Laboratory Test of Newton’s Second Law for Small Accelerations

We have tested the proportionality of force and acceleration in Newton’s second law, $F=ma$, in the limit of small forces and accelerations. Our tests reach well below the acceleration scales relevant to understanding several current astrophysical puzzles such as the flatness of galactic rotation curves, the Pioneer anomaly, and the Hubble acceleration. We find good agreement with Newton’s second law at accelerations as small as $5\times {10}^{-14}\mathrm{m}/{\mathrm{s}}^2$.

Figure 1

The solid line shows a typical measurement of the angular position of the torsion pendulum. The upward drift in the data is due to a slow unwinding of the torsion fiber. The noise in the measured trace is due to Brownian excitation and readout noise. For purposes of comparison the simulated motion of the pendulum in a MOND-like regime with $a_0=1.0\times {10}^{-12}\mathrm{m}/{\mathrm{s}}^2$ is plotted with a dashed line. Note that the MOND-like dynamics would result in a much shorter period of oscillation.

Figure 2

The measured force versus the measured acceleration. The solid line is the best fit for acceleration, $a$ being exactly proportional to force $F$. Our data agree very well with the curve. The insets on the right and top of the main graph give the residuals of the data to the fitted line.