## Springs in Series

Consider three springs in series are attached to a fixed point on a ceiling as shown in the figure. How much force is required to stretch the springs 5 m?

##
__
__**Hint**

**Hint**

The force in a spring is:

$$$F_s=k\delta $$$

where
$$k$$
is the spring constant, and
$$\delta$$
is the change in spring length from the un-deformed spring length.

##
__
__**Hint 2**

**Hint 2**

The spring constant for springs in series:

$$$\frac{1}{k_{eq}}=\sum_{i}\frac{1}{k_i}$$$

First, let’s determine the equivalent spring constant for several springs in series:

$$$\frac{1}{k_{eq}}=\sum_{i}\frac{1}{k_i}$$$

Since the problem statement has three springs in series:

$$$\frac{1}{k_{eq}}=\frac{1}{k_1}+\frac{1}{k_2}+\frac{1}{k_3}=\frac{1}{1N/m}+\frac{1}{2N/m}+\frac{1}{3N/m}$$$

$$$\frac{1}{k_{eq}}=\frac{6}{6N/m}+\frac{3}{6N/m}+\frac{2}{6N/m}=\frac{11m}{6N}$$$

$$$\frac{1}{\frac{1}{k_{eq}}}=\frac{1}{\frac{11m}{6N}}$$$

$$$k_{eq}=\frac{6}{11}\:N/m$$$

A spring’s deflection and force are related by:

$$$F_s=k_{eq}\delta $$$

where
$$F_s$$
is the forced applied to the spring,
$$k_{eq}$$
is the equivalent spring constant, and
$$\delta$$
is the change in spring length from the un-deformed spring length.

Therefore, the force required to stretch the springs 5 m is:

$$$F=\frac{6N}{11m}\times 5m=2.7\:N$$$

2.7 N