Math 142 Practice problems for midterm exam
1. Phase plane sketch Consider a system with the potential V(x) = x2−x4/4.
a. Find the stable and unstable equilibrium points.
Answer: Equilibrium points are where the derivative of the potential (and thus the force) is zero. This
occurs when 2x−x3=0, which means x=0 and x=±√2.
The equilibrium point is stable if the potential is concave up and unstable if it is concave down. With
V′′(x) = 2−3x2, we see that x=0 is stable and x=±√2 is unstable.
b. Sketch the trajectories in the phase plane.
Answer:
c. Consider the potential V(x) = x2−x4/4+2. Why does this potential produce the same dynamics?
Answer: The force is given by the derivative of the potential, which in both cases is f(x) = 2x−x3. If the
force is the same, then Newton’s equation is the same, and so the dynamics are the same.
2. Damped harmonic oscillator If you were designing a suspension system for a car, would you want the
spring system to be underdamped, overdamped, or critically damped and why?
Answer: If the system is underdamped, then the car will continue to bounce after it hits a bump. If the
car is overdamped, the wheels will return to equilibrium slowly, so that two bumps in a row could cause
problems. Critically damped springs will return to equilibrium as fast as possible without oscillating.
3. Nonlinear pendulum Consider the non-linear pendulum ¨
θ=−g
ℓsinθ.
a. Compute the potential energy, using the above equation.
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