A spring of force constant k is cut into two pieces of lengths `l_(1)` and `l_(2).` Calculate force constant of each part.

Explanation

Spring of length `l` is cut into two pieces of length `l_1 and l_2`.
Force constant `prop (1)/("length of spring")`
`K_1 prop (1)/(l_1) , K_2 prop (1)/(l_2)`

because `l= l_1 + l_2`
`K prop (1)/(l_1 + l_2)`
If C is spring constant
`K_1 = ( C)/(l_1 )................(A)`
`K_2 = (C )/(l_2) ....................(B)`
and `K= (C )/(l_1 + l_2).....................(4)`
Dividing (A) by (4)
`(K_1)/(K)= (l_1+ l_2)/(l_1)`
`rArr K_1 = K (1+ (l_2)/(l_1))`
Dividing (B) by (4)
`(K_2)/(K )= (l_1 + l_2)/(l_2) rArr K_2= K (1 + (l_1)/(l_2))`
`because (l_1)/(l_2)= (n_1)/(n_2)`
So. `K_1 = K (1+ (n_2)/(n_1)) rArr K_1 = K ((n_1 + n_2)/(n_1))`
and `K_2 = K (1+ (n_1)/(n_2)) rArr K_2 = K ((n_1+n_2)/(n_2))`