Laboratory Test of Newton ’ s Second Law for Small Accelerations

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Laboratory Test of Newton ’ s Second Law for Small Accelerations. Ki-Young Choi Sogang University Stephan Schlamminger, Chris Spitzer, Jens Gundlach, University of Washington, Seattle Brian Woodahl, Jennifer Coy, Ephraim Fischbach Purdue University. PRINCIPIA: Law II.

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### Laboratory Test of Newton’s Second Law for Small Accelerations

Ki-Young Choi

Sogang University

Stephan Schlamminger, Chris Spitzer, Jens Gundlach,

University of Washington, Seattle

Brian Woodahl, Jennifer Coy, Ephraim Fischbach

Purdue University

PRINCIPIA: Law II

The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

PRINCIPIA: Law II

The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

Reasons to Doubt

Flatness of the galactic rotation curves imply more acceleration for very small forces, than Newton’s 2nd law (NSL).

Rotation velocity (km/s)

Distance from center (kpc)

Reasons to Doubt

Flatness of the galactic rotation curves imply more acceleration for very small forces, than Newton’s 2nd law (NSL).

Rotation velocity (km/s)

Milgrom in 1983 suggested this modification of NSL: Astrophys. J. 270, 371 (1987)

Distance from center (kpc)

For a>>a0μ=1 => recover NSL

For a<<a0μ=a/a0

Slope=1

Slope=2

a0=1.2 x 10-10 m/s2

a0

MOdified Newtonian Dynamics

Reasons to Doubt

Flatness of the galactic rotation curves imply more acceleration for very small forces, than Newton’s 2nd law (NSL).

Rotation velocity (km/s)

Milgrom in 1983 suggested this modification of NSL: Astrophys. J. 270, 371 (1987)

Distance from center (kpc)

For a>>a0μ=1 => recover NSL

For a<<a0μ=a/a0

Slope=1

Slope=2

a0=1.2 x 10-10 m/s2

a0

MOdified Newtonian Dynamics

A Torsion Balance to Measure F=ma

Simulated trace for a0=10-11 m/s2

Assume:

Hooke’s law is valid

Measured trace

The Experimental Setup

Allows to damp the pendulum to without storing energy in the fiber.

Measures the excursion of the pendulum. Dynamic range:

Noise:

The Torsion Pendulum

20 m diameter tungsten fiber

(length: 108 cm)

=2.39 nNm

8test masses (4 Be & 4 Ti )

4.84 g each (within 0.1 mg)

(can be removed)

4 mirrors

tiny asymmetries

torsional frequency: 1.266 mHz

quality factor: ~ 4000

decay time: ~ 12 days

machining tolerance: 5 m

total mass : 70 g

5 cm

Power Spectral Amplitude

Thermal Noise

Data

Amplitude Gain

Probability, that the measured amplitude < x after 1600 s (2 torsional periods)

In order to measure low amplitudes we have to repeatedly damp the pendulum and measure.

326 traces

Results

PRL 98, 150801 (2007)

Conclusion & Outlook
• Good agreement with Newton’s 2nd law at accelerations as small as 5x10-14 m/s2.
• Our result does not constrain MOND, since for MOND |a|=0.
• In the future we want to test Fg=ma, where Fg is a gravitational force.

### THE END

More reasons to be doubtful

a0=(8.7 ± 1.3) x 10-10 m/s2

More reasons to be doubtful

a0=(8.7 ± 1.3) x 10-10 m/s2

Coincidentally:

c H0=3 x 108 m/s x 71 km/s /Mpc =7 x 10-10 m/s2