Lecture 2
Outline
• Vertical oscillations of mass on spring
• Pendulum
• Damped and Driven oscillations (more
realistic)
Vertical Oscillations (I)
• At equilibrium (no net force),
spring is stretched (cf. horizontal
spring): spring force balances
gravity
Hooke’s law: (Fsp )y = −k∆y = +k∆L
Newton’s law: (Fnet )y = (Fsp )y + (FG )y = k∆L − mg = 0
mg
⇒ ∆L = k
Vertical Oscillations (II)
•
Oscillation around equilibrium, y = 0
(spring stretched)
block moves upward, spring still stretched
(Fnet )y = (Fsp )y + (FG )y = k (∆L − y) − mg
Using k∆L − mg = 0 (equilibrium), (Fnet )y = −ky
•
...as before
y(t) = A cos (ωt + φ0 )
Example
• A 8 kg mass is attached to a spring and allowed
to hang in the Earth’s gravitational field. The
spring stretches 2.4 cm. before it reaches its
equilibrium position. If allowed to oscillate, what
would be its frequency?
Pendulum (I)
• Two forces: tension (along string) and gravity
• Divide into tangential and radial...
(Fnet )tangent = (FG )tangent = −mg sin θ = matangent
acceleration around circle
d2 s
dt2
= −g sin θ
more complicated
Pendulum (II)
• Small-angle approximation
sin θ ≈ θ (θ in radians)
(Fnet )tangent ≈
d2 s
dt2
mg
− L s
(same as mass on spring)
⇒
=
⇒ s(t) = A cos (ωt + φ0 ) or θ(t) = θmax cos (ωt + φ0 )
g
−Ls
(independent of m, cf. spring)
Example
• The period of a simple pendulum on another planet is
1.67 s. What is the acceleration due to gravity on this
planet? Assume that the length of the pendulum is 1m.
Summary
•
linear restoring force ( ∝ displacement from equilibrium)
e.g. mass on spring, pendulum (for small angle)
• (x y for vertical): x(t) = A cos (ωt + φ ) v (t) = −ωA sin (ωt + φ )
• A, φ0 determined by initial conditions (t=0)
0
•
•
x
x0 = A cos φ0 , v0 x = −ωA sin φ0
!
!
ω depends on physics ( k/m or g/L ), not on A, φ0
conservation of energy (similarly for pendulum):
2
1/2 mvx2 + 1/2 kx2 = 1/2 kA2 = 1/2 mvmax
KE
PE
turning point equilibrium
0
Pendulum (III)
• Physical pendulum (mass on string is
simple pendulum)
(restoring) torque
moment arm
τ = −M gd = −M gl sin θ ≈ −M glθ (small angle)
d2 θ
τ
α (angular acceleration) = dt2 =
⇒
I (moment of inertia)
!
SHM equation of motion: ω = 2πf = MIgl
d2 θ
dt2
=
−M gl
I θ
Damped Oscillations (I)
•
dissipative forces transform mechanical
into heat e.g. friction
•
model of air resistance (b is damping coefficient, units: kg/s)
energy
D̄ = −bv̄ (drag force) ⇒
(Fnet )x = (Fsp )x + Dx = −kx − bvx = max
d2 x
b dx
k
(equation
of
motion
for
damped
oscillator)
+
+
x
=
0
2
dt
m dt
m
•
Check that solution is (reduces to earlier for b = 0)
Damped Oscillations (II)
•
Lightly damped: b/(2m) ! ω0
xmax (t) = Ae−bt/(2m)
•
Energy not conserved:
τ≡m
(time constant)
b
E(t) =
•
2
1
2 k (xmax )
!1
"
= 2 kA2 e−t/τ = E0 e−t/τ
measures characteristic time of energy
dissipation (or “lifetime”): oscillation not
over in finite time, but “almost” over in
time
Resonance
•
Driven oscillations
(cf. free with damping so far):
periodic external force
e.g. pushing on a swing
•
natural frequency
of
oscillation
f0 :!
!
e.g. k/m or g/L
• fext : driving frequency of external force
pushes oscillator
• amplitude rises as fext → f0: external forces
f != f →
at same point in cycle, adding energy (
add, other times remove, not in sync)
•
ext
amplitude very large: fext = f0 (resonance)
0
sometimes