A spring of force constant k is cut into lengths of ratio

A spring of force constant k is cut into lengths of ratio 1 : 2 : 3. They are connected in series and the new force constant is k'. Then they are connected in parallel and force constant is k′′ . Then k' : k′′ is

Let ℓ be the complete length of the spring.

Length when cut in ratio, 1 : 2 : 3 are	ℓ	,	ℓ	and	ℓ
	6		3		2

Spring constant (k) ∝	1
	length (ℓ)

Spring constant for given segments k₁ = 6k, k₂ = 3k and k₃ = 2k
When they are connected in series

	1	=	1	+	1	+	1
	k'		6k		3k		2k

⇒	1	=	6
	k'		6k

∴ Force constant k' = k
And when they are connected in parallel k" = 6k + 3k + 2k
⇒ k" = 11k
Then the ratios

	k'	=	1	i.e. k' : k" = 1 : 11
	k"		11