Question

Write short notes on two springs connected in series.

Answer 1

1. When two or more springs are connected in series, we can replace (by removing) all the springs in series with an equivalent spring (effective spring) whose net effect is the same as if all the springs are in series connection.
2. Given the value of individual spring constants k₁, k₂, k₃, ... (known quantity), we can establish a mathematical relationship to find out an effective (or equivalent) spring constant k_s (unknown quantity).
3. For simplicity, let us consider only two springs whose spring constants are k₁ and k₂ and which can be attached to a mass m as shown in Figure.
4. The results thus obtained can be generalised for any number of springs in series.
   

   Springs are connected in series

5. Let F be the applied force towards right as shown in Figure. Since the spring constants for different springs are different and the connection points between them are not rigidly fixed, the strings can stretch in different lengths.
6. Let x₁ and x₂ be the elongation of springs from their equilibrium position (un-stretched position) due to the applied force F. Then, the net displacement of the mass point is x = x₁ + x₂    ...(i)
7. From Hooke’s law, the net force
   

   Effective spring constant in series connection

   
   `F = -k_s(x_1 + x_2) => x_1 + x_2 = -F/K_s`    ...(ii)
8. For springs in series connection 
    −k₁ x₁ = −k₂ x₂ = F   
   `=> x_1 = -F/k_1` and `x_2 = -F/k_2`    ...(iii)
   Therefore, substituting equation (iii) in equation (ii), the effective spring constant can be calculated as
   `-F/k_1 - F/k_2 = -F/k_s`
   `1/k_s = 1/k_1 + 1/k_2`
   Or,
   `k_s = (k_1k_2)/(k_1 + k_2)Nm^-1`