Y2, C8 - Modelling with Differential Equations

Question 1

Write v and a in terms of x (displacement)

Answer 1

v = dx/dt
a = d2x/dt2

Question 2

What does simple harmonic motion mean

Answer 2

The acceleration is proportional to the displacement (x) of the particle from O

Question 3

What is O in SHM

Answer 3

The centre of oscillation

Question 4

What is the equation for SHM

Answer 4

a = -ω^2 * x
a = dv / dt = dv/dx * dx/dt = v * dv/dx

Question 5

Where is the acceleration of SHM always towards

Answer 5

The centre O

Question 6

What is ω

Answer 6

The angular velocity of the particle
(number of oscillations per 2pi seconds)

Question 7

By the chain rule, what is v * dv/dx equal to

Answer 7

Acceleration a

Question 8

In SHM, when x is maximum, what are a and v equal to

Answer 8

a = x
v = 0

Question 9

Using the harmonic identity x = a * sin(ωt + α), what is a equal to

Answer 9

Amplitude

Question 10

Using the harmonic identity x = a * sin(ωt + α), what is the period equal to

Answer 10

(2 * pi) / ω

Question 11

What methods are there for finding maximum displacement

Answer 11

Using the harmonic identity (find R)
v = 0

Question 12

x = Rsin(2t + α)
What is the period

Answer 12

One oscillation = 2pi / ω
ω = 2
Therefore:
One oscillation = pi

Question 13

What is the equation for a particle moving with damped harmonic motion

Answer 13

d2x/dt2 = -k * dx/dt - ω^2 * x
Thus
d2x/dt2 + k * dx/dt + ω^2 * x = 0

Question 14

What type of damping is caused by x = Ae^-αt + Be^-βt
(distinct roots of AE)

Answer 14

Heavy damping (no oscillations)

Question 15

What type of damping is caused by x = (A + Bt)e^-αt (equal roots of AE)

Answer 15

Critical damping (the limit for which there are no oscillations)

Question 16

What type of damping is caused by x = Ae^-αt sin(bt)
(no roots of AE) (imaginary)

Answer 16

Light damping (oscillates)

Question 17

What is the period of oscillations of the G.S. e^-kt * (Pcos(kt) + Qsin(kt))

Answer 17

period = 2pi/ω
ω = k
Therefore:
period = 2pi/k seconds

Question 18

What is the definition of coupled first-order linear differential equations

Answer 18

dx/dt = ax + by + f(t)
dy/dt = cx +dy +g(t)

Question 19

When are coupled first-order linear differential equations homogenous

Answer 19

If f(t) = g(t) = 0 for all t

Question 20

When given simultaneous differential equations, how do you find an equation in terms of x

Answer 20

Rearrange to find y
Differentiate to find dy/dt
Sub into equation to have everything in terms of x

Question 21

What is a restoring force

Answer 21

A force acting against displacement in order to try to bring the system back to equilibrium