Unit 3.2 - Vibrations Flashcards
2 categories of all motions
Periodic motion
Non-periodic motion
Periodic motion
An object repeats the same pattern of motion over certain periods of time (e.g - butterfly wings and machines)
What’s the problem with the pattern of periodic motion
Often complex and difficult to analyse
What do we use to help us analyse periodic motion? Why?
Use motion in a circle, as it’s also periodic and is easier to understand
Oscillator
An object moving backwards and forwards in a periodic motion
Oscillation
The motion of the oscillator
Period
The time taken for one oscillation (s)
Frequency
The number of oscillations in a fixed amount of time (per second)
Amplitude (when describing simple harmonic motion)
The largest displacement form the undisturbed position (m)
Phase (when describing simple harmonic motion)
The relationship cycle between 2 systems oscillating or between 2 points in a single oscillation
How could you describe 1 complete oscillation?
Motion from one point, passing through all the other points, passing through all the other points on the line and returning to the original position
What is independent of the amplitude of the oscillator during simple harmonic motion?
Period
Equation for acceleration towards the centre of a circle and how it’s adapted for simple harmonic motion
a = ω^2r
In simple harmonic motion, the acceleration has an inverse relationship with the distance from the midpoint, so the RHS becomes negative, and r is replaced with x (displacement)
a = -ω^2x
What does acceleration have an inverse relationship with in simple harmonic motio?
Distance from the midpoint
What 3 things can be deduced from the equation a = -ω^2x?
The acceleration is proportional to the displacement
The direction of the acceleration is opposite to the direction of displacement
The motion’s acceleration is only dependent on the period and the displacement
What is simple harmonic motion’s acceleration dependent on?
The period and the displacement (not the amplitude)
Why is simple harmonic motion called this?
Sinusoidal motion and a pretty straightforward equation
Simple harmonic motion definition
Simple harmonic motion occurs when an object moves such that its acceleration is always directed towards a fixed point (equilibrium position) and proportional to its distance from the fixed point
In which direction is the force causing simple harmonic motion?
In the same direction as the acceleration
Relationship between the force and displacement from the centre point in simple harmonic motion
The force is directly proportional to the displacement from the centre point and is always directed towards it
How do we know if an x vs t graph uses a cos function?
It starts at a maximum (x=A when t=0), then at t = T/2 the displacement is A again. The displacement is at zero exactly halfway between t=0 and t=T/2, then is at A again at t=T
How does the graph of displacement against time repeat its motion for simple harmonic motion?
With a period of T
How do we model the displacement (x) sinusoidally from a cos graph of displacement against time?
We need to relate time t to an angle
The period of a sine graph = 2pi radian
So, to construct the model, we multiply t by 2pi/T radian s-1
So, when t=T —> 2pi/T = 2pi radian
How come x=A when t=0, T and 2T on the displacement against time graph?
Using x ∝ cos (2pi/T x t)
The equation gives +1, which is the maximum value of cos
How come x = -A when t=0.5T, 1.5T and 2.5T on a displacement against time graph?
Using x ∝ cos (2pi/T x t)
Gives -1
What is the maximum value of cos?
1
What can we write as x = A when cos (2pi/T x t) = 1?
x = Acos (2pi/T x t)
Derive the equation x = Acos(ωt)
x = Acos (2pi/T x t)
ω = 2pi/T
So, x = Acos(ωt)
Under what conditions is it okay to use the equation x=Acos(ωt)?
As long as the timing starts at the maximum oscillation (amplitude)
I.e. - when t=0, x=A
How is the equation for simple harmonic motion altered when the displacement starts being timed with the amplitude at the bottom?
x = -Acos(ωt)
How is the equation for simple harmonic motion altered when the displacements starts being timed with the amplitude in the middle, going up?
x = Asin(ωt)
How is the equation for simple harmonic motion altered when the displacement starts being timed with the amplitude in the middle, going down?
x = -Asin(ωt)
When can we not just alter the simple harmonic motion equation?
If timing wasn’t started exactly at a middle or maximum
If timing wasn’t started exactly at a middle or maximum, what must we do to the simple harmonic motion equation?
A term must be added to the ωt to compensate for the difference in phase —> ε
General equation for simple harmonic motion
x = Acos (wt + ε)
What does ε stand for in the simple harmonic motion equation?
The phase difference that is given in radians
What is ε basically used for?
