2.2 Pendulum

Question 1

How can we deal with the 2nd order ODE of the pendulum?

Answer 1

Treat as a coupled first order

y = dΘ/dt, dy/dt = -w^2 sin(Θ)

Question 2

Where are the fixed points for the coupled first order pendulum equation?

Answer 2

At y bar = 0, sin (Θ bar) = 0 so Θ bar = n pi

Question 3

What is the linearised equation equal to for the pendulum?

Answer 3

d^2/dt^2 𝛿Θ = -w^2 (-1)^N dΘ

Question 4

Briefly describe the two solutions for the linearised pendulum equation, d^2/dt^2 𝛿Θ = -w^2 (-1)^N dΘ

Answer 4

N = even: Oscillatory solution
d^2/dt^2 𝛿Θ = Ae^iwt + Be^-iwt

N = odd: 4 solution separatrix
d^2/dt^2 𝛿Θ = Ae^wt + Be^-wt

Question 5

State the full energy equation for the pendulum and the terms

Answer 5

E = y^2 / 2 - w^2 cos(Θ_0)

• For a pendulum which is being held at a maximum angle Θ_0 and y_0 = 0
• Also symmetric in y and theta

Question 6

Describe the phase plane topology of the y-Θ graph for the pendulum

Answer 6

BOUND ORBITS:
Circular orbits on even n
Separatrix shapes on odd n which connect

UNBOUND ORBITS:
General wave that oscillates above and below the bound orbits

Question 7

What is the core property about the trajectories of the pendulum?

Answer 7

The y(Θ) trajectories cannot cross unless at a fixed point

Question 8

What is the name for even and odd n for the y-Θ phase plane?

Answer 8

Even n are called centres

Odd n are called saddle points

Question 9

Describe how we obtain the critical value of the energy

Answer 9

y is bounded for a given Θ_0, so y = dΘ/dt = 0
E = E_0 = -w^2 cosΘ
Θ_0 = pi and y = 0 from separatrix
E_c = w^2

Question 10

Describe the pendulum behaviour when below, at, and above the critical energy

Answer 10

E < E_c: Oscillations about a stable fixed point (circular orbits on phase plane) where Θ < pi
E = E_c: Theta = pi (pendulum upside down) at a saddle unstable fixed point
E > E_c: Swinging the pendulum and we have unbounded motion

Question 11

Describe the shape and properties of the potential-theta graph for the pendulum, and the different types of motion

Answer 11

• A negative cosine graph
• Small Θ, we get oscillatory, circular motion where E < E_c
• At Θ = pi, we are at the positive peak of the separatrix i.e. it can fall down from a small perturbation. This is at E=E_c
• If E > E_c, we get unbounded motion

Question 12

What sort of timescale are we referring to in regards to “weak” damping?

Answer 12

A timescale > 1/w

Question 13

Describe the graph for the weak damping case for the V-Θ graph for a pendulum

Answer 13

Have the negative cosine graph

• Pendulum comes in with wide oscillations which get narrower as it comes into closer to the trough
• Slowly loses energy

Question 14

Describe the phase plane for the damped pendulum graph

Answer 14

If a particle starts at a separatrix, it will spiral into the centres in damped motion - Centres are spiral fixed points
If a particle starts from an unbounded state, it will move towards the separatrix
TRAJECTORIES CANNOT CROSS