ABC is an equilateral triangular frame of mass m and side r. It is at rest under the action of horizontal magnetic field B (as shown) and the gravitational field.

Options

A. The frame remains at rest if the current in the frame is `(2mg)/(rB)`
B. The frame remains at rest if the current in the frame is `(2mg)/(rBsqrt(3))`
C. The frame is in simple harmonic motion when frame is slightly displaced in its plane perpendicular to AB. The period of oscillation is `pi[(rsqrt(3))/(g)]^(-1//2)`
D. For same as in above option, teh period of oscillation is `pi[(3r)/(2g)]^(1//2)`.

Answer

A, C

Explanation

According to Lenz's law, current I is in clockwise direction
`A'B'=2A'O=(2)/(tan 60^(@)) ((rsqrt(3))/(4))=(r )/(2)`
For equilibrium `mg-I((r)/(2))B`
`I=(2mg)/(rB)`
If loop is displaced by x,
F= Restoring force
`-I[((2)/(tan 60^(@)))(rsqrt(3)/(4)+x)]B+mg`
`=-(IrB)/(2) + mg - (2IB)/(sqrt(3))x`
`=-(2IB)/(sqrt(3))*x`
`F prop (-x) x`, motion is SHM.
`a=-(2IB)/(msqrt(3))*x`
`T=2pi[(msqrt(3))/(2IB)]^(1//2) = pi[(rsqrt(3))/(g)]^(1//2)`.