An arm making an angle of `120^(@)` at the center of ring of mass m and radius r is cut from the ring. The arc is made to rotate about z-axis perpendicular to its plane and passing through the center of the ring. The moment of inertia of the arc about the z-axis is

Options

A. `mr^(2)`
B. `(mr^(2))/3`
C. `(mr^(2))/2`
D. `(mr^(2)/4`

Explanation

b) The mass of ring of per unit angle is `lambda= m/(2pi)`
`therefore` dm-= (m)/(2pi) dtheta`
`therefore` The moment of inertia of considered element about z-axis passing through point O is
dl = `r^(2)dm`
`therefore` the moment of inertia of the arc is
`I = r^(2) = (mr^(2)/3 int_(theta=0)^(theta=(2pi//3)))` dm = `(mr^(2)/3)`