A microcantilever, fabricated from a single crystal of silicon, is being used to test the inverse square law of gravity on micron scales (Weld et al., 2008). It is clamped horizontally at one end, and its horizontal length is ℓ = 300 μm, its horizontal width is w = 12 μm, and its vertical height is h = 1μm. (The density and Young’s modulus for silicon are ρ = 2,000 kgm⁻³ and E = 100 GPa, respectively.) The cantilever is loaded at its free end with a m = 10 μg gold mass.

(a) Show that the static deflection of the end of the cantilever is η(ℓ) = mgℓ³/(3D) = 9 μm, where g = 10 ms⁻² is the acceleration due to gravity. Explain why it is permissible to ignore the weight of the cantilever.

(b) Next suppose the mass is displaced slightly vertically and then released. Show that the natural frequency of oscillation of the cantilever is

(c) A second, similar gold mass is placed 100 μm away from the first. Estimate roughly the Newtonian gravitational attraction between these two masses, and compare with the attraction of Earth. Suggest a method that exploits the natural oscillation of the cantilever to measure the tiny gravitational attraction between the two gold masses.

The motivation for developing this technique was to seek departures from Newton’s inverse-square law of gravitation on ∼micron scales, which had been predicted if our universe is a membrane (“brane”) in a higher-dimensional space (“bulk”) with at least one macroscopic extra dimension. No such departures have been found as of 2016.

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f = 1/(27)√/g/n(l) ~ 170 Hz.