# Can we measure $10^{-12}\ \mathrm{N}$ force?

I would be interested to measure a very small force, say in the order of $$10^{-12}\ \mathrm{N}$$? Is this possible? What equipment is needed?

## My setup

Assume that I have a relatively heavy machine say between 5-10 kg that I want to measure if it produces this thrust, which according to calculations should be of this feeble magnitude. But (according to the predictions) this should be periodic, with a frequency of about 200Hz and it should last for about a quarter of the time period. I should also mention that this apparatus is expected to vibrate (a little), since inside a disk is supposed to be rotated at about 12k rpm.

## My research

I have read about torsion balance as a possible method. I am also thinking about some piezo-electric crystals. Would be feasible?

What piezoelectric cells would be recommended? I read that the Atomic Force Microscopy devices are also implemented using piezoelectric materials.

• Just one man's "wild" idea: don't atomic force microscopes use this technique? Nov 26, 2021 at 2:54

The question is, a $$10^{-12}\rm\,N$$ force applied to what. A force of $$10^{-12}\rm\,N$$ applied to a hydrogen atom, with mass $$10^{-27}\rm\,kg$$, would produce an acceleration $$F/m = 10^{+15}\rm\,m/s^2$$.

A torsion pendulum is absolutely a way to allow very feeble forces to cause observable, macroscopic motion. My favorite underrated classic paper is Beth’s 1936 experiment which transferred angular momentum from a beam of circularly polarized light to a torsion pendulum. There was a parity-violation experiment in the 1960s that used a torsion pendulum as a detector for circular polarization in photons emitted from a parity-violating weak interaction process. And the 2001-ish proposal that gravity might be non-Newtonian at short distances has been mostly ruled out by torsion-pendulum measurements of gravitational attraction between coin-sized test masses.

In all of those cases, you accumulate the very small force by doing the experiment many times, repeating at a frequency near the resonant frequency of the pendulum.

For more direct measurements of very tiny forces, you might read about the operation of an atomic force microscope.

• Another very notorious but perhaps more mundane example is the Einstein-de Haas effect, which establishes a direct connection between "intrinsic" angular momentum and spatial/rotational angular momentum Nov 26, 2021 at 1:42
• Just to add to something you said, often measuring a small effect that is varying in time at a known frequency is easier than measuring a small effect that is constant. Nov 26, 2021 at 2:17
• For an excellent of the story that @lurscher refers to, read Galison's "How Experiments End."
– rob
Nov 26, 2021 at 2:18

Not to answer your question completely but on the AFM point- we can model the AFM cantilever as a spring in contact mode. The spring constant can be ~0.2 N/m or lower.

Now you can, without much work get z-direction sensitivities of 0.1 nm without too much trouble, taking into account experimental noise.

Very roughly speaking this gives a resolution of 0.02 nN or 20 pN. So, not far off for measuring forces on a tiny cantilever. However, for a mass that size I'm not sure it's possible.