IN03 · A laboratory measurement of gravity

Cavendish Gravitational Balance

How can a slowly twisting wire measure the attraction between two small spheres? Nearby spheres pull the suspended pair sideways and twist a soft ribbon. A mirror turns that tiny angle into a light-spot displacement. Measure the ribbon’s free oscillation and the shift after the source spheres change sides to estimate their gravitational attraction.

Move the source spheres, follow the reflected light and record both equilibrium positions. Calibrate the wire from a separate free oscillation, then estimate the gravitational constant from the stored optical readings.

A very soft spring for gravity

Live spot $0.20\,\mathrm{mm}$

Live rotor angle $0\,\mathrm{mrad}$

Gravity from saved data $6.67\times10^{-11}$

\mathrm{m^3\,kg^{-1}\,s^{-2}}

Calibrated free period $486\,\mathrm{s}$

Closest center spacing

b/\mathrm{mm}

46.5

Damping ratio

\zeta

0.16

Each source mass

M/\mathrm{kg}

1.5

Drag either turquoise source sphere. The gold test masses and mirror respond slowly.

Detector and calibration

Detector zero drift

v_0/(\mathrm{mm}\,\mathrm{h}^{-1})

0

Spot noise scale

\sigma_x/\mathrm{mm}

0.015

Mirror to screen

L/\mathrm{m}

1.5

Sampling seed

s_0

176

The locked-rotor zero is an ideal noiseless reference. Detector drift acts only during the two-position measurement; it is suppressed during the separate free-oscillation calibration.

Model assumptions

Simulated Newtonian gravity between four homogeneous, non-overlapping spheres drives one torsion coordinate. All four source–test interactions and sphere rotational inertia are included. The ribbon is linear with viscous damping; the crossbar and mirror are massless. An ideal noiseless locked-rotor reference fixes the optical zero. A separate source-free oscillation calibrates stiffness; the drift control acts only on the subsequent two-position measurement. Long traces are time-compressed; all timestamps retain physical seconds. The apparatus keeps the true tiny rotation; the optical path is compressed in the drawing. Thermal motion, electrostatic forces, air currents, anelasticity and geometric calibration uncertainty are omitted. Adjustable detector drift is deliberately left uncorrected in the inferred result.

Recorded spot after the source changes sides

Default simulated measurement · detector samples

$t/\mathrm{min},\quad x/\mathrm{mm}$

Teal: simulated detector samples. Gold: means of the final fifth of each position. Changing live settings leaves this record intact.

Free-oscillation calibration and model

Move the source away and time the ribbon

$t/\mathrm{min},\quad x/\mathrm{mm}$

Homogeneous spheres, one torsion angle, linear ribbon and viscous damping. Four source