To “shift” the function back to the cos function
Equation for the period of a system having stiffness (force per unit extension) k, and mass m
T = 2pi √m/k
What is k in T = 2pi √m/k
Spring constant
Spring constant
Force per unit extension (stiffness)
Unit of k
Nm-1
Give 2 examples of simple harmonic motion
A mass-spring system
A pendulum
When is the the velocity of an object undergoing SHM at its maximum?
When the object travels through the midpoint of oscillation
What happens to the velocity of an object undergoing SHM as it reaches its maxima?
Decelerates
How do we know that the velocity of an object undergoing SHM is at its maximum when travelling through the midpoint of oscillation?
The steepest gradient on a displacement against time graph
Velocity
The rate of change of displacement
What is the gradient of a displacement against time graph?
Velocity
How do we know if a graph is a cos graph?
Values between -1 and +1 only
Starts at a maximum
T in an angle
2pi
T/2 in an angle
Pi
T/4 in an angle
Pi/2
3/4T in an angle
3pi/2
What is the velocity of an object undergoing SHM at the maximum position?
Zero
How can we prove that acceleration and displacement are out of phase during SHM?
Using a displacement against time graph
At maximum displacement, the velocity is zero
What are out of phase during SHM?
Displacement and acceleration
Velocity at t = T/4 on a cos graph
-Vmax
Velocity at T/2 on a cos graph
0
Velocity at T3/4 on a cos graph
+Vmax
Velocity at T on a cos graph
O
Velocity at t=0 on a cos graph
0
How do we derive the velocity in SHM equation?
Equation for displacement during SHM:
x = Acos (wt + ε)
From this equation, we see that the maximum displacement (A) occurs when the value of the cos term is 0 or pi
We require an expression that returns a value of zero at these points and a maximum when x=0
The sin function gives us this result - it is exactly the same shape as a cosine wave, but pi/2 out of phase
That is, when sin x=1, cos x=0 and when sin x=0, cos x = +-1
So, the velocity for an object at any time t undergoing SHM is given by V= -Asin (wt + ε)
Equation for displacement during SHM
x = Acos (wt + ε)
Equation for velocity in SHM
V = -Awsin (wt + ε)
Can the equation for velocity in SHM be adapted to get rid of the phase factor (ε)?
Yes
Prove that the equation for the velocity of an object in SHM works
At t = T/4…
(ε = 0)
V = -Asin (wt + ε)
V = -Awsin (2pi/2 x T/4)
(Simplified and calculated)
V = -Aw
So, -Vmax = -Aw
When do we need to ensure that the calculator is in radians?
Whenever using sin or cos
How do we put the calculator in radians?
1.) shift
2.) menu setup
3.) 2 (angle unit)
4.) 2 (radians)
ε
Phase angle
What value does ε usually have?
+ pi/2 or -pi/2
What is T also equivalent to? (2 different things)
1/f
2pi/w
What can both the displacement and acceleration during SHM motion equations be used to derive? How?
By inputting the displacement equation into the acceleration equation
a = -w^2x
Where does the equation a = -w^2x come from?
Inputting x=Acos(wt + ε) into a = -Awcos(wt + ε)
Where would friction come from during SHM?
Air resistance or at the point of suspension
What does an object in SHM do if there’s no friction at all?
Will preserve all energy that is has ben given to start moving
What happens to energy during SHM in the real world? Give an example of how we know this is true
Energy is lost
A pendulum doesn’t swing forever
In which situation can we disregard energy losses in oscillation calculations? Why?
Over a few oscillations to a close approximation, since the energy losses are only small
When a dynamic trolley is tethered between 2 identical springs, what can this pretty much be taken as?
It’s like fixing the trolley in the middle of a spring twice the length of either of the 2 springs, so the 2 springs together be regarded as one spring which obey’s Hooke’s law
What will the resultant force exerted by the springs be when a trolley tethered between 2 identical springs is pulled to one side of the equilibrium position so that its displacement is x?
-kx
What do we need to calculate to calculate the energy stored in a stretched spring?
The work done on the spring in stretching it
How do we calculate the work done in extending a spring from extension 0 to extension x?
By calculating the energy transferred to it
Equation for calculating the work done on spring (energy transferred to it)
Average resultant force applied x extension
Under which conditions can we use the equation…
Work done on spring (energy transferred to it) = average resultant force applied x extension
?
Assuming that Hooke’s law is obeyed
Apply force Fapplied
Causes extension x
Derive the equation W = 1/2Kx^2
Energy stored in a spring
W = 1/2k
Force applied
F = kx
Energy stored (elastic potential energy)
W = 1/2kx^2
Equation for energy stored in a spring
W = 1/2kx
Equation for force applied to a spring
F = kx
Why is the equation for the energy stored in a spring W = 1/2kx
Area under the force-extension graph
Equation for energy stored in a spring (elastic potential energy)
W = 1/2kx^2
If frictional forces are neglected, what happens to the potential energy stored in a spring when the trolley between 2 springs is release?
