## Properties of Elastic Solids

### Young's Modulus

*Young's modulus* can be thought of as the *spring constant
for solids*. Consider an ideal *rod* (or bar) of length and
cross-sectional area . Suppose we apply a force to the face of
area , causing a displacement along the axis of the rod.
Then Young's modulus is given by

For wood, Young's modulus is on the order of N/m. For aluminum, it is around (a bit higher than glass which is near ), and structural steel has [180].

#### Young's Modulus as a Spring Constant

Recall (§B.1.3) that *Hooke's Law* defines a spring
constant as the applied force divided by the spring
displacement , or . An elastic solid can be viewed as a
*bundle* of ideal springs. Consider, for example, an *ideal
bar* (a rectangular solid in which one dimension, usually its
longest, is designated its length ), and consider compression by
along the length dimension. The length of each spring in
the bundle is the length of the bar, so that each spring constant
must be inversely proportional to ; in particular, each doubling of
length doubles the length of each ``spring'' in the bundle, and
therefore halves its stiffness. As a result, it is useful to
normalize displacement by length and use *relative
displacement*
. We need displacement per unit length
because we have a constant spring compliance per unit length.

The *number* of springs in parallel is proportional to the
cross-sectional area of the bar. Therefore, the force applied to
each spring is proportional to the total applied force divided by
the cross-sectional area . Thus, Hooke's law for each spring in the
bundle can be written

We may say that Young's modulus is the Hooke's-law spring constant for the spring made from a specifically cut section of the solid material, cut to length 1 and cross-sectional area 1. The shape of the cross-sectional area does not matter since all displacement is assumed to be longitudinal in this model.

### String Tension

The *tension* of a vibrating string is the *force* used
to stretch it. It is therefore directed along the axis of the string.
A force must be applied at the endpoint on the right, and a force
is applied at the endpoint on the left. Each point interior to
the string is pulled equally to the left and right,
*i.e.*, the net force on an interior point is
. (A nonzero
force on a massless point would produce an infinite acceleration.)

If the cross-sectional area of the string is , then the tension is given by the stress on the string times .

**Next Section:**

Wave Equation for the Vibrating String

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Rigid-Body Dynamics