Lecture #13: Large Oscillations

Question 1

Energy of a simple harmonic oscillator in phase space

Answer 1

Ellipses

Question 2

Energy of projectile motion in phase space

Answer 2

Parabolas on their side

Question 3

Non-linear pendulum

Answer 3

We do not have small angles

Question 4

For the nonlinear pendulum, what happens when E < 2mgl?

Answer 4

• Pendulum does not roll
• Pendulum motion is oscillatory
• Bottom of the wells are approximately elliptical and we have SHM

Question 5

Requirement for rolling motion of a pendulum

Answer 5

v0 needs to be greater than the escape velocity

Question 5

For the nonlinear pendulum, what happens when E > 2mgl?

Answer 5

• Pendulum rolls

Question 6

Separatrix

Answer 6

Curve that is the boundary between the two types of motion for nonlinear pendulum

Question 7

For a nonlinear pendulum, what does the angle versus time graph look like?

Answer 7

• Periodic
• a-sinusoidal, with flatter slopes and pointier tops
  -distortions increase as you increase theta max

Question 8

What does the phase plot look like for large angles, nonlinear pendulum?

Answer 8

Diamond-like (no longer ellipses)