Simple Harmonic Motion

Question 1

What is the equation for the restoring force for simple harmonic motion?

Answer 1

F = -kx
-kx = m × d^2x/dt^2
k = Constant for the material

Question 2

What is an equation to find the acceleration during simple harmonic motion?

Answer 2

a = -ω^2x
a = -Bω^2sin(ωt)

Question 3

What is the equation to find velocity during simple harmonic motion?

Answer 3

v = Bωcos(ωt)

Question 4

What are the equations to find the displacement during simple harmonic motion? (3)

Answer 4

x = Acos(ωt)
x = Bsin(ωt)
x = Ccos(ωt) + Dsin(ωt)

Question 5

Describe the simple harmonic motion of a mass on a spring

Answer 5

-kx = m × d^2x/dt^2
==>
d^2x/dt^2 = -(k/m)x
ω = √(k/m) [DO NOT LEARN THIS EQUATION]

Question 6

Describe the simple harmonic motion of a capacitor and an inductor

Answer 6

Vc + VL = 0
q/c + LdI/dt = 0
q/c = -L(dq/dt)/dt
q/c = -L(d^2q/dt^2)
d^2q/dt^2 = -q/Lc
ω = 1 / √ Lc [DO NOT LEARN THIS EQUATION]

Question 7

What happens to an atom with ‘Z’ number of electrons with a centre of mass ‘c’ when an electric field is applied to it?

Answer 7

The centre of mass of the electrons (‘c’) moves by a distance of x from the origin position

Question 8

What is the equation for the force on the electrons around an atom when an electric field is applied to it?

Answer 8

F = QE = ZeE

Question 9

What is the equation for the restoring force on electrons around an atom when an electric field is applied?

Answer 9

F = -βx
β = Constant for restoring force

Question 10

What is the equation when the restoring force is in equilibrium force with the force of the electric field?

Answer 10

ZeE = -βx

Question 11

You need to know what happens to the dipole moment when an electric field is on an atom. The equation folows:

Answer 11

p = Zex = Ze ( ZeE / β ) = (( Z^2 e^2 ) / β ) E

[YOU DO NOT NEAD TO LEARN THIS]

Question 12

What happens when an electric field applied to an atom is suddenly removed? What are the equations for this?

Answer 12

Only the restoring force is present
> F = Ma 
> Mass Electrons  = ZMe
> -βx = ZMe
> ω = √ ( β / ( ZMe )) [DO NOT LEARN THIS]

Question 13

How is the polarizibility calculated from the SHM of the atom when an electric field is removed?

Answer 13

a = -(x0)ω^2sin(ωt) = d^2x/dt^2
Where x0 = initial displacement
-βx = M(d^2x/dt^2) => -βx/M = d^2x/dt^2
-(x0)ω^2sin(ωt) = -βx/M

Question 14

Describe SHM with a mass on a spring with a resistive force

Answer 14

Resistive force = -βv = -β × dx/dt
Restoring force: -kx - = m × d^2x/dt^2
Combining them: -kx -β(dx/dt) = m(d^2x/dt^2)

Question 15

Describe SHM for a capacitor and inductor and a resistor

Answer 15

Capacitor Voltage: V = q/c
Inductance Voltage: V = L(dI/dt)
Resistor Voltage: V = IR
L(dI/dt) + IR + q/c = 0 
L(d^2q/dt^2) + (dq/dt)R + q/c = 0

Question 16

How do the equation for SHM with a mass on a spring with a resistive force and SHM for a capacitor and inductor and a resistor seem related.

Answer 16

0 = (d^2x/dt^2)m + (dx/dt)β + kx
0 = (d^2q/dt^2)L + (dq/dt)R + q/c

Question 17

What happens to the dipole moment across an atom when an electric field across an atom is instantly becomes 0?

Answer 17

It does not immediately become 0. There is a relaxation time because the charges need to move back into their origin position.

Question 18

If the induced dipole moment was graphed against time when the electric field was removed what would it look like? What is the symbol associated with the time?

Answer 18

> Exponential decay

> Symbol: τ

Question 19

What is the equation for the induced dipole moment when the electric field is removed? [DO NOT NEED TO KNOW JUST NEED TO KNOW HOW TO USE]

Answer 19

dp/dt = - ( p - α(0)E ) / τ

Question 20

What happens when the electric field oscillates with a sinusoidal AC signal?

Answer 20

> The induced dipole moments try to follow the electric field direction
If they manage to follow the electric field then at any moment

Question 21

What happens when the electric field oscillates at a low frequency sinusoidal AC signal?

Answer 21

> The induced dipole moment follows the value of the AC signal reliably. (DC)

Question 22

What happens when the electric field oscillates at a high frequency sinusoidal AC signal?

Answer 22

> The induced dipole moment cannot keep up

> The induced dipole moment is therefore 0

Question 23

How does α(ω) = α(0) / 1 + jωτ explain the actions of a dielectric at different frequencies?
j = imaginary number
[DO NOT NEED TO KNOW EQUATION]

Answer 23

At low frequencies where ω ≈ 0 and jω &laquo_space;1:
α(ω) = α(0) becuase 1 + jwτ ≈ 1
At high frequencies where ω ≈ ∞ and jω&raquo_space; 1:
α(ω) = 0 because 1 + jωτ ≈ ∞ and 1/∞ = 0

Question 24

What does the equation [α(ω) = α(0) / 1 + jωτ] tell you about the relationship of of ω and j with amplitude?
[DO NOT NEED TO KNOW EQUATION]

Answer 24

At high but not super high frequencies:
Amplitude falls with the relationship of: ∝ 1/ω
Amplitude lags behind by 1/j = 90°

Question 25

If a dielectric that is not suited to high frequency is placed in a high frequency AC electric field, what happens?

Answer 25

It might as well be that there is no dielectric. Treat it like a vacuum.