Oscillations and Waves 1 Flashcards
Simple Harmonic Motion
Definition
Fx = -kx = max
In simple harmonic motion, the acceleration is both proportional to and oppositely directed from, the displacement from the equilibrium position
Simple Harmonic Motion
Position Function
x = A cos(ωt + 𝛿)
Simple Harmonic Motion
Angular Frequency
ω = 2πf = 2π/T
Simple Harmonic Motion
Mechanical Energy
E = K + U = 1/2 kA²
Simple Harmonic Motion and Circular Motion
If a particle moves in a circle with constant speed, the projection of the particle onto a diameter of the circle moves in simple harmonic motion.
Simple Harmonic Motion
General Motion and Equilibrium
If an object is given a small displacement from a position of stable equilibrium, it typically oscillates about this position with simple harmonic motion.
Natural Frequency
Mass on a String
ω0 = √(k/m)
Natural Frequency
Simple Pendulum
ω0 = √(g/L)
Natural Frequency
Torsional Oscillator
ω0 = √(κ/I)
I = moment of inertia κ = torsional constant
Torsional Constant
For small oscillations of a physical pendulum
κ = MgD
where D is the distance of the axis of rotation from the centre of mass
Damped Oscillations
Definition
In the oscillations of real systems, motion is damped because of dissipative forces.
If the damping is greater than some critical value, the system does not oscillate when disturbed it simply returns to its equilibrium position.
The motion of a weakly damped system is nearly simple harmonic motion with an amplitude that decreases exponentially with time.
Damped Oscillations
Frequency
ω’ = ω0 √(1 - 1/4Q²)
Damped Oscillations
Energy
E = E0 e^(-t/τ)
Damped Oscillations
Amplitude
A = A0 e^-(t/2τ)
Damped Oscillations
Decay Time
τ = m / b
