A particle is moving uniformly in a circular path of radius r. When it moves through an angular displacement `theta`, then the magnitude of the corresponding linear displacement will be

Options

A. `2r cos (theta/2)`
B. `2r cos (theta/2)`
C. `2r tan (theta/2)`
D. `2rsin(theta/2)`

Answer

D

Explanation

According to cosine formula

`cos theta = (r^(2) + r^(2) -d^(2))/(2r^(2))`
`2^(2) cos theta = r^(2) + r^(2) -d^(2)`
`d^(2) = 2r^(2) cos theta = 2r^(2) [1- cos theta]`
`[therefore cos theta =1-2 sin^(2) theta/2]`
or `d = 2r sin (theta)/2`
`therefore` Linear displacement, `AB = 2r sin (theta)/2`