Question

Two pendulums begin to swing simultaneously. The first pendulum makes 9 full oscillations when the other makes 7. Find the ratio of the length of the two pendulums.
(A) $\dfrac{{49}}{{81}}$
(B) $\dfrac{7}{9}$
(C) $\dfrac{{50}}{{11}}$
(D) $\dfrac{1}{2}$

Answer 1

Hint: To calculate the ratio of the length of the two pendulums, as we know that the time period is directly proportional to the root square of the length, and also we know that one complete oscillation is equal to the time period.

Complete answer:
Let the time period of oscillation be $T$ .
So, the time period of the first pendulum is ${T_1}$ .
And the time period of the second pendulum is ${T_2}$ .
Now, as we know that the time period is directly proportional to the root square of the length-
So, the time period of the first pendulum is directly proportional to the root square of the length of the first pendulum:
$\therefore {T_1}\alpha \sqrt {{l_1}} $ ……….(i)
where, ${l_1}$ is the length of the first pendulum.
Similarly, the time period of the second pendulum is directly proportional to the root square of the length of the second pendulum:
$\therefore {T_2}\alpha \sqrt {{l_2}} $ ……..(ii)
where, ${l_2}$ is the length of the second pendulum.
Now, as per the question, the first pendulum makes 9 full oscillations when the other makes 7, that means, 9 times of the time period of first pendulum is equal to the 7 times of the time period of the second pendulum:
$\therefore T = 9{T_1} = 7{T_2}$
So, now we will solve the equation in the form of ratio:
$ \Rightarrow (\dfrac{{{T_1}}}{{{T_2}}}) = \dfrac{7}{9}$
So, by equation(i) and eq(ii), we get:-
$
   \Rightarrow \sqrt {\dfrac{{{l_1}}}{{{l_2}}}} = \dfrac{7}{9} \\
   \Rightarrow \dfrac{{{l_1}}}{{{l_2}}} = {(\dfrac{7}{9})^2} = \dfrac{{49}}{{81}} \\
 $
Therefore, the ratio of the length of the two pendulums is $\dfrac{{49}}{{81}}$ .
Hence, the correct option is (A) $\dfrac{{49}}{{81}}$ .

Note:
The relationship between time period and length of a simple pendulum is ${T^2} = 4{\pi ^2}(\dfrac{l}{g})$ . Where $l$ is the length of a simple pendulum and $g$ is the value of acceleration due to gravity at a place where $T$ is measured.