| MadSci Network: Physics |
Extensive data on the mechanical properties of wood is available. One good source is Mechanical Properties of Wood by Green, Winandy and Kretschmann which is taken from The Wood Handbook. The values for the bulk Young's Modulus of a wide variety of woods are listed in their Table 4-3, and a few selected values are given below. It is important to note that the modulus and other mechanical properties are highly anisotropic for wood due to the structure of the wood or wood grain. They will generally be best in the direction of the grain and worst perpendicular to the grain. Fortunately most meter sticks have the grains oriented along the length. The values obtained in the longitudinal direction are approximately 10% higher than the bulk values listed below.
| Species | Modulus of Elasticity (MPa) 12% moisture content |
|---|---|
| Ash (White) | 12,000 |
| Cherry (Black) | 10,300 |
| Elm (American) | 9,200 |
| Hickory (Mockernut) | 15,600 |
| Maple (Red) | 11,300 |
| Oak (White) | 12,300 |
| Walnut (Black) | 11,600 |
| Cedar (Northern white) | 5,500 |
| Pine (Eastern white) | 8,500 |
| Redwood (Young-growth) | 7,600 |
Meter sticks used in schools are typically made from a hard wood such as maple or oak. Since the value for Young's moduius can vary so much, it is a good idea to either learn which wood was used to make the meter stick or make a rectangular beam from a known wood. By using several different woods the dependence of Young's modulus upon material can be highlighted.
The Institute of Physics (IOP) has a lesson on measuring the Young's modulus of rulers and other common items that is informative. It includes suggestions for testing.
In general, the maximum deflection which occurs at the end of a cantilever beam when a load is applied to the end is given by the equation
where
In this context, the moment of inertia is Second Momment, or Moment of
Inertia, of an Area (1). The moment of inertia used is a moment of inertia of a
geometric shape, e.g., a rectangle. Its full formal name is the Second
Moment of the Area. This moment of inertia does not involve a mass,
only lengths and areas. The mathematical equation for the Second
Moments of the Area are
Ix = (Integral) y^2 dA
Iy = (Integral) x^2 dA
The integral gives a value with a fourth order length tem (m^2 * m^2)
when the integrals are integrated over the area. Multiplying
the second moment (m^4) by density (kg/m^2) yields the units that many people
are familiar with for a moment of inertia: (kg * m^2).
This is for a weightless beam or one whose weight is very small relative to the applied load. If the beam weight is not negligible relative to the applied load, the weight must be considered as a distributed load. The maximum deflection of a cantilever beam subjected to a uniform load along its entire length occurs at the free end and is given by the general equation
where
The total deflection is a summation of the two deformations. For a uniform rectangular beam such as a typical meter stick, the equation for calculating the deflection is
where
By hanging a known weight from the end or pulling directly downward with a known force and measuring the deflection of the tip of the meter stick it is possible to get all the information needed to determine E. For best results, the deflections should be measured at several loads and the results averaged. It is also a good idea to plot the deflection versus load to be sure that the data points form a straight line. Outliers can be found and eliminated in this manner. Depending on the class level, statistics such as mean/average, standard deviation and confidence intervals can be introduced by using multiple readings.
How accurately a measurement of the Young's modulus can be made depends upon the accuracy to which Mg, mg, y, L, b and d can be measured. A classroom ruler might only measure y, b and d to the nearest millimeter. In the case of the thickness of a meter stick that resolution would introduce a significant error into b, perhaps as much as 17% for a meter stick 6 mm thick. A typical error in the measurement of the height would be 4% for a 25 mm tall meter stick. If micrometers are used, the accuracy of measuring b and d can be improved 1,000X with the ability to resolve the dimensions down to as little as 1 micrometer. The error in measuring b would correspondingly decrease to 0.0017%, a truly negligible number. Similar improvements would be seen in the errors for the other measurements.
Likewise the accuracy of the measurement of Mg and mg will affect the accuracy of measuring the Young's modulus. If a typical spring scale is used, an error of 5% to 10% is easily introduced. Dead weights measured with an analytical scale may have errors of 0.01% or even less.
If one combines the errors of using a typical ruler with a spring scale the total error in the measurement of Young's modulus could be as large as 56%. If micrometers and calipers are used to measure the smaller dimensions, a machinist's ruler is used to measure the length, and dead weights are used to apply the load, the error diminishes to a fraction of a percent. This highlights why it is important to use the best analytical techniques for measuring scientific phenomena.
For a well run classroom experiment with multiple measurements of deflection at various loads and repeats, the accuracy should be on the order of 1% to 5% of the wood's modulus. If the meter stick is assumed to be red maple with the grain oriented in the direction of the length of the beam and the moisture content to be around 12%, the modulus should be 12.4 GPa. A 5% error would be 0.6 GPa, and an error around 0.1 GPa should be achievable.
1. Mechanics of Materialsm F.P. Beer and E.R. Johnston, Jr., McGraw-Hill Book Company, New York, NY, (1981) pp.579-581
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