A magnetic needle performs `20` oscillations per minute in a horizontal plane. If the angle of dip be `30^@`, then how many oscillation per minute will this needle perform in vertical, north south plane and in vertical east -west plane?

Explanation

In horizontal plane the magnetic needle oscillates in horizontal component `H`.
`:. T=2pisqrt((MH))`
In the vertical north-south plane (magnetic meridian) the needle oscillates in the total earth's magnetic field `B_e` and in vertical east -west plane (plane perpendicular to the magnetic meridian) it oscillates only in earth's vertical component `V`. If its time period be `T_1` and `T_2`, then
`T_1=2pisqrt(I/(MB_3))` and `T_2=2pisqrt(I/(MV))`
From above equations, we can find
`T_1^2/T^2=H/B_e` or `n_1^2/n^2=B_e/H`
Similarly, `n_2^2/n^2=V/H`
Further, `B_e/H=sectheta=sec30^@=2/sqrt3`
and `V/H=tantheta=tan30^@=1/sqrt3`
`:. n_1^2=(n)^2(B_e/H)=(20)^2(2/sqrt3)`
or `n_1=21.5 oscillation//min`
and `n_2^2=(n)^2(V/H)=(20)^2(1/sqrt3)`
`:. n_2=15.2oscillation//min`