Consider a swing whose one end is fixed above the other rope by ''b'. The distance between the poles of swing is `a`. The lengths `l_(1)` and `l_(2)` are such that `l_(1)^(2) +l_(2)^(2) =a^(2) +b^(2)` as shown. The period of the smaller oscillations of the swing, by neglecting the height of the swinging person, is.....

Options

A. `2pi sqrt((l_(1)+l_(2))/(g))`
B. `2pi sqrt((l_(1)l_(2))/(ga))`
C. `2pi sqrt((l_(1)l_(2))/(g(a+b)))`
D. `2pi sqrt((l_(1)^(2)+l_(2)^(2))/(g(a+b)))`

Answer

B

Explanation

From the given information, it is inferred that two ropes are perpendicular to each other:
`:. T_(1) cos theta +T_(2) sin alpha = mg …………(1)`
`T_(1) sin alpha = T_(2) cos alpha ……………(2)`
`rArr T_(1) = Mg cos alpha` and `T_(2) = mg sin alpha`

When the swing is given a displacement, `x`, along outward normal to plane of the paper as:

`F_("restoring") = T_(1)cos theta +T_(2) cos phi, cos theta =(x)/(l_(1)), cos phi =(x)/(l_(2))`
Solve for time period.