ADVANCED DYNAMICS — PHY 4241/5227

HOME AND CLASS WORK – SET 10

(March 2, 2009)

(33) One of the Apollo missions left a mirror on the moon. The McDonald Observatory in Texas flashes a laser wave at this mirror, which is received back after an elapsed time △t. At what distance was the mirror, when it reflected the signal? Due March 5 in class (1 point).

(34) Let (in arbitrary units)

(a) ct = 5, x^{1} = 1, x^{2} = 2, x^{3} = 3 ,
(b) ct = 5, x_{1} = 1, x^{2} = 2, x^{3} = 3 ,
(c) ct = 5, x^{1} = 1, x^{2} = 2, x^{3} = −3 ,
(d) ct = 5, x^{1} = 0, x^{2} = 3, x^{3} = 4 ,
(e) ct = 5, x_{1} = 0, x_{2} = 3, x_{3} = 4 ,
(f) ct = 5, x^{1} = 2, x^{2} = 3, x^{3} = 4 ,
(g) ct = 5, x^{1} = 0, x^{2} = 3, x^{3} = −4 .

Write down the row and column vector for each case and calculate x^{α} x_{α}. Due March 5 in class (7 points).

(35) A particle of mass m and electric charge q moves under the influence of a constant magnetic field of the form B(r) = B_{0}ẑ. Obtain the most general solution for the velocity v(t) using Newton’s second law of motion in combination with the Lorentz force

F = mv̇ = \frac{q}{c}v × B .

Homework, due March 15 before class (10 points).

(36) Consider the 2D rotation

( \begin{matrix}x^{′ 1} \\ x^{′ 4} \end{matrix} ) = ( \begin{matrix}cos(ϕ) & sin(ϕ) \\ − sin(ϕ) & cos(ϕ) \end{matrix} ) ( \begin{matrix}x^{1} \\ x^{4} \end{matrix} )

and substitute ϕ = i ζ, x^{4} = i x^{0}, x^{′ 4} = i x^{′ 0}. Write out the equations for x^{′ 1} and x^{′ 0}. Due March 24 in class (8 points).

Read the Notes on Relativity up to section 1.1.3 (due March 15).