Question

two pendulums of lengths 100cm and 110.25cm start oscillating in phase. after how many oscillations will they again be in same phase?

Answer 1

Angular frequency of pendulum is given by

$\omega = \sqrt{\frac{g}{l}}$

for both pendulum we have

$\omega_1 = \sqrt{\frac{9.81}{1.00}}$

$\omega_1 = 3.13 rad/s$

For other pendulum

$\omega_2 = \sqrt{\frac{9.81}{1.1025}}$

$\omega_2 = 2.98 rad/s$

now we have relate angular frequency given as

[tex\omega_1 - \omega_2 = 3.13 - 2.98 = 0.15 rad/s[/tex]

now time taken to become in phase again is given as

$t = \frac{2\pi}{\omega_1 - \omega_2}$

$t = \frac{2\pi}{0.15} = 41.88 s$

now number of oscillations complete in above time

$N = \frac{t}{\frac{2\pi}{\omega_1}}$

$N = \frac{41.88}{\frac{2\pi}{3.13}}$

$N = 21 oscillation$