Question 364: Oscillations

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

Accepted Answer

As two pendulums begin to swing simultaneously, then
$$n_1 T_1=n_2 T_2\quad	ext{.... (i)}$$
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 periods.
The time period of simple pendulum is given by
$T=2 \pi \sqrt{\frac{l}{g}}$
where, $I=$ length of pendulum
and $g=$ acceleration due to gravity
$\Rightarrow \quad T^2 \propto l\quad	ext{.... (ii)}$
So, from Eqs. (i) and (ii), we get
$$\frac{l_1}{l_2}=\frac{T_1^2}{T_2^2}=\frac{n_2^2}{n_1^2}$$
Here, $n_1=9, n_2=7$
$$\Rightarrow \quad \frac{l_1}{l_2}=\frac{(7)^2}{(9)^2}=\frac{49}{81}$$
Hence, the ratio of pendulum lengths $I_1: I_2=49: 81$.

Answer

$\frac{49}{81}$

Answer

$\frac{64}{81}$

Answer

$\frac{8}{9}$

Answer

$\frac{7}{9}$

Oscillations

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

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