Two spring of spring constant `k_(1)` and `k_(2)` are joined in series The effective spring constant of the combination is given by

Options

A. `sqrt(k_(1)k_(2))`
B. `(k_(1) + k_(2))//2`
C. `k_(1) + k_(2)`
D. `(k_(1)k_(2))/(k_(1) +k_(2))`

Answer

D

Explanation

Let as consider two spring constants `k_(1)` and `k_(2)` joined in series as shown in the figure

Under a force `F` they will strach by `y_(1)` and `y_(2)` so `y = y_(1) + y_(2)`
or `(F)/(k) = (F_(1))/(k_(1)) = (F_(2))/(k_(2))`
but as spring and massless so force on then must be the same i.e. `F_(1) = F_(2) = F`
so `(1)/(k) = (1)/(k_(1)) + (1)/(k_(2))` or `k = (k_(1)k_(2))/(k_(1) +k_(2))`
After in series conbination
`(1)/(k_(2)) = (1)/(k_(1)) + (1)/(k_(2))+ (k_(2) + k_(1))/(k_(1)k_(2)) rArr k_(3) = (k_(1)k_(2))/(k_(1))+ (k_(2)) rArr k_(s) = (k_(1)k_(2))/(k_(1) + k_(2))`