Page:Elementary Text-book of Physics (Anthony, 1897).djvu/136

122 where $$\tau$$ is the amount of torsion, $$l$$ the length, $$r$$ the radius of the wire, $$C$$ the moment of couple, and $$n$$ the modulus of rigidity. No general formula can be found for wires with sections of variable form.

The laws of torsion in wires were first investigated by Coulomb, who applied them in the construction of an apparatus called the torsion balance, of great value for the measurement of small forces.

The apparatus consists essentially of a small cylindrical wire, suspended firmly from the centre of a disk, upon which is cut a graduated circle. By the rotation of this disk any required amount of torsion may be given to the wire. On the other extremity of the wire is fixed, horizontally, a bar, to the ends of which the forces constituting the couple are applied. Arrangements are also made by which the angular deviation of this bar from the point of equilibrium may be determined. When forces are applied to the bar, it may be brought back to its former point of equilibrium by rotation of the upper disk. Let $$\Theta$$ represent the moment of torsion; that is, the couple which, acting on an arm of unit length, will give the wire an amount of torsion equal to a radian, $$C$$ the moment of couple acting on the bar, $$\tau$$ the amount of torsion measured in radians; then $$C = \Theta \tau$$. We may find the value of $$\Theta$$ in absolute measure by a method of oscillations analogous to that used to determine $$g$$ with the pendulum.

A body of which the moment of inertia can be determined by calculation is substituted for the bar, and the time $$T$$ of one of its oscillations about the position of equilibrium observed.

Since the amount of torsion is proportional to the moment of couple, the oscillating body has a simple harmonic motion.

The angular acceleration a of the oscillating body is given by the equation $$C = \Theta \tau = I \alpha$$ (§ 39). Now, since every point in the body has a simple harmonic motion, in which its displacement is proportional to its acceleration, and since its displacement and acceleration are proportional respectively to the angular displacement $$\tau$$ and the angular acceleration $$\alpha$$, we may set $$\alpha = \frac{4 \pi^2}{T^2} \tau .$$ Making