Two apdulums begin to swing simultaneosuly. The first pendulum makes `9` full oscillations when the other makes `7`. Find the ratio of length of the two pendulums

Options

A. `(49)/(81)`
B. `(64)/(81)`
C. `(8)/(9)`
D. `(7)/(9)`

Answer

A

Explanation

As tow pendulums being to swing simultaneoulsy then
`n_(1)T_(1) = n_(2)T_(2)`
where `n_(1)` and `n_(2)` are the number of oscillations of first and second pendulum respectively and `T_(1)` and `T_(2)` be their respective time periosds .
The time period of simple pendulum is given by
`T = 2pi sqrt((l)/(g))`
where l = lenght of pendulum
and g = acceleration due to gravity
`rArr " " T^(2) prop l`
So , from Eqs. (i) and (ii) , we get
`(l_(1))/(l_(2)) = (T_(1)^(2))/(T_(2)^(2)) = (n_(2)^(2))/(n_(1)^(2))`
Here `n_(1) = 9 , n_(2) = 7`
`rArr " "(l_(1))/(l_(2)) =((7)^(2))/((9)^(2)) = (49)/(81)`
Hence the ratio of pendulum lenghts `l_(1) : l_(2) = 49 : 81`