A force `vec F = prop hat i + 3 hat j + 6 hat k` is acting at a point `vec r = 2 hat i - 6 hat j - 12 hat k`. The value of `prop` for which angular momentum about origin is conserved is.

Options

A. `-1`
B. 2
C. Zero
D. 1

Answer

A

Explanation

When the resultant external torque acting on a system is zero , the taotal angular momentum of a system remains constant . This is the principle of the conservation of angular momen tum .
Given force `F = alpha hat(i) + 3hat(j) + 6hat(k)` is acting at a point `bar(r) = 2hat(i) - 6hat(j) - 12hat(k)`
As angular momentum about origin is conserved i.e , `tau = "cons "tan t `
` rArr ` Torque ` , tau = 0 rArr bar(r ) xx bar(F) = 0 `
`|{:(hat(i),hat(j),hat(k),),(2,-6,-12,),(alpha,3,6,):}|=0`
`rArr 0 hat(i) - 12 (1 + alpha) hat (j) + 6 (1+alpha) hat(k) = 0 `
`rArr 6 ( 1+alpha) = 0 rArr alpha = - 1 `
So , value of of `alpha` for angular momentum about origin is conserved , `alpha =1`