Simple Harmonic Motion Flashcards
What is the equation for the restoring force for simple harmonic motion?
F = -kx -kx = m × d^2x/dt^2 k = Constant for the material
What is an equation to find the acceleration during simple harmonic motion?
a = -ω^2x a = -Bω^2sin(ωt)
What is the equation to find velocity during simple harmonic motion?
v = Bωcos(ωt)
What are the equations to find the displacement during simple harmonic motion? (3)
x = Acos(ωt) x = Bsin(ωt) x = Ccos(ωt) + Dsin(ωt)
Describe the simple harmonic motion of a mass on a spring
-kx = m × d^2x/dt^2
==>
d^2x/dt^2 = -(k/m)x
ω = √(k/m) [DO NOT LEARN THIS EQUATION]
Describe the simple harmonic motion of a capacitor and an inductor
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]
What happens to an atom with ‘Z’ number of electrons with a centre of mass ‘c’ when an electric field is applied to it?
The centre of mass of the electrons (‘c’) moves by a distance of x from the origin position
What is the equation for the force on the electrons around an atom when an electric field is applied to it?
F = QE = ZeE
What is the equation for the restoring force on electrons around an atom when an electric field is applied?
F = -βx β = Constant for restoring force
What is the equation when the restoring force is in equilibrium force with the force of the electric field?
ZeE = -βx
You need to know what happens to the dipole moment when an electric field is on an atom. The equation folows:
p = Zex = Ze ( ZeE / β ) = (( Z^2 e^2 ) / β ) E
[YOU DO NOT NEAD TO LEARN THIS]
What happens when an electric field applied to an atom is suddenly removed? What are the equations for this?
Only the restoring force is present > F = Ma > Mass Electrons = ZMe > -βx = ZMe > ω = √ ( β / ( ZMe )) [DO NOT LEARN THIS]
How is the polarizibility calculated from the SHM of the atom when an electric field is removed?
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
Describe SHM with a mass on a spring with a resistive force
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)
Describe SHM for a capacitor and inductor and a resistor
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
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.
0 = (d^2x/dt^2)m + (dx/dt)β + kx 0 = (d^2q/dt^2)L + (dq/dt)R + q/c
What happens to the dipole moment across an atom when an electric field across an atom is instantly becomes 0?
It does not immediately become 0. There is a relaxation time because the charges need to move back into their origin position.
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?
> Exponential decay
> Symbol: τ
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]
dp/dt = - ( p - α(0)E ) / τ
What happens when the electric field oscillates with a sinusoidal AC signal?
> The induced dipole moments try to follow the electric field direction
If they manage to follow the electric field then at any moment
What happens when the electric field oscillates at a low frequency sinusoidal AC signal?
> The induced dipole moment follows the value of the AC signal reliably. (DC)
What happens when the electric field oscillates at a high frequency sinusoidal AC signal?
> The induced dipole moment cannot keep up
> The induced dipole moment is therefore 0
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]
At low frequencies where ω ≈ 0 and jω «_space;1:
α(ω) = α(0) becuase 1 + jwτ ≈ 1
At high frequencies where ω ≈ ∞ and jω»_space; 1:
α(ω) = 0 because 1 + jωτ ≈ ∞ and 1/∞ = 0
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]
At high but not super high frequencies:
Amplitude falls with the relationship of: ∝ 1/ω
Amplitude lags behind by 1/j = 90°
If a dielectric that is not suited to high frequency is placed in a high frequency AC electric field, what happens?
It might as well be that there is no dielectric. Treat it like a vacuum.
