# Simple Harmonic Motion Flashcards

1
Q

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

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

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

A
```a = -ω^2x
a = -Bω^2sin(ωt)```
3
Q

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

A

v = Bωcos(ωt)

4
Q

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

A
```x = Acos(ωt)
x = Bsin(ωt)
x = Ccos(ωt) + Dsin(ωt)```
5
Q

Describe the simple harmonic motion of a mass on a spring

A

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

6
Q

Describe the simple harmonic motion of a capacitor and an inductor

A
```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]```
7
Q

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

A

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

8
Q

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

A

F = QE = ZeE

9
Q

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

A
```F = -βx
β = Constant for restoring force```
10
Q

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

A

ZeE = -βx

11
Q

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

A

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

[YOU DO NOT NEAD TO LEARN THIS]

12
Q

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

A
```Only the restoring force is present
> F = Ma
> Mass Electrons  = ZMe
> -βx = ZMe
> ω = √ ( β / ( ZMe )) [DO NOT LEARN THIS]```
13
Q

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

A

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

14
Q

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

A

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)

15
Q

Describe SHM for a capacitor and inductor and a resistor

A
```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```
16
Q

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.

A
```0 = (d^2x/dt^2)m + (dx/dt)β + kx
0 = (d^2q/dt^2)L + (dq/dt)R + q/c```
17
Q

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

A

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

18
Q

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?

A

> Exponential decay

> Symbol: τ

19
Q

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]

A

dp/dt = - ( p - α(0)E ) / τ

20
Q

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

A

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

21
Q

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

A

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

22
Q

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

A

> The induced dipole moment cannot keep up

> The induced dipole moment is therefore 0

23
Q

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]

A

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

24
Q

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]

A

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

25
Q

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

A

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