Physics 152 Homework #7

Read my notes on Complex Numbers at

http://www.phys.virginia.edu/classes/152.mf1i.spring02/ComplexNumbers.htm

before trying the following!

1. Using the identity e^{i(A + B)} = e^iA e^iB, find cos(A + B) and sin(A + B) in terms of sinA, sinB, cosA and cosB. How would you find tan(A + B)?

2. Thinking of a simple pendulum as a ball on a string of length l, let us consider small oscillations in which the pendulum doesn’t just swing back and forth along a line, but moves more generally in the plane (for example, it could be moving in a circle). But we will restrict the oscillations to be small enough that we can take the force acting on the pendulum to be just linear towards the center (that is, we take sinq = q). We now represent the position of the pendulum in this plane by a complex number z, with the pendulum’s equilibrium position as the origin.

Show that the equation of motion of the point in the complex plane is

.

Show that Ae^iwt + Be^-iwt is a solution to this equation, where A, B may be complex numbers.

Sketch the pendulum motion corresponding to:

(i) A = 1, B = 0, (ii) A = 0, B = 1, (iii) A = 1, B = 1, (iv) A = 1, B = -1, (v) A = i, B = 0,

(vi) A = 1, B = i, (vii) A = 1, B = 2, (viii) A = 1, B = 2i.

Halliday: Chapter 16: 30P, 46E, 62P .