The Optical Lever

An optical lever is a convenient device to magnify a small displacement and thus to make possible an accurate measurement of the displacement. Experiment M-11, Young's modulus, uses an optical lever to magnify the extension of a wire produced by a series of different loads.

The plate P carries a mirror M. The mirror mount has two points resting in a fixed groove, F, and at the other end has a single point resting on the object whose displacement one is measuring. Raising the object through a distance $\Delta L$ will tilt the mirror through an angle $\theta$ or $\Delta L/d$ radians (approximately) but will turn the light beam through an angle $2\theta$.

Figure 1: Schematic of the optical lever.

Hence

$$\theta \; = \; \frac{\Delta L}{d} \sim \frac{\frac{1}{2} ( y_1-y_0)}{D}$$

if $\theta$ is small so that $\theta \sim \tan \theta$. Therefore

$$2\theta = \frac{2\Delta L}{d} = \frac{y_1-y_0}{D}, \; \mbox{and} \; \Delta l = (y_1-y_0)\left[ \frac{d}{2D}\right]  .$$

Note that with the telescope nearly perpendicular to the scale at the beginning then $y_0$ is close to the telescope, and the difference between two elongations ( $\Delta L_2 -\Delta L_1$) is very accurately given by

$$\Delta L_2 - \Delta L_1 = \frac{y_2 d}{2D} - \frac{y_1 d}{2D} = \frac{( y_2-y_1) d}{2D}$$

where $y_{i}$ is the scale reading. This relation holds so long as $2\theta$ is small enough that $\tan 2\theta  \sim  2 \theta$.