Discussion Forum

 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 

Answer:

Explanation:

 As two pendulums begin to swing  simultaneously , then

    n₁T₁   = n₂ T₂    ...........(i)

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

  The time period of simple  pendulum is given by

  $T=2\pi\sqrt{\frac{l}{g}}$

 where, l= length  of pendulum

and g= acceleration  due to gravity

  $\Rightarrow$    $T^{2}\propto l$  .......(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, n₁=9, n₂=7

$\Rightarrow$   $\frac{l_{1}}{l_{2}}=\frac{(7)^2}{(9)^2}=\frac{49}{81}$

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