Aa simple pendulum is suspended from the ceiling of a vehicle which moves down an inclined plane o inclination `Theta`. The period of oscilation of the pendulum is

Options

A. `2pisqrt(L/(gcostheta))`
B. `2pisqrt(Lcostheta/g)`
C. `2pisqrt(L/(gsintheta))`
D. `2pisqrt(Lsintheta/g)`

Answer

A

Explanation

Acceleration of trolley is g `sin theta` along the plane.

We can resolve acceleration of trolley along horizontal and vertical.
`a_(x) = (g sin theta) cos theta = g sin theta cos theta`
`a_(y) = (g sin theta) sin theta = g sin^(2) theta `
Effective acceleration due to gravity for hte pendulum can be calculated as follows :
`g_(eff)^(2) = a_(x)^(2) + (g-a_(y))^(2)`
` = g^(2) sin^(2)theta cos^(2) theta + g^(2) + g^(2) sin^(4) theta - 2g^(2) sin^(2)theta`
` = g^(2) sin^(2)theta [cos^(2)theta + sin^(2)theta] +g^(2) - 2g^(2) sin^(2) theta`
` = g^(2) sin^(2)theta g^(2) - 2 g^(2) sin^(2)theta`
` = g^(2) - g^(2) sin^(2) theta`
` g^(2) (1-sin^(2) theta) = g^(2) cos^(2) theta `
`rArr g_(eff) = g cos theta `
`T = 2pi sqrt((L)/(g_(eff)))`
`rArr T = 2pi sqrt((L)/(g cos theta))`
Hence option (a) is correct.