Question

Question is given below ?

The length of a magnet is large as compared to its other dimensions. The time period of vibration in vibration magnetometer is 2 s. The magnet is cut into three equal parts by length and three parts are placed together with their like poles together. Find new time period of this combination.

(1) 2 s

(2)2/3 s

(3) 2`sqrt3` s

(4)`2/sqrt3` s

Answer 1

`2 . 2/3 s`

Explanation:

Time period for a vibration magnetometer `T= 2pi sqrt(I/(MB))` (where, `I` is the moment of inertia and `M` is magnetic moment, `B` is the magnetic field strength)

Given, `T=2s`

On cutting the magnet into three pieces their mass becomes `m/3` and length also becomes `l/3`

Now,if initial moment of inertia was `I=(ml^2)/12`

New,moment of inertia `I'=(m/3)(l/3)^2(1/12)=(1/27)(ml^2)/12`

So,moment of inertia of the combination is `I_1=I'*3=(ml^2)/12(1/9)=I/9`

Now,magnetic moment for the system remains unchanged. (as, `M prop l`,so each will have magnetic moment of `M/3` again for the whole system it will be `M/3 *3=M`)

So,new time period is `2pi sqrt((I_1)/(MB))=2pi sqrt(I/(9MB)`

`=1/3 T=2/3` (as, `T=2pi sqrt(I/(MB)))`