Question

Two pendulums begin to swing simultaneously. The first pendulum makes nine full oscillations when the other makes seven. The ratio of the lengths of the two pendulums is ______.

पर्याय

• `49/81`

• `64/81`

• `8/9`

• `7/9`

Answer 1

Two pendulums begin to swing simultaneously. The first pendulum makes nine full oscillations when the other makes seven. The ratio of the lengths of the two pendulums is `underline(49/81)`.

Explanation:

As two pendulums begin to swing simultaneously, then

n₁T₁ = n₂T₂      ....(i)

where, n₁ and n₂ are the number of oscillations of first and second pendulum respectively and T₁ and T₂ be their respective time periods.

The time period of simple pendulum is given by

T = `2pi sqrt("l"/"g")`

where, I = length of pendulum

and g = acceleration due to gravity

⇒ T² ∝ l     ...(ii)

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, n₁ = 9, n₂ = 7

`=> "l"_1/"l"_2 = (7)^2/(9)^2 = 49/81`

Hence, the ratio of pendulum lengths l₁ : l₂ = 49 : 81.