Simple Harmonic Motion: Flashcards
Define ‘free vibrations’
Amplitude is constant and no frictional forces are present.
Define time period of oscillation:
The time it takes to have passed through 1 position and then return to the same position.
Define angular frequency (equation)
= 2pi/T
Give the equation for phase difference of 2 oscillating objects:
= 2pi*change in t/T where change in t is the time between them reaching the same position or the amplitude.
Describe acceleration time graph for SHM:
Object starts at maximum negative acceleration, at T/2 it is max. positive value and then carries on in sine graph shape. Greatest acceleration occurs when velocity is 0.
Define acceleration for SHM:
1) Proportional to displacement.
2) Always in opposite direction to displacement.
a=-w^2*x
How can the frequency of a mass spring system be changed?
Change the mass or change the spring i.e. change k
Prove T = 2pi*sqr(rt) m/k
Since restoring force ,T, is proportional and in opposite direction to displacement, then T=-kx. therefore the acceleration is -kx/m. This can be rewritten as w^2 = k/m. Therefore (2pif)^2 = k/m. Therefore T = 2pisqr(rt)m/k
What is centripetal velocity squared equal to in a simple pendulum system?
g/L
How is SHM speed equation derived?
total energy =Ek +Ep therefore Ek =Et-Ep =1/2k(A^2-x^2) which can be rewritten as v^2=w^2(A^2-x^2)
What is the equation for maximum energy in SHM?
Et = 1/2kA^2
Define damping:
Motion when dissipative forces are present so the oscillations reduce to zero.
Define light damping:
Each oscillation takes the same amount of time regardless but it reduces by a fraction of the cycle each time.
Define critical damping:
When the oscillating system returns to zero in the shortest possible time. This is important for suspension in vehicles.
Define heavy damping:
System returns to equilibrium very slowly but doesn’t oscillate. Different to critical as critical returns to zero immediately.
What does graph of kinetic energy against displacement look like in SHM? How does this compare to a potential energy / displacement graph?
An inverted parabola. Starts at 0 and curves to maximum as velocity hits a maximum and then curves back to zero.
Potential energy starts at maximum when displacement is at a maximum and then follows parabolic shape. Inverse of kinetic energy
Hence total energy = kinetic energy + potential energy.
Define periodic force:
A force applied at regular intervals.
Define natural frequency:
The frequency a system oscillates at with no periodic force applied. This becomes a forced vibration when a periodic force is applied to it.
Example of forced vibrations: What does the graph show?
Mass held vertically between 2 springs. Top end is fixed. At the end of the bottom spring a mechanical oscillator provides a periodic force when connected to a signal generator (provides an applied frequency).
-Graph of amplitude against frequency shows that it reaches a maximum amplitude at a certain frequency
- The phase difference between the displacement and periodic force increases from zero to pi / 2 as it increases to max amplitude and then increases to pi as it increases further.
Define resonance:
When the phase difference between displacement and periodic force is pi/2 and it is in phase with velocity.
-This occurs when applied frequency is equal to the natural frequency of the system.
-The frequency this occurs at is the resonant frequency (this only occurs when there is little to no damping. At damping, resonant frequency is slightly below natural frequency)
-The periodic force is applied at the same point in each cycle.
-At max amplitude, the energy gained by the force is lost at the same rate because of damping.
How does resonance change as damping changes?
The lighter the damping:
-the larger the max amplitude is
-the closer the resonant frequency is to the natural frequency.
What happens as the applied frequency becomes larger than the resonant frequency?
-Amplitude of oscillations decrease.
-Phase difference between displacement and periodic force increases from pi/2 until the displacement is pi radians out of phase.
In Barton’s pendulum, why does the weight with the same length as the driver pendulum oscillate the most?
The length is the same therefore the time period is the same and therefore the natural frequency is the same. Therefore they both oscillate in resonance.
What is another example of resonance?
Stationary waves on a string. The string has a natural frequency and when the oscillator moves the string at a multiple of the first harmonic it is in resonance.
