1. Using the identity
ei(A + B) = eiA eiB, find cos(A +
B) and sin(A + B) in terms of sinA,
sinB, cosA
and cosB.
How would you find tan(A + B)?
2. Imagine a pendulum, a heavy ball on a long thin string, swinging backwards and forwards over distances small compared with the length of the string. This pendulum, being on a string rather than a rigid rod, can swing around in a plane rather than just along one line. Denote its position in the plane by the complex number z, where z = 0 is the rest position of the pendulum.
It can be shown that the motion of the pendulum is given by
z = Aeiwt + Be-iwt, 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.
3. In the complex plane, sketch the curves of points z = x + iy satisfying the following equations. In each case, state the name of the curve you find.
(i) |z – 1| + |z| =2
(ii) |z - 1| = x
(iii) (z -1)/(z + 1) is pure imaginary.
A critical approach to
Tipler…
4. Tipler Ch 13, Problem 50. Do this problem, then think about how reasonable it is to take the air to have uniform density. How much does the air density change in 10 km, approximately? How does that affect the drag force? What about the buoyancy? ( We’re also assuming here the radius of the weather balloon stays constant—is that reasonable? Google “weather balloons” to find out.) How, roughly, will these considerations affect the answer?
5. Tipler Problem 63. Don’t do this problem—just read it. Note the assumption of nonviscous flow. The problem states that the liquids will flow until the surfaces are at the same level. If that really happens, and there’s no viscosity (that is, no friction) is energy conserved? What do you think would happen if we tried this experiment, assuming there was very little viscosity?
6. Problem 33: (done already, I know): You are asked to
assume the part of your chest in front of your lungs is about 0.09 square
meters. How accurately do you think this could be measured? To
10%? 1%?
The answer in the back of the book (how deep in the water your chest is
when breathing becomes difficult) is 0.453m. What would be a reasonable
accuracy level in this answer?