Four springs, with k values of
k = | G d ^{4}
8 n D ^{3} |
The real values of the spring constant were then measured using the second equation shown.
k = | F
Dx |
The predicted values of k varied from the real values by
as much as
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 |
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.
Material | Insulated Copper | Bare Copper | Bare Copper | Steel | ||
---|---|---|---|---|---|---|
Wire Diameter | d | m | 0.00153 | 0.00117 | 0.00203 | 0.00069 |
Wire Length | L | m | 0.684 | 0.750 | 0.668 | 0.740 |
Shear Modulus | G | Pa | 37.6×10^{9} | 47.3×10^{9} | 36.5×10^{9} | 85.7×10^{9} |
Spring Diameter | D | m | 0.01300 | 0.00978 | 0.01914 | 0.00724 |
Number of Turns | n | 23 | 28 | 11 | 28 | |
Predicted Spring Constant | k′ | Nm^{-1} | 510.2 | 422.7 | 1004.4 | 228.6 |
Actual Spring Constant | k | Nm^{-1} | 930.3 | 712.4 | 1618.0 | 330.2 |
Maximum Load | F_{max} | N | ? | 6.5 | 25 | 23 |
Quantity | Error | ||
---|---|---|---|
Wire Length | DL = | ±2×10^{-3}m | |
Mass Radii | DR = | ±0.11×10^{-3}m | |
Wire Diameter | Dd = | ±0.25×10^{-3}m | |
Spring Diameter | |||
Commercial | DD = | ±0.06×10^{-3}m | |
Constructed | DD = | ±4×10^{-3}kg | |
Spring Length | |||
Commercial | D(Dx) = | ±2×10^{-3}m | |
Constructed | D(Dx) = | ±1.2×10^{-3}kg |
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.