Answer to Question #278636 in Civil and Environmental Engineering for Carol

Question

When a pendulum of length 10 cm has swung so that 𝜃 is the radian measure of the angle formed by the pendulum and a vertical line, then if ℎ(𝜃) cm is the vertical height of the end of the pendulum above its lowest position, ℎ(𝜃) = 20 𝑠𝑖𝑛^2 𝜃/2 . Find the instantaneous rate of change of ℎ(𝜃) with respect to 𝜃 when (a) 𝜃 = 𝜋/3 and (b) 𝜃 = 𝜋/2

Answer 1

h = L (1-cos T) geometry exact

cos 2A = 1 - 2 sin^2 A identity

so

cos T = 1 - 2 sin^2 (T/2)

h = L [ 1 - (1 -2 sin^2 (T/2) ]

= L [2 sin^2 (T/2) ]

= 20 sin^2 (T/2)