Periodic Motion: Simple Harmonic Motion Flashcards
What is the amplitude of oscillations?
The max displacement of oscillating object from eqm.
For two objects oscillating at the same frequency, phase difference is radians is?
phase difference = 2πΔt/T,
where Δt is the difference between successive instances when the two objects are at max displacement in same direction.
(bc 2π = 360)
In degrees = 360Δt/T
Acceleration is greatest when gradient of velocity-time graph is greatest. This is when velocity is..
and displacement is..
..zero and occurs at max displacement in the OPPOSITE direction.
What are the conditions of SHM?
If the acceleration of a body is always directly proportional to its displacement FROM a fixed point, and always directed TOWARDS that point, the motion is SH.
a ∝ - x a = - ω^2 x
Time period is independent of?
Amplitude of oscillations.
What’s the restoring force?
For any oscillating object, the resultant force acting on the object acts towards the eqm position. This is the restoring force.
So if restoring force, F ∝ - x?
If F ∝ - x then a ∝ - x ∴ SHM
Mass-spring system: when does frequency decrease? Explain fully. (2)
- mass increases bc increases inertia ∴ at given x slower oscillations therefore takes more time.
- weaker spring used bc restoring force at given x decreases ∴ a and v decrease ∴ takes longer time.
SHM:
a =F/m= -kx/m = -k/m x = -ω^2x
Simple pendulum: what’s restoring force ( think weight) and ∴ what’s a?
Assumption?
Restoring force = -mgsinθ
∴ a = -mgsinθ/m = -gsinθ
Small angle approx sinθ = x/L
∴ a = -g/L x = -ω^2x
(at eqm T-mg = mv^2/L)
See sheet for derivation of T eqs!!!!!!!!!!!!!!!!!!!!!!!!!!!
Spring-mass system:
Et = Ek +Ep.
What’s Ek? What’s v?
Ek = Et - Ep = 1/2 k (A^2 - x^2)
= 1/2 mv^2
Bc ω^2 = k/m, above can be written as v^2 = ω^2 (A^2 - x^2)
What are dissipative forces?
What are damped oscillations?
Pendulum oscillations gradually die away because of air resistance gradually reduces the total energy of the system. The forces causing the amplitude to decrease are dissipative forces.
The motion is damped if dissipative forces are present.
Light damping?
Amplitude gradually decreases, reducing by same fraction each cycle.
T is independent of A, so each cycle takes same time as the oscillations die away.
Critical damping?
System returns to eqm position in shortest possible time without oscillating.
Damping is just enough to stop the system oscillating after its been displaced from eqm position and released.
Heavy damping?
Occurs when damping is so strong that the displaced object returns to eqm much more slowly than if the system were critically damped.
Eg a spring in thick, viscous oil.
How can we see if the damping /amplitude has exponential decay?
See if it has a constant half life on the graph.
What’s a free oscillating object?
A free oscillating object oscillates with a constant amplitude because there’s no friction/external force acting on it. The only forces acting on it combine to provide the restoring force.
What’s natural frequency?
When a system oscillates without a periodic force being applied/oscillates freely, the system’s frequency is called it’s natural frequency.
What’s a periodic force?
An external force repeatedly applied at regular intervals. System undergoes forced vibrations.
As we increase the applied frequency from zero: (2)
1 - amplitude
2 - phase difference
1 - amplitude of oscillations increase until it reaches maximum amplitude at a particular frequency, and then A decreases again.
2 - phase difference between displacement and periodic force increases from zero to 1/2π to π as the f increases further.
At max A, phase difference is 1/2π ∴ periodic force is exactly in phase with the velocity of the oscillating system, and the system is…..
is in resonance. The f at max A is called the resonant frequency.
The lighter the damping:
-the larger the max A becomes at resonance, and
-the closer the resonant f is to natural f of the system.
So for an oscillating system with little to no damping at resonance:
the applied frequency of the periodic driving force= the natural frequency of the system
At resonance, the periodic force acts on the system..
at the same point in each cycle, causing the A to increase to a max value limited only by damping.
(At max A energy supplied by the periodic force is lost at the same rate bc of the effect of damping.)
If we plot a graph of amplitude of forced oscillations against driving frequency, we see a very large peak at..
the natural frequency/resonant frequency.
Increased damping causes above graph to
Have a lower peak bc lower max A due to energy losses from system, less sharp peak/broader peak due to damping, reduced frequency (bc longer to complete each oscillation so resonant f gets further and further from natural f)
Resonance eg 1: Barton’s pendulums:
Pendulums along horizontal string, pendulum B is on one side and DEFH on other side, all of different length strings.
Pendulum E has same length as B. B is oscillated. What is observed and why?
Effect of oscillating motion transferred to rest of pendulums along horizontal string. E oscillates in resonance with B bc it’s subjected to forced oscillations of the same f as its natural f.
Resonance eg 2: Bridge oscillations:
If a bridge span is not fitted with —–, it can be made to oscillate at resonance if the bridge span is subjected to a suitable periodic force.
dampers
Can also: stiffen structure,
increase mass of bridge,
restrict number of pedestrians,
redesign to change natural f
