13. Oscillations Flashcards
What is simple harmonic motion (SHM)?
An object experiencing SHM experiences a restoring force
The force is directly proportional to the objects distance
As it’s a restoring force, it acts away from the distance
Hence,
F=-kx
What is a restoring force?
A force that acts towards the centre of equilibrium
How does the acceleration correlate with displacement in SHM?
As force is directly proportional to acceleration, a must be directly proportional to -x
It is linked by angular speed, ω^2
a= -(ω^2)x
What is angular speed, ω in SHM?
The angle refers to between the displacement and the equilibrium
Angular speed refers to how far the angle moves through a period of time
v=s/t
So, ω=2π/T
ω = 2πf
How do we find the displacement, x at certain time periods?
When the mass of the spring starts oscillating from 0, it creates a sine graph, hence
x = Asin(ωt)
When the mass of the string starts oscillating from max displacement: the amplitude, A
x = Acos(ωt)
Where A = amplitude, m
Not given the sine formula in exam
How do we find the velocity of the object in SHM?
When object starts from amplitude, A
The derivative of cosx = -sinx
and v =dx/dt
Hence:
v = -A ωsin(ωt)
Given to in an exam
However, when object starts at equilibrium
The derivative of sinx = cosx
Hence:
v = Aωcos(ωt)
This one is not given in an exam
How do we find the acceleration of the object in SHM through the graph?
acceleration = dv/dt
Meaning for when object starts from amplitude, A
a = -A(ω^2)cos(ωt)
Given in exam
When object starts from equilibrium
a = -A(ω^2)sin(ωt)
This is nor given in an exam
How do we find the maximum acceleration, a(max)?
As a=-(ω^2)x
When x = A (amplitude)
a will be max, hence
a(max) = -(ω^2)A
How do we find the maximum velocity, v(max)?
As v = -Aωsin(ωt)
the max sin(ωt) an be = 1
Hence,
v(max) = -Aω
What are the 2 types of SHM and how can we find out their time periods, T?
A mass on a string, with a length of string and gravity affecting the time period only
- like a pendulum
T = 2π √(l/g)
A mass-spring system, with the mass of the object and spring constant affecting the time period only
T = 2π √(m/k)
Get both of these formulas, just need to know when which one applies
How do we draw the displacement, velocity and acceleration graphs for non-damping SHM?
Displacement moves away, the towards equilibrium, creating either a sine or cos graph
v = dx/dt, meaning the graph is it’s derivative
a=dv/dt/ meaning the graph is v’s derivative
What is resonance?
Where amplitude of oscillations of a system drastically increases due to gaining an increased amount of energy from a driving force
Resonance occurs when the driving frequency is equal to the natural frequency of a system
What is a driving force?
When vibrations are forced, so forced vibrations are present, the system experiences an external driving force
What is a natural frequency?
An object has free vibrations when its acted on from internal disturbance
All objects have internal disturbance
What is damping?
When a SHM system looses energy to its environment
Leading to reduced amplitude of oscillations
How does the conservation of energy apply to SHM?
An oscillating system can’t gain or loose energy without external forces acting upon it
How does kinetic energy and potential energy change during SHM?
At amplitude, A, potential energy is maximum and kinetic energy = 0
At equilibrium, kinetic energy is maximum and potential energy = 0
In an undamped system, the total energy stays a constant the whole way through the motion
What are the 3 types of damping depending on how quickly the amplitude decreases?
Light damping
Heavy damping
Critical damping
What is light damping?
Oscillations are lightly damped
The amplitude decreases exponentially
The frequency and time period of oscillations remains a constant, keeping it in SHM
What is heavy damping?
There are no oscillations
But takes a while until reaches equilibrium
The gradient decreases slower and slower as time goes on
E.g. door dampers to prevent them from slamming shut
What is critical damping?
Returns to rest at its equilibrium position in the shortest possible time without oscillating
E.g. car suspension systems prevent the car from oscillating after travelling over a bump in the road
What happens to the natural frequency, when there is more damping?
The natural frequency slightly decreases, shifting the amplitude graph left
The graph also spreads out more (broadens)
Why are ductile materials used to damp oscillations?
Ductile materials are able to undergo a large amount of plastic deformation
As plastic deformation takes the energy out of oscillations, it enables them to damp
Why does a material plastically deform during damping?
Energy is absorbed by internal friction, taking out energy from the oscillation, therefore damping the oscillation
The oscillation causes stress on the material. Going past the materials elastic limit the material will deform