Be converted into kinetic energy and the trolley returns to its equilibrium position
When a trolley between 2 springs is in its equilibrium position, what are the values of displacement and potential energy?
x = 0
So Ep = 0
Relationship between kinetic energy and potential energy
Ek + Ep = constant = Ep(max)
When is the maximum potential energy of a spring system?
When the mass has x=A
Equation for working out Ep max
Ep (max) = 1/2kA^2
Equation for working out the kinetic energy of a spring with extension x + show how you get to this
Ek = Ep(max) - Ep(x)
1/2kA^2 - 1/2kx^2
1/2k(A^2 - x^2)
What happens to the total energy in a system undergoing SHM?
Remains constant with no losses
Ek at extremes during SHM
0
Ep at extremes during SHM
Maximum
Are there negative energies plotted on a graph? Why?
No
Energies are scalar
No negative values
What have most of the systems examined in this unit been, which is theoretical?
Assumed to be oscillating freely, without friction acting
What is any real oscillation system subject to?
Frictional forces of many kinds
What do frictional forces do to an oscillating system?
Decrease the frequency and amplitude of the oscillations
Is damping deliberate?
In many cases, yes
Types of damping
Light damping
Heavy damping
Critical damping
What is light damping?
The system oscillates about the equilibrium point with decreasing amplitude
When does light damping occur?
Naturally, when an oscillating object is subjected to air resistance or to friction
What is heavy damping?
The system takes a long time to return to its equilibrium point
The object only completes 1/4 of an oscillation and doesn’t continue through the equilibrium position
Example of heavy damping
A pendulum swinging in oil
What is critical damping?
The system reaches equilibrium and comes to rest in the shortest time possible, about T/4
What type of damping is critical damping similar to but how is it different?
Heavy damping, but the object comes to rest in the same time as it would in a free oscillation (nearly) but doesn’t proceed through the equilibrium position
What’s the most useful case of damping?
Critical damping
What is critical damping used in?
Vehicle suspensions
Why is critical damping used in vehicle suspensions and how does this work?
A driver wants to absorb a bump in the road, but doesn’t want to go bouncing along the road afterwards
By damping the spring critically, it means the wheel is pushed straight back onto the road without subsequent bounces
How could a system oscillating freely be described?
Has only the initial energy input given to it and then continues to repeat the same motion “for ever” (assuming no energy loss)
Under which conditions will a system resonate?
When given extra energy, if its given at a certain frequency , it will be absorbed and the system will resonate
When does resonance occur?
When a system or object is subjected to an external force or vibration that matches its natural frequency
Forced oscillations
When systems are oscillating in response to an input of energy from an external force
When systems are oscillating in response to an input of energy from an external force
Forced oscillations
In a spring set-up, how is the electric motor connected and why?
Connected via variable resistor
The speed can be varied continuously
In the spring set up, why is the mass in a transparent tube?
To prevent the mass from moving excessively from side to side when the amplitude of the oscillations is large
When would a mass connected to a spring move excessively from side to side?
When the amplitude of the oscillations is large
What happens to the spring set up when the speed of the motor is high? Why?
Top of the spring is moved up and down rapidly
The mass hardly moves at all (the amplitude of the drive oscillations is small)
What happens as the speed of the motor in the spring set up is reduced?
Th amplitude of the driven oscillations increases until it reaches a maximum
What happens to a system when the amplitude of driven oscillations is at a maximum?
Resonance
What is happening when the amplitude of the driven oscillations on a spring reaches a maximum?
This driving frequency is the same as the natural frequency of oscillation of the mass on the spring = resonance occurs
Which frequencies are the same for resonance to occur?
The driving frequency is the same as the natural frequency of oscillation of the object
What happens as the driving frequency on an object decreases further past the point of maximum amplitude?
The amplitude of the driven oscillations again decreases
What type of oscillating systems can exhibit resonance?
Any oscillating system
What in particular frequently result in resonance?
The interaction of waves and oscillating systems
Which frequencies get amplified?
Resonant frequencies
Can amplitudes increase infinitely with resonance?
No, there’s a limit to the amplitude of the oscillations, even when the driving frequency and driven frequency are the same
Why is there a limit to the amplitude of the oscillations that a mass makes, even when the driving frequency and driven frequency are the same?
