Question

Two pendulums of different lengths are in phase at the mean position at a certain instant. The minimum time
after which they will be again in phase is 5T/4, where T is the time period of shorter pendulum. Find the
ratios of lengths of the two pendulums:
1) 1:16
2) 1:4
3) 1:2
4) 1:25
​

Answer 1

Answer:

Ratio of the length of two pendulums is 1 : 25

Explanation:

We know that the time period of the pendulum is given by the formula

$T = 2\pi \sqrt{\frac{L_1}{g}}$

also for other pendulum we have

$T_1 = 2\pi \sqrt{\frac{L_2}{g}}$

now from above two equations we have

$T_2 = \sqrt{\frac{L_1}{L_2}}$

now we have

$L_1 : L_2 = T_1^2 : T_2^2$

now we know that two pendulum comes again in same phase after 5T/4

so we have

$\frac{2\pi}{\omega_r} = 5T/4$

here we know that

$\omega_r = \frac{2\pi}{T_1} - \frac{2\pi}{T_2}$

so we have

$\frac{1}{\frac{1}{T_1} - \frac{1}{T_2}} = 5T/4$

$\frac{1}{\frac{1}{T} - \frac{1}{T_2}} = 5T/4$

$\frac{4}{5T} = \frac{1}{T} - \frac{1}{T_2}$

$T_2 = 5T$

so we have

$L_1 : L_2 = T^2 : 25T^2$

$L_1 : L_2 = 1 : 25$

#LEarn

Topic : Simple Pendulum

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