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

Answer

B

Explanation

Consider two symmetrically located elements `dI=rd theta`, Here `pir=l`. Net field due to the two elements is along y-axis.

`dE=2dE,costheta=(2G(m//l)rd theta)/(r^2)costheta`
`E=intdE=(2Gm)/(lr)overset(90^@)underset(0)(int)costhetad theta`
`=(2Gm)/(lr)(sintheta)_(0)^(90^@)`
`=(2Gm)/(l(l//pi))(1-0)`
`=(2piGm)/(l^2)` along y-axis.