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Phys141 – Fri 11/
TODAY: Oscillations (Ch 15)
Mon: Start Ch 16, Waves
Wed Ch 16 waves – end
Pre-class quizzes Mon/Wed
HW due Fri 11/
Example 2: Spring-Mass System
A block of mass m is attached to a spring, the block is free to move on a frictionless horizontal surface
When the spring is neither stretched nor compressed, the block is at the equilibrium position x = 0 Note: Force pointed toward equilibrium position
Simple Harmonic Motion
The force described by Hooke’s Law is the net force in Newton’s Second Law 2 x^ =^2 = −
d x k a x dt m
Hooke = Newton
− = x
F F
kx ma
Acceleration
Which function satisfies this differential equation?
2 2 2 = −^ ω
d x x dt
2 k m
ω =
Simple harmonic motion
x ( t ) = A cos ( ω t + φ)
A , ω,φ are all constants
A amplitude of the motion
ω angular frequency
φ phase constant or the initial phase angle
T 2
ω = k m^2
m T k
= π
frequency
Period
Graphical Representation
x ( t ) = A cos ( ω t + φ) 2 m T k
= π
Phase constant (initial phase angle)
determined by initial conditions
When is max amplitude reached? -> cos ( ω t + φ) =1 -> ω t + φ = 0,2π,4π,6π, ...
Forced Oscillations
- What happens when we apply a driving force: F (^) o sin (ωt)?
For damped harmonic motion:
- k x – bv (^) x + F (^) o sin (ωt) = max
- Many systems settle into a steady state, where the amplitude of the motion remains constant – I.e. the energy input per cycle exactly equals the decrease in energy in each cycle due to nonconservative forces
Forced Oscillations, 2
- The amplitude of a driven oscillation is
ω 0 is the natural frequency of the undamped oscillator
0 2 2 22 0
F A m b m
ω ω ω
= ⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠
Resonance, Final
The shape of the resonance curve depends on Damping:
(1) The amplitude increases with decreased damping
(2) The curve broadens as the damping increases
