THE PLASTICITY OF SOLIDS

227

portional to the torque, and there is a very considerable magni-
tude to the value which can be assigned to the friction in his
experiments with pitch.

Trouton assumes from symmetry that any two planes in the
material, lying at right angles to the axis of the cylinder, move
over each other, about the common axis, remaining plane all the
while.

Let dx be the distance apart of the two planes, and 5co be the
relative angular velocity of the planes, then

~
F

« 2m" 8(1) I
— / = — -r I

At dX Jo

where F is the torque applied and / is the force used up in Over-
coming the friction, obtained by extrapolation.
4 Thus for a solid cylinder we have

(91)

and for a tube of material this becomes

"20F-7)

(92)

where Ri and Rz are the external and internal radii respectively.
Trouton proved the validity of the fourth-power law by using
two cylinders of pitch whose radii were 0.36 and 0.67 cm and
obtained mobilities of L01 X 1Q-11 and 0.99 X 10"-11 respectively
which is excellent agreement.

The Sagging Beam Method.—The rate of sagging U of a rod
at its center is found to be

5 gmL* ,

384 XI W

U =

where m is the mass of the rod between the supports, L is its
length and I is the moment of inertia of the cross-section of the
rod, and g is the gravitation constant. This does not take
account of the friction.
In order to prove that the rate of sagging of a beam varies as
the fourth power of the length, Trouton measured the times T
which beams of different lengths required to sag a certain dis-
tance. Table LIX shows that TL4 is very nearly constant.