A wire of length l and mass m is bent in the form of a semicircle. The gravitational field intensity at the centre of semicircle is

Options

A. `(Gm)/(pil)` along x- axis
B. `(Gm)/(pil)` along y-axis
C. `(2pGm)/(l^(2))` along y-axis
D. `(2piGm)/(l^(2))` along x-axis

Answer

C

Explanation

Consider two small elements each of length `dl=r d theta` symmetrically.
`pir=l`
Resolve the elemental field intensites
`because |dE_(1)|=|dE_(2)|`

`dE=2dE_(1) cos theta=(2Grd theta)/(r^(2))cos theta`
G = Gravitational constant
`lambda `=Linear mass density `=(m)/(l)`
or `dE=(2lambdaG)/(r)int_(0)^(90^(@))cos theta d theta =(2lambdaG)/(r)=(2piGm)/(l^(2))`