The length of the string is a physical limit to the amplitude of the oscillations of the mass
There’s some damping of the oscillations, which absorbs energy
What happens when the driving frequency approaches the natural frequency of a system?
The amplitude of the oscillations increase rapidly
Under which conditions do the amplitude of oscillations increase rapidly if the driving frequency approaches the natural frequency of a system?
If the system is only lightly damped
When is the increase in amplitude of something oscillating less?
As the damping of the system increases
Quality factor/ Q-factor
The width of the resonance curves
The width of resonance curves
Quality factor/Q factor
When is the Q-factor better?
The sharper the curve (the narrower)
What does increasing the damping do to the Q factor and why?
Broadens the resonance curve
Reduced the Q-factor
What does light damping affect?
Amplitude
What does light damping have little effect on?
The natural frequency of oscillation of a system
What does heavier damping reduce?
The amplitude of the driven oscillations
Also reduces the natural frequency
Difference between light and heavier damping
Light - little effect on the natural frequency of oscillation of a system, although it does affect the amplitude
Heavy - reduces both the amplitude and the natural frequency
How could we damp a mass on a spring?
Put it into a liquid such as oil
Uses of resonance
Circuit tuning
Microwave heating
MRI scanners
Problems associated with resonance
The design of bridges
Buildings collapsing
Example of circuit tuning (use of resonance)
Tuning a circuit to resonate at certain radio frequencies
How does microwave heating use resonance?
Water molecules resonate at microwave frequencies, the microwaves cause oscillations in the molecules which heat up the water
Why does resonance need to be considered in the design of bridges?
Wind at the right frequency an causes them to collapse (e.g - Tacoma Narrows Bridge)
What needs to be considered relating to resonance when constructing buildings?
Short buildings respond to high frequency oscillation and vice versa
What type of damping is happening if the object still oscillates more than once?
Light
Purpose of damping in a car suspension system
Return quickly to equilibrium
Avoid resonance/large amplitudes
Reduce oscillations
Dissipating energy
Words to include when describing resonance
Certain (as in certain frequency)
Frequency
Maximum amplitude
What type of damping prevents oscillations occurring at all when the system is displaced and released?
Critical damping
If calculating maximum acceleration using a=-w^2x, how can we use the equation if we don’t have a value for x?
Remember that xmax = A
So, the equation is altered to a max = -w^2A
How do we show damping on a graph and why?
All lower down and a bit to the left
Lower peak
Natural vibrating frequency decreases
How is the equation v = -Awsin(wt + E) altered when working out the maximum velocity and why?
Vmax = Aw
(Since the maximum value of sin is 1)
What’s the assumption always made with calculations involving springs?
That the spring obeys Hooke’s law
Extension of a spring
The difference in length when the force is applied
How does damping occur in the air?
Transfer of kinetic energy to air molecules from the moving object
How do we measure T on a graph?
Peak to peak
What’s the phase angle when an object is moving through the equilibrium position towards the maximum positive displacement at t=0?
-pi/2
What’s the phase angle when an object is moving away from the maximum possible displacement at t=0?
Pi/2
Why is there a minus sign in v= -Awsin(wt + e)?
Ensures that v is initially negative as it moves away from the point of maximum possible displacement
Compare acceleration in SHM and displacement in SHM graphs and explain why this is the case
Mirror images of eachother
Since a = -w^2x
What happens in terms of energy during damping? Why?
Energy lost to surroundings due to resistive forces
What do we need to ensure that we do when using V = -Awsin(wt + e) and x = Acos(wt + e)?
Make sure that the calculator is in radians
How would we rearrange x = Acos (wt + e) to find t?
When moving cos to the other side, it’ll be cos-1
Forced oscillations
Periodic input of energy during the oscillation
Compare free and forced oscillations
Free —> energy input at start and then no further input energy
Forced oscillations —> periodic input of energy during oscillation. As driving force approaches the resonant frequency (natural frequency) the amplitude of oscillation increase
Explain how a graph of a v.s x would show simple harmonic motion
Straight line: acceleration is proportional to displacement
Negative gradient: acceleration in the opposite direction to displacement
What kind of forces cause damping?
Resistive forces
When does resonance occur?
When a system or object is subjected to a periodic external force or vibration that matches its natural frequency
What shows that acceleration is proportional to displacement on a SHM graph?
Straight line through the origin
What will happen when heavy damping decreases the natural vibrating frequency of something?
The amplitude of oscillation will decrease too
What do we use when a question asks us to give the maximum height reached by a pendulum mass above the level of its lowest point?
1/2mv^2 = mgh
