A pendulum consists of a point mass swinging back and forth at the end of a mass

Question

A pendulum consists of a point mass swinging back and forth at the end of a massless string of length R. The motion is in a vertical plane. The string makes a maximum angle of 90 with respect to the vertical direction. First, derive the answers in the form of definite integrals. Then evaluate those integrals using some numerical method. I recommend Mathematica.

(a) How long does it take the pendulum to travel from = 90 (where it is at rest) to = 45?
(b) How long does it take the pendulum to travel from = 45 to = 0 (still with the initial condition that it was at rest at = 90?

Explanation / Answer

suppose time period of oscillation is T then

w = 2 pi / T

and T = 2pi sqrt(R/g)

angular displacement,

theta = theta_max cos(w t )

for theta = 45 deg

45 = 90 cos(w t)

w t = cos^-1(0.5) = pi / 3

t = (pi / 3)(1/w) = ( pi / 3) (T /2pi) = T / 6

t = ( pi / 3) sqrt(R/g) ........Ans(A)

(b) total time from 90 deg to 0 deg = T/4

hence for 45 to 0 deg.

t' = (T/4) - (T/6) = T / 12

t' = ( pi / 6) sqrt(R/g)

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