## 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
Mass Radii DR = ±0.11×10-3m
Wire Diameter Dd = ±0.25×10-3m
Spring Diameter
Commercial DD = ±0.06×10-3m
Constructed DD = ±4×10-3kg
Spring Length
Commercial D(Dx) = ±2×10-3m
Constructed D(Dx) = ±1.2×10-3kg

The micrometer is inaccurate in measuring wire thicknesses due to the inherent ductability of wire. A better method for measuring the wire diameters would have been to weigh a known length of wire and find its volume (e.g. by fluid displacement). This would allow us to calculate the density and thus the diameter of the wire.

The error in the length measurements is due to the difficulty of measuring long lengths, vertically, with the wire clamped into a frame. The obvious solution to this problem, measuring the wire horizontally, requires that the wire be marked (or cut) at the points where it is clamped, which introduces a greater error than the measuring actually involves.