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## Problem

Consider two springs with different spring constants and .

Part 1: Determine the equivalent spring constant when the two springs are connected in series.

Part 2: Determine the equivalent spring constant when the two springs are connected in parallel.

## Solution

Part 1: Springs connected in series (same forces, different length)

When two springs are connected in series, the result is essentially a longer and flimsier spring. When a force is applied to the combined spring, the same force is applied to each individual spring. Since the springs have different spring constants, the displacements are different. Thus we get three equations:

First equate equations 2 and 3 and isolate for

.

Substitute into equation 1.

To solve for the equivalent spring constant, set the above equation with equation 2. The strategy is to divide away .

In general, given springs added in series

.

Part 2: Springs connected in parallel (same length, different forces)

When two springs are connected in parallel, the result is essentially two springs working together. When a force is applied to the combined spring, the force applied on each individual spring is different. Thus we get three equations:

Thus

.

Therefore,

.

In general, given springs added in parallel

.