Everyday experience tells us that force equals mass times acceleration, but does Newton's second law still hold for very small accelerations? By observing a torsion pendulum oscillating with a very long period, physicists in the US have now found that the law is valid down to accelerations of about 5 x 10-14 m/s2 -- a thousand times smaller than the previous lower limit. The researchers say that the results could give further credence to the existence of dark matter (Phys. Rev. Lett. 98 150801).
While Newton’s second law has been proven over and over again here on Earth, astronomical evidence has led some physicists to suggest that it may not hold for all values of acceleration. Stars at the outer edges of galaxies, for example, rotate faster than predicted by the second law. This rotation can be explained by either accepting that Newton’s law breaks down at very small accelerations or by the introduction of dark matter, which has yet to be observed directly. In our own solar system, the Pioneer 10 and 11 spacecraft appear to be affected by an acceleration as yet unexplained by Newton’s law as they travel away from the sun.
It turns out that both of these gravitational anomalies could be explained by introducing a characteristic acceleration below which Newton’s law breaks down. For rotating galaxies this acceleration is about 1 x 10-10 m/s2 and for Pioneer 10 and 11 it is about 9 x 10-10 m/s2.
In 1986, physicists verified the second law to about 10-11 m/s2, which suggested that researchers should look elsewhere for explanations. Now, Jens Gundlach and colleagues at the University of Washington along with co-workers in Indiana have used a torsion pendulum to confirm Newton’s law for accelerations down to 5 x 10-14 m/s2 — providing even stronger evidence that a breakdown of Newton’s law is not responsible for these anomalies.
The pendulum weighed 70 g and was suspended from a metre-long tungsten wire that was 20 µm in diameter. When twisted, the wire exerted a restorative force on the weight, causing it to oscillate with a period of about 13 minutes. The weight was made to oscillate over a range of extremely small amplitudes (13 nrad to 19 µrad), which meant that weight experienced extremely small accelerations for relatively long periods of time. Any breakdown in Newton’s law during these periods of small acceleration should have caused the oscillation frequency of the pendulum to deviate from that predicted by Newton’s law.
The researchers measured pairs of amplitudes and frequencies over a wide range of amplitudes. The amplitude was then used to calculate the maximum force on the weight and the oscillation and amplitude were used to calculate the maximum acceleration. The physicists found that force did indeed equal mass times acceleration down to an acceleration of 5 x 10 -14 m/s2.
While this shows that Newton’s law holds at very low accelerations, the measurements were made using a mechanical restoring force and not gravity. Gundlach and colleagues are now devising an experiment that will test Newton’s law at very small accelerations in which the force is gravitational.
