## 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

#### 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

### 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