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Question
Write short notes on two springs connected in series.
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Solution
- 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.
- 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 ks (unknown quantity).
- 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.
- The results thus obtained can be generalised for any number of springs in series.
Springs are connected in series - 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.
- 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)
- 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) - 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`
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