Oscillations

Question 1

What are the equations for w^2 for a massless pendulum and a massless spring?

Answer 1

• w^2 = g/L

- w^2 = k/m

Question 2

What is are the relationships for resonant frequency and period of oscillation ?

Answer 2

• w(o) = 2 Pi/T

- T = 1 / f(o) = 2 Pi/ w(o)

Question 3

What is the definition of simple harmonic motion ?

When can we say a system is simple harmonic

Answer 3

When the displacement of an object is a sinusoidal function of time
This is a characteristic of any system where the restoring force is proportional to the displacement

Question 4

How do you set up the second order differential equations for oscillating systems ?

Answer 4

Take all your restoring forces and set them equal to f=ma and then switch you acceleration and velocity to be derivatives of displacement

Question 5

What is the differential equation for an undampened simple harmonic oscillation ?

Answer 5

d^2 x/ d t^2 = - w^2 x(t)

Question 6

What form does the equation for elastic potential energy , and kinetic energy of an elastic object take ?

Answer 6

Elastic potential - U = 1/2 k x^2

Elastic Kinetic - K = 1/2 m v^2

Question 7

What is the total mechanical energy of a system ?

Answer 7

It is the elastic potential + the kinetic energy of a system
E = 1/2 k x^2 + 1/2 m v^2

Question 8

What is the equation for the angular velocity, and period in terms of moment of inertia ?

Answer 8

• w(o) = sqrt( mgh / I )

- T = 2Pi sqrt( I / mgh )

Question 9

What are the small angle approximations for sin, cos and tan ?

Answer 9

Sin x => x
Tan x => x
Cos x => 1
(This for a pendulum would mean that x = s / L)

Question 10

What theorem describes an oscillating system about a pivot point that is not the centre of mass?

Answer 10

The parallel axis theorem
This is given by
I = I(com) + m d^2

Question 11

What are the four types of damping ?

Answer 11

Un-damped
Under-damped
Critically-damped
Over-damped

Question 12

What is the equation for the force of damping ?

Answer 12

F(damp) = -b v

Question 13

What is special about critical damping in comparison to under or over damping ?

Answer 13

Critical damping is where the system returns to its equilibrium point in the shortest amount of time (w(o) = gamma)

Over damped takes longer (w(o) < gamma) and under damped will slowly decay as it loses energy (w(o) > gamma)

Question 14

What is the solution to the differential equation in the under damped case ?

Answer 14

x(t) = A sin(wt+phi) exp( -gamma t)

Question 15

In what direction does F(damp) apply?

Answer 15

F(damp) is always applied so that it opposed the velocity of the system

Question 16

What is gamma equal too ?

Answer 16

gamma = b/2m

Question 17

How do you find the damped angular frequency?

Answer 17

w(1) = sqrt(w(o)^2 - gamma^2)

Question 18

What is the Q factor for a system ?

Answer 18

The Q factor or quality factor describes the amount of damping in a system
The higher the Q value, the weaker the damping in the system
This will be given by
Q= 2Pi (E(stored)/E(lost per cycle))

Question 19

What is the solution to the differential equation in the critically damped case?

Answer 19

x(t) = (A+Bt) exp( -gamma t)

Question 20

What can the Q factor be reduced to for under damped systems?

Answer 20

Q= w(o) m / b

Question 21

How do you find the roots to second order differential equations for oscillating systems with some form damping ?

Answer 21

r = -gamma +/- sqrt( gamma^2 - w(o)^2 )

Question 22

What is Tao for oscillating systems and what is it given by ?

Answer 22

Tao is the time constant of a system
it is given by
Tao = m / b

Question 23

Describe what is meant by a forced oscillator

Answer 23

A forced oscillator is an oscillator subjected to a periodic external force
This is mainly be by a force that is sinusoidal
F(ext) = F(o) cos( w(dr) t )

Question 24

What is w(dr) ?

Answer 24

It is the angular frequency for a driven oscillation in the steady state

Question 25

What is the angular resonant frequency defined as ?

Answer 25

w(res) = sqrt( w(o)^2 - 2.gamma^2 )

Question 26

When damping is small, what equations of Q can we equate in terms of decay rate and resonance?

Answer 26

Q(decay) = w(1) m / b
Q(res) = w(res) / E(FWHM)

Question 27

What can we say about the size of the value of Q if we see a sharp resonance curve and a large resonance amplitude ?

Answer 27

Q is large

Question 28

True or False:

One normal mode can be excited by multiple resonant frequencies

Answer 28

False

one normal mode can only be excited by one resonant frequency

Question 29

True or False:

One mode will never decay into another mode

Answer 29

True

Question 30

How many modes will there be if three oscillators are coupled together?
What is the general formula for number of modes to number of oscillators?
What can we say about a system that has more modes than coupled oscillators?

Answer 30

• There will be 3 modes
• The number of modes equals the number of coupled oscillators
• The other modes that are greater than the number of coupled oscillators are just superposition of the initial modes

Question 31

What can we say about the dependence of normal modes?

Answer 31

Normal modes are independent of one another

Question 32

What is beating ?

Answer 32

Beating is an interference effect
In beating there is an exchange of energy between individual oscillators but there is no exchange of energy between normal modes

Question 33

What can we say about the first normal mode in a coupled oscillating system ?

Answer 33

It is the same as the resonant frequency for the system as if it were one mass at its centre

Question 34

What is the definition of a normal mode?

Answer 34

A collective motion at a single frequency that does not exchange energy with other modes