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