Page:Elementary Text-book of Physics (Anthony, 1897).djvu/135

§ 109] tension causes an increase of volume given by $$\frac{P}{3k}\cdot$$ This is equivalent to an elongation of each side of the cube equal to $$\frac{P}{9k},$$ since the changes of form are supposed infinitesimal (§ 98). One of the shears produces an elongation between the upper and lower faces equal to $$\frac{P}{6n}$$, and a negative elongation or contraction equal to $$\frac{P}{6n}$$ between one pair of the other faces. The other shear produces an equal elongation between the upper and lower faces and an equal contraction between the remaining pair of faces. The total elongation between the upper and lower faces is therefore $$P \left(\frac{1}{9k} + \frac{1}{3n}\right)$$, and the total contraction between either pair of the other faces is given by $$P \left(\frac{1}{6n} - \frac{1}{9k}\right)\cdot$$

Since the two shears involved in the longitudinal traction cause no change of volume, the change of volume experienced by the body is due to the hydrostatic tension alone. It is therefore equal to $$\frac{P}{3k}\cdot$$ A body under longitudinal traction will therefore experience an increase of volume unless it is practically incompressible, that is, unless the ratio $$\frac{P}{k}$$ is negligible.

109. Elasticity of Torsion.—When a cylindrical wire, clamped at one end, is subjected at the other to the action of a couple, the axis of which is the axis of the cylinder, it is found that the amount of torsion, measured by the angle of displacement of the arm of the couple, is proportional to the moment of the couple, to the length of the wire, and inversely to the fourth power of its radius. It also depends on the modulus of rigidity. The relation among these magnitudes may be shown to be represented by the formula