Oscillations Flashcards
Define displacement of wave
The distance of a point on a wave from its undisturbed position
Define amplitude
The maximum displacement from its undisturbed position
Define time period
The time taken for one complete oscillation
Define frequency
The number of vibrations per second of any oscillating system
f = 1/T
Define angular velocity, and state the equation to calculate it
Angular velocity is the rate of change of angle for a circulating object
ω = dθ/dt
ω = 2π/T
or ω = 2πf
Define angular frequency, and state the equation to calculate it
Angular frequency is the number of cycles per unit time
It uses the same equations as angular velocity
ω = 2π/T
or ω = 2πf
What is phase difference and what is it measured in?
Phase difference is how out of step two identical waves are. It is measured in degrees or radians.
How do you calculate phase difference?
t/T x 360 = degrees
or
t/T x 2π = rad
Where t is the time interval between two corresponding points on the waves, and T is the time period
State the conditions for a particle to perform simple harmonic motion
- the acceleration of a particle is directly proportional to its displacement
- the acceleration of the particle is directed in the opposite direction to the displacement
What are the shapes of the displacement/time, velocity/time and acceleration/time graphs of a particle in SHM, starting from rest position
- displacement/time graph is a sine wave
- velocity/time graph is a cosine wave
- acceleration/time graph is a sine wave reflected in the x-axis
What are the relationships between displacement, velocity and acceleration for a particle in SHM?
- the velocity is a maximum when the acceleration is zero and the displacement is zero
- the acceleration is a maximum when the displacement is at a maximum and the velocity is zero
State the formal relationship between acceleration and displacement for a particle in SHM
acceleration ∝ -displacement
a ∝ -x
What is the constant of proportionality added to the relationship, a ∝ -x, and what equation does it form?
Constant of proportionality = ω² or (2πf)²
a = -ω²x
or a = -(2πf)²x
This is the general equation for SHM
Write the equation and it’s variations that can be used to calculate displacement, x, when x = 0 at t = 0
x = Asinθ
x = Asin(ωt)
x = Asin(2πft)
Write the equation and it’s variations that can be used to calculate displacement, x, when x = A at t = 0
x = Acosθ
x = Acos(ωt)
x = Acos(2πft)
or x = Asin(2πft + φ)
What are the equations for velocity and acceleration given by differentiating x = Asin(2πft)
v = 2πfAcos(2πft)
or v = ωAcos(ωt)
a = -(2πft)²Asin(2πft)
or a = -ω²t²Asin(ωt)
State the equation for velocity containing amplitude and displacement, and hence state the equation for maximum velocity
v = +/- ω√(A² - x²)
Velocity is a maximum when displacement is zero
Therefore v max = +/- ωA
or v max = +/- 2πfA
State what will happen to the period of a simple harmonic oscillator if its amplitude is changed
The period of a simple harmonic oscillator is independent of its amplitude. This means that the period stays the same when the amplitude is changed.
What are the energy changes in SHM?
In an oscillation there is a constant interchange of energy between potential and kinetic, and if the system is undamped its total energy is constant
What are the relationships between displacement, kinetic energy and potential energy for a system in SHM?
- the kinetic energy is a maximum when the displacement is zero and potential energy is zero
- the potential energy is a maximum when the displacement is at a maximum and the kinetic energy is zero
State the equation for potential energy in a spring system
Ep = 1/2kx²
State the equation for total energy
E total = 1/2kA²
State the equation for kinetic energy, using the equations for E total and Ep
Ek = E total - Ep
Ek = 1/2kA² - 1/2kx²
Ek = 1/2k(A² - x²)
Define the total potential energy of a system for a spring oscillating vertically
For a spring oscillating vertically, the total potential energy will be the sum of the elastic and gravitational potential energy of the system
State and describe the three types of damped motion
Under-damped motion - there is an exponential envelope
Critical damping - no oscillation occurs, but the motion ceases in as short a time as possible
Over-damped motion - no oscillation occurs but the time to return to the equilibrium position is longer than for the critical case
Define critical damping
Critical damping occurs when the motion ceases after the shortest possible time
What is a feature of an exponential envelope?
In equal times the amplitude is seen to fall by equal fractions
Eg. if A1 A2 A3 etc are amplitudes after equal intervals of time then
A1/A2 = A2/A3 etc
What is the difference between natural frequency and forcing frequency?
All systems have a natural frequency of oscillation, which could be SHM, a more complex harmonic motion or damped harmonic motion.
An outside agent can apply a forcing frequency of any value, f, to the system. This forcing frequency is a source of energy, passed on to the system.
Define resonant frequency
When the forcing frequency is equal to the natural frequency of the system, then resonance is said to occur, and amplitude is a maximum
State the relationships between resonance and damping
- as we increase damping, the amplitude of the resonant vibration decreases
- the resonant peak also becomes broader, and resonance occurs at a lower frequency
