## Introduction

Four springs, with k values of 930 Nm-1, 712 Nm-1, 1620 Nm-1 and 330 Nm-1, were constructed, using copper and steel. Values for the spring constants were predicted based on the equation shown.

(1)
 k = G d 4 8 n D 3

The real values of the spring constant were then measured using the second equation shown.

(2)
 k = F Dx

The predicted values of k varied from the real values by as much as 620 Nm-1. However, this was still within the error margins (±637 Nm-1 in the extreme case).

### Variables

The variables in equations 1 and 2 are defined below.

 k: spring constant G: shear modulus of wire d: diameter of wire n: number of turns in spring D: diameter of spring F: force applied to spring Dx: extension in spring

## Background

Mechanical springs are elastic bodies, in that when a load is applied, they change shape to absorb the energy, and importantly when the load is removed, the original shape is recovered.

Above a certain load, plastic deformation occurs, and the spring is stretched. Within the elastic region (below the critical load) the relationship between load and extension is usually linear. This is known as Hooke's Law.

Our springs demonstrated a very linear relationship between load and extension, closely following Hooke's Law.

## Results

Material Wire Diameter Wire Length Shear Modulus Spring Diameter Number of Turns Insulated Copper Bare Copper Bare Copper Steel d m 0.00153 0.00117 0.00203 0.00069 L m 0.684 0.750 0.668 0.740 G Pa 37.6×109 47.3×109 36.5×109 85.7×109 D m 0.01300 0.00978 0.01914 0.00724 n 23 28 11 28 k′ Nm-1 510.2 422.7 1004.4 228.6 k Nm-1 930.3 712.4 1618.0 330.2 Fmax N ? 6.5 25 23

## Sources of Error

QuantityError
Wire Length DL = ±2×10-3m