Lecture 20
Goals:
• Wrap-up Chapter 14 (oscillatory motion)
• Start discussion of Chapter 15 (fluids)
• Assignment
HW-8 due Tuesday, Nov 16
Monday: Read through Chapter 15
Physics 207: Lecture 19, Pg 1
SHM Solution...
The general solution is: x(t) = A cos (ωt + φ)
where A = amplitude
ω = (angular) frequency
φ = phase constant
k
-A
m
0(Xeq)
A
Physics 207: Lecture 19, Pg 2
SHM Solution...
The mechanical energy is conserved:
U = ½ k x2 = ½ k A2 cos2(ωt + φ)
K = ½ m v2 = ½ k A2 sin2(ωt+φ)
U+K = ½ k A2
E = ½ kA2
U~cos2
K~sin2
Physics 207: Lecture 19, Pg 3
k
-A
m
0(Xeq)
A
For simple harmonic motion, the kinetic energy of the
mass is largest when:
A)at x=A
B)at x=-A
C)at x=0
D)Neither of the above
Physics 207: Lecture 19, Pg 4
k
-A
m
0(Xeq)
k
-1.5A
T=1s
A
m
0(Xeq)
1.5A
T is:
A)T > 1 s
B)T < 1 s
C) T=1 s
Physics 207: Lecture 19, Pg 5
SHM So Far
For SHM without friction
l The frequency does not depend on the amplitude !
lThe oscillation occurs around the equilibrium point
where the force is zero!
lMechanical Energy is constant, it transfers between
potential and kinetic energies.
Physics 207: Lecture 19, Pg 6
What about Vertical Springs?
k
k
k
equilibrium
L
m
new
equilibrium
mg=k L
Fnet= -k (y+ L)+mg=-ky
L
y=0
y
m
Fnet =-ky=ma=m d2y/dt2
Which has the solution y(t) = A cos( ωt + φ)
Physics 207: Lecture 19, Pg 7
The “Simple” Pendulum
l A pendulum is made by suspending a mass m at the end
of a string of length L. Find the frequency of oscillation for
small displacements.
Σ Fy = may = T – mg cos(θ)
z
Σ Fx = max = -mg sin(θ)
If θ small then x ≅ L θ and sin(θ) ≅ θ
dx/dt = L dθ/dt
ax = d2x/dt2 = L d2θ/dt2
so ax = -g θ = L d2θ / dt2
and θ = θ0 cos(ωt + φ)
with
y
θL
x
T
m
ω = (g/L)½
T = 2 (L/g)½
mg
Physics 207: Lecture 19, Pg 8
Energy in SHM
U = ½ k x2
U
l The total energy (K + U) of a
system undergoing SHM will
always be constant!
K
E
U
-A
0
x
A
Physics 207: Lecture 19, Pg 9
SHM and quadratic potentials
l SHM will occur whenever the potential is quadratic.
l For small oscillations this will be true:
l For example, the potential between
H atoms in an H2 molecule looks
something like this:
U
E
U
x
K
U
-A
0
A
Physics 207: Lecture 19, Pg 10
x
What about friction?
lOne way to include friction into the model is to assume
velocity dependent drag.
Fdrag= -bv=-b dx/dt
Fnet=-kx-b dx/dt = m d2x/dt2
d2x/dt2=-(k/m)x –(b/m) dx/dt
a new differential
equation for x(t) !
Physics 207: Lecture 19, Pg 11
Damped oscillations
lFor small drag (under-damped) one gets:
x(t) = A exp(-bt/2m) cos (ωt + φ)
x(t)
1.2
1
0.8
0.6
0.4
A
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
ωt
t
Physics 207: Lecture 19, Pg 12
Driven oscillations, resonance
lSo far we have considered free oscillations.
lOscillations can also be driven by an external force.
oscillation amplitude
ω0: natural frequency
ωext ≅ ω0
ωext
Physics 207: Lecture 19, Pg 13
Chapter 15, Fluids
l An actual photo of an iceberg
Physics 207: Lecture 19, Pg 14
l At ordinary temperature, matter exists
in one of three states
Solid - has a shape and forms a
surface
Liquid - has no shape but forms a
surface
Gas - has no shape and forms no
surface
l What do we mean by “fluids”?
Fluids are “substances that
flow”…. “substances that take the
shape of the container”
Atoms and molecules are free to
move.
Physics 207: Lecture 19, Pg 15
Fluids
lAn intrinsic parameter of a fluid
Density (mass per unit volume)
=m/V
units :
kg/m = 10-3 g/cm3
3
ρ(water) = 1.000 x 103 kg/m3
= 1.000 g/cm3
ρ(ice)
= 0.917 x 103 kg/m3
= 0.917 g/cm3
ρ(air)
= 1.29 kg/m3
= 1.29 x 10-3 g/cm3
ρ(Hg)
= 13.6 x103 kg/m3
= 13.6 g/cm3
Physics 207: Lecture 19, Pg 16
Fluids
lAnother parameter
Pressure (force per unit area)
P=F/A
SI unit for pressure is 1 Pascal = 1 N/m2
lThe atmospheric pressure at sea-level is
1 atm = 1.013 x105 Pa
= 1013 mbar
= 760 Torr
= 14.7 lb/ in2 (=PSI)
Physics 207: Lecture 19, Pg 17
Pressure vs. Depth
l For a uniform fluid in an open
container pressure same at a
y
given depth independent of the
container
l Fluid level is the same everywhere in
a connected container, assuming no
surface forces
p(y)
l Why is this so? Why does the
pressure below the surface depend
only on depth if it is in equilibrium?
F
Imagine a tube that would connect two regions at the same
depth.
F If the pressures were different, fluid would flow in the tube!
Physics 207: Lecture 19, Pg 18