Question

What is the difference between the series and parallel combination of springs? Calculate the value of effective spring constant for every combination.

Answer 1

(a) Series Combination

As shown in Fig. spring S₁ and S₂ are joined in series. A mass hanging from the combination is stretched y distance downwards and left, then the working force on both the springs is same; where as the increase in lengths is y₁ and y₂ respectively (suppose) are different. The restoring force for spring Si in this combination is;
F₁ = -K₁ y₁ 

Fig: Series combination of the springs

and restoring force in the spring S₂ is;

F₂ = -k₂y₂ 

Since, the tension on both the springs is same hence the working restoring force on the body.

where, k is the effective spring constant of combination. If n springs of spring constant k₁ = k₂ = k₃ =….= k’ are joined in series, then k will be the effective spring constant of combination.

(b) Parallel Combination
Two springs having spring constant k₁ and k₂ are combined in parallel from same base and mass m is hanged from the free end of the springs together. The weight is given y displacement from the equilibrium position, then both springs are extended equally in length; where as the restoring force on both the springs is different.
Hence the restoring force working on both the springs is;
F₁ = -k₁y
and F₂ = -k₂ y respectively;
The effective restoring force on the combination in this position is;
F = F₁ + F₂
– ky = -k₁y – k₂y

and effective spring constant k = k₁ + k₂

Fig: Series combination of the springs

Time period of oscillating body;

\(T = 2 \pi \sqrt{\frac{m}{k}}\)

where, effective spring constant

k = k₁ + k₂