Download HON152 – Physics of Time – Homework 2 This homework is due

HON152 – Physics of Time – Homework 2
This homework is due Thursday 9th October in class.
1. In class I told you about the Fibonacci sequence of numbers. The nth Fibonacci
number is Fn . They satisfy a two-term recurrence relation which says that
Fn+2 = Fn+1 + Fn
As in any two-term recurrence, you need to know the first two numbers, which are F0 = 0,
and F1 = 1.
(1a.) Try a solution of the form Fn = Cxn , where C is a constant indepedent of n, and
x is an unknown number. What equation must be satisfied by x in order for this to
be a solution to the Fibonacci recurrence relation? Solve this equation to find the
allowable values of x.
(1b.) There will be two solutions for x, one of which will be the Goldean Mean beloved of
Greek architects
√
1+ 5
x1 = φ =
2
and the other is x2 = 1 − φ. Write down the complete solution as a linear combination
of the two solutions.
(1c.) Now fix the constants in the linear combination by demanding that F0 = 0 and F1 = 1
(called applying the initial conditions), and thereby derive the general formula for
an arbitrary Fibonacci number. Using your formula compute the Fibonacci number
corresponding to your date of birth (only the day of the month, a number between 1
and 31).
2. In class I showed you that if I discretized time such that tn = n∆t, then the
equation for the angle θ(tn ) = θn of a simple pendulum became a two-term recurrence
relation
θn+2 = (2 − (ω0 ∆t)2 )θn+1 − θn
(2a.) Again try a solution of the form θn = Cz n where C is a constant and z is now a
(possibly complex) number. Write down the equation that must be satisfied by z and
solve it to find the two values z± .
(2b.) Show that if ω0 ∆t < 2 both solutions are complex, that they are complex conjugates
of each other, and that their product is 1.
(2c.) Show that z+ can be written as ei∆ψ where ∆ψ is a small angle when ∆t is small.
Show that as ∆t → 0 ∆ψ → ω0 ∆t.
(2d.) Write down the general solution as a linear combination of the two solutions with
arbitrary constants. Fix these constants by demanding the initial conditions θ0 =
θ1 = 1.0. Thereby find θn at any time and plot it.
3. Three coordinate systems (frames) are related in the following way. Alice has the
standard set of 3-dimensional coordinates with perpendicular x, y, and z axes.
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(3a) Bob has a frame related to Alice’s in the following way: Rotate by π/4 around Alice’s
z axis, then rotate by π/4 around the new x axis, and finally rotate by π/4 around the
new z axis. Find the matrix MBA that takes the components of a vector in Alice’s
frame and give the components in Bob’s frame.
(3b.) Cathy has a frame related to Alice’s as well: To get to Cathy’s frame from Alice’s,
first rotate by π/4 around the x axis, then by π/4 around the new z axis, and finally
by π/4 around the new x axis. Find the matrix MCA taking the components of a
vector in Alice’s frame and giving back the components in Cathy’s frame.
(3c.) Consider a vector that has components (1, 1, 1) in Alice’s frame. What are its components in Bob’s and Cathy’s frames?
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