1.1 Universality

Question 1

What is the full equation of motion for a pendulum?

Answer 1

d^2 Θ / dt^2 + w^2 sin(Θ) = 0

Question 2

What makes the pendulum equation non-linear, and how can we fix this?

Answer 2

The sin(Θ) term is non linear, so assume Θ is small

Question 3

State the linearised pendulum equation, and its solution

Answer 3

d^2 Θ / dt^2 + w^2 (Θ) = 0

Θ = Acos (wt + Φ)

Question 4

How can we describe a partice moving in a potential V?

Answer 4

There is a minimum at x(0)

- Expect small oscillations about this minimum

Question 5

What is the equation of motion for a particle moving in a potential V?

Answer 5

m d^2 x / dt^2 = -dV/dx

Question 6

What is the proper way to find the frequency of oscillations for a particle moving in a potential V?

Answer 6

Taylor expand about the minimum

x = x(0) + 𝛿x

Question 7

What do we mean by linearising an equation in terms of a taylor expansion?

Answer 7

Assume all the terms of the order of (𝛿x)^2 are &laquo_space;than 𝛿x

- Essentially ignoring higher order terms and all functions are linear is 𝛿x

Question 8

For what systems is the pendulum solution also valid for?

Answer 8

Any conservative system where we have a small displacement x(0) - UNIVERSAL

Question 9

What solution for w do we get for the pendulum equation?

Answer 9

w^2 = V’’( x (0) ) / m

Question 10

How would we solve the equation for a system of linear coupled oscillations, and why can we do this?

Answer 10

Linear systems obey superposition

Take the simple solution of Θ = Acos (wt + Φ) and just rewrite as a sum over A, w and Φ