Question

A light spring with spring constant k1 hangs from an elevated support. From its lower end hangs a second light spring, which has spring constant k1. An object of mass m is hung at rest from the lower end of the second spring.

Find the effective spring constant of the pair of springs as a system. We describe these springs as being in series.

Answer 1

The effective spring constant of two identical springs connected in series, each with spring constant k1, is half of the spring constant of one spring, which is keff = k1/2.

Step-by-step explanation:

The student has asked for the effective spring constant of a pair of light springs connected in series, each with a spring constant k1, with an object of mass m hanging from the lower spring.

When two springs are connected in series, the overall extension is the sum of the extensions of each spring under the same force. The effective spring constant keff for springs in series can be found using the formula:

1/keff = 1/k1 + 1/k2

Since both springs have the same spring constant k1, this simplifies to:

1/keff = 1/k1 + 1/k1

1/keff = 2/k1

Hence, keff = k1/2

The effective spring constant of the pair is therefore half the spring constant of one of the springs.

Answer 2

The effective spring constant of the pair of springs in series is half of each individual spring constant.

Step-by-step explanation:

The effective spring constant of the pair of springs in series can be found by adding the reciprocals of the individual spring constants. In this case, let's denote the spring constant of the first spring as k1 and the spring constant of the second spring as k2. The effective spring constant (keff) is given by:

1/keff = 1/k1 + 1/k2

Since the two springs are identical with the same spring constant k, we can substitute k for both k1 and k2:

1/keff = 1/k + 1/k = 2/k

Simplifying the equation, we get:

keff = k/2

Therefore, the effective spring constant of the pair of springs in series is half of each individual spring constant.