Question

(i) a series combination

(ii) a parallel combination of springs. Obtain the spring constant in each case.

Answer 1

(i) Series combination of springs : When two light springs obeying Hooke’s law are connected as shown in below figure and both the springs experience the same force applied to the free end of the combination, they are said to be connected in series.

Light spring connected in(a) series (b) parallel

Consider two springs, 1 and 2, with respective spring constants k₁ and k₂ connected in series and supporting a load F = mg so that the springs are extended. Since the same force acts on each spring, by Hooke’s law, F = k₁ x₁ (for spring 1) and F = k₂ x₂ (for spring 2) The system of two springs in series is equivalent to a single spring, of spring constant k_s such that F = k_s x, where the total extension x of the combination is the sum x₁+ x₂ of their elongations.

x = x₁ + x₂

∴ \(\frac{F}{kg}\) = \(\frac{F}{k_1}\) + \(\frac{F}{k_2}\)

 ∴ \(\frac{1}{kg}\) = \(\frac{1}{k_1}\) + \(\frac{1}{k_2}\)

For a series combination of N such springs, of spring constants, k₁, k₂, k₃, … k_N

  \(\frac{1}{k_s}\) = \(\frac{1}{k_1}\) + \(\frac{1}{k_2}\) + \(\frac{1}{k_3}\)+.....+\(\frac{1}{k_N}\) = \(\sum^N_{i=1}\)\(\frac{1}{k_i}\)

(ii) Parallel combination of springs : When two light springs obeying Hooke’s law are connected via a thin vertical rod as shown, they are said to be connected in parallel. If a constant force \(\vec F\)is exerted on the rod such that the rod remains perpendicular to the direction of the force, the springs undergo the same extension.

Consider two springs, 1 and 2, with respective spring constants k₁ and k₂ connected in series and supporting a load F = mg so that the springs are extended. The two springs stretch by the same amount x but share the load.

F = F₁ + F₂

The system of two springs in parallel is equivalent to a single spring, of spring constant kF such that F = k_PX,

∴ k_Px = k₁x + k₂x 

∴ k_P = k₁ + k₂

For a parallel combination of N such springs, of spring constants k₁ , k₂ , k₃ , … k_N

k_P = k₁ + k₂ + k₃ + … + k_N = \(\sum^N_{i=1}\)k_i

Therefore, for a parallel combination of N identical light springs, each of spring constant

k, k_P = Nk