Chapter 15 Oscillations
Simple Harmonic Motion (SHM)
Examples
• Mass on spring
• Simple Pendulum
• Not-so-simple pendulum (physical pendulum)
• Car shocks
• Earthquake-region buildings
• Atomic clocks
Forces in SHM
• mass + spring system
• Simple pendulum
• Physical pendulum
All have same mathematical form
SHM occurs when…
• F  -x
• PE  x2
(like spring)
(like spring)
• {x, v, a} are sinusoidal (sines, cosines)
• Period T independent of amplitude
• Any one  all the others
Mass & spring
Fnet = -k x & Fnet = m a
 ma = -k x
Rearrange  diff-eq is
𝑑2𝑥
𝑑𝑡2
= - w2 x
Solution is
x(t) = A cos (wt + f)
v(t) = ?
a(t) = ?
Mass & spring: x(t) = A cos (wt + f)
• Maximum x = xmax =
• Maximum v = vmax =
• Maximum a = amax =
• Amplitude A = xmax
• w = (k/m)1/2 units [=] rad/s
• w=2pf
f = 1/T
T depends on? - Demo with 2 m’s ,same spring: measure T
Energy in SHM
• KE =
• PE =
• Etot = KE + PE = …
=
=
CQs: 15-0,1,2,4
Phase, phase constant
• x(t) = A cos (wt + f)
• Phase = wt + f
depends on time t
• Phase constant = f does NOT depend on t
Relation between SHM and circular
motion
• Why does SHM have angular frequency w?
(demo: spinning wheel and its shadow)
phaseCQ
The figure shows four oscillators at t=0. Which
one has the phase constant f0 = -3p/4 rad?
Simple pendulum
• t=Ia
• Restoring force = mg sin q
w, T for pendulum
• t=Ia
• Restoring force = mg sin q
• For small angle, sin q  q
Some math…
How does T depend on m, L?
(on board)
(demos)
Simple, not-so-simple pendulum
Simple pendulum
• T=2p
𝐼
𝑚𝑔𝐿
=2p
𝐿
𝑔
Not-so-simple pendulum
(Lh)
• T=2p
𝐼
𝑚𝑔ℎ
where h = distance from cm
to pivot point
CQs: 15-5,6,7abc
Damped oscillations
• Car shocks
• Oscillations are slower  lower w, larger T
• Oscillations get smaller and smaller and smaller…
Damping and time constant
• b=damping constant
• Large b  heavy damping
• Small b  light damping
(x vs t pictures on board)