Oscillations and Waves

Question 1

Differential equation for mass on a spring

Answer 1

d^2x/dt^2=-k/m*x
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Question 2

Differential equation for pendulum

Answer 2

d^2(theta)/dt^2=-g/l*theta

Question 3

Differential equation for LC circuit

Answer 3

d^2i/dt^2=-1/LC*i

Question 4

What is the general solution for a harmonic oscillator

Answer 4

Acos(w0*t+phi)
for any displacement eg. x, theta, etc.

Comes from the superposition from the two general solutions.

Question 5

How are circular motion and harmonic motion linked

Answer 5

harmonic motion is circular motion projected onto the real (x) axis

Question 6

What is the total energy of a naturally oscillating system

Answer 6

1/2mw0^2*A^2

Question 7

what is w0 for a spring

Answer 7

sqrt(k/m)

Question 8

what is w0 for a pendulum

Answer 8

sqrt(g/l)

Question 9

what is the damping coefficient

Answer 9

gamma=b/m where b is the constant of proportionality of the linear damping term

Question 10

what is the term added to the equation for a damped harmonic oscillator

Answer 10

-b*velocity [x dot]

Question 11

what is the quadratic solution for w for a damped oscillating system

Answer 11

w=i(gamma)/2 ± sqrt((wo)^2-(gamma/2)^2)

Question 12

what is the case for light damping

Answer 12

gamma/2 < w0

Question 13

what is the light damping solution

Answer 13

Ae^(-gammat/2) cos(wdt+phi)

Question 14

what does an increased damping coefficient mean for a lightly damped system

Answer 14

a decreased time to get to equilibrium

Question 15

what is the case for heavy damping

Answer 15

gamma/2 > w0

Question 16

what is the heavy damping solution

Answer 16

Be^-t(gamma+)/2 + Ce^-t(gamma-)/2

Question 17

what does an increased damping co-efficient mean for a heavy damped system

Answer 17

an increased time to get to equilibrium

Question 18

what is the case for critical damping

Answer 18

gamma/2=w0

Question 19

what is the critical damping solution

Answer 19

(B+C)e^-t(gamma)/2

Question 20

what is particular about critical damping

Answer 20

there is no oscillation, there is just a decay

Question 21

what is the form of the external driver force

Answer 21

Focos(wt)

Question 22

what is the solution, and parameters, doe the driven oscillator
what type of solution is this

Answer 22

x=Acos(wt+phi)

STEADY STATE SOLUTION
where:

A=(Fo/m)/sqrt((w0^2-w^2)^2+w^2(gamma)^2)

phi=tan^-1(-w(gamma)/w0^2-w^2)

Question 23

when does resonance occur

Answer 23

w=w0

Question 24

What happens to phi over the resonance ‘cycle’

Answer 24

below resonance, phi=0 [driving force too slow]
at resonance, phi=-pi/2
above resonance, phi=-pi [driving force too fast]

Question 25

what determines the width of the resonance amplitude peak

Answer 25

gamma (damping co-efficient)

Question 26

what is a special case for the driven harmonic oscillator and what are the solutions

Answer 26

at resonance, w=w0:

A=(Fo/2mw0)/sqrt((w-w0)^2+(gamma/2)^2)

phi=tan^-1(gamma/w-w0)

Question 27

what is the equation for Q

Answer 27

Q=w0/gamma