462 CHAPTER 24. MOVING COORDINATE SYSTEMS
the pendulum will be vibrating in a plane determined by k and j. (Recall k points awayfrom the center of the earth and j points East. ) At this instant in time, defined as t = 0,the conditions of 24.27 will hold for some value of c and so the solution to 24.25 havingthese initial conditions will be those of 24.26. (Some interesting mathematical details arebeing ignored here. Such initial value problems as 24.26 and 24.27 have only one solutionso if you have found one, then you have found the solution. This is a general fact shown indifferential equations courses. However, for the above system of equations see Problem 13on Page 464 found below.) Writing these solutions differently,(
x(t)y(t)
)= c
(sin( bt
2
)cos( bt
2
) )sin
(√b2 +4a2
2t
)
This is very interesting! The vector, c
(sin( bt
2
)cos( bt
2
) ) always has magnitude equal to |c| but
its direction changes very slowly because b is very small. The plane of vibration is deter-
mined by this vector and the vector k. The term sin(√
b2+4a2
2 t)
changes relatively fast
and takes values between −1 and 1. This is what describes the actual observed vibrationsof the pendulum. Thus the plane of vibration will have made one complete revolution whent = T for
bT2≡ 2π.
Therefore, the time it takes for the earth to turn out from under the pendulum is
T =4π
2ω cosφ=
2π
ωsecφ .
Since ω is the angular speed of the rotating earth, it follows ω = 2π
24 = π
12 in radians perhour. Therefore, the above formula implies
T = 24secφ .
I think this is really amazing. You could determine latitude, not by taking readings withinstruments using the North star but by doing an experiment with a big pendulum. Youwould set it vibrating, observe T in hours, and then solve the above equation for φ . Alsonote the pendulum would not appear to change its plane of vibration at the equator becauselimφ→π/2 secφ = ∞.
24.8 Exercises1. Find the length of the cardioid, r = 1+ cosθ ,θ ∈ [0,2π]. Hint: A parametrization
is x(θ) = (1+ cosθ)cosθ ,y(θ) = (1+ cosθ)sinθ .
2. In general, show that the length of the curve given in polar coordinates by r =
f (θ) ,θ ∈ [a,b] equals∫ b
a
√f ′ (θ)2 + f (θ)2dθ .
3. Using the above problem, find the lengths of graphs of the following polar curves.
(a) r = θ , θ ∈ [0,3]