## Parallel Springs

Consider three springs in parallel have the following spring constants (from left to right): 1 N/m, 2 N/m, & 3 N/m. If a 10 kg mass is placed on top, how many meters will the springs compress?

##
__
__**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 parallel:

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

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

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

Since the problem statement has three springs in parallel:

$$$k_{eq}=k_1+k_2+k_3=1\frac{N}{m}+2\frac{N}{m}+3\frac{N}{m}=6\:N/m$$$

A spring’s deflection and force are related by:

$$$F_s=k\delta $$$

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

Before proceeding, we need to determine the spring force by multiplying the mass by gravity:

$$$F_s=mass\times acceleration=(10kg)(9.8m/s^2)=98\:N$$$

Now our units should cancel out when we solve for the spring’s compression distance:

$$$\delta=\frac{F_s}{k_{eq}}=\frac{98N}{6N/m}=16.3\:m$$$

16.3 m