHM
kxdx
dUF
constkxU
~
2
1~ 2
Near a position of stable equilibrium, U can be approximated by a parabola, i.e. an harmonic potential
x =0
U 2
2
1~ kxU
Motion near an equilibrium position can be approximated by SHM
SHM for a diatomic molecule
)1()(2)( 0rreDrU
Morse potential
r = distance between atoms
N2
The Morse potential, named after physicist P.M. Morse, is a convenient model for the potential energy of a diatomic molecule.
SHM for a diatomic molecule
m mr
)1()(2)( 0rreDrU
00
U
r
0
0
F
r
0r
0r
20 )()( rrDrU
For r~r0
00
U
r
)(2)( 0rrDrF
dr
dUrF )(
D
Show that for small amplitude oscillations, the motion can be approximated by a SHM
Determine T for small amplitude oscillations
Problem: Particle of mass m sliding without friction in a spherical bowl of radius r
r
m
g
rT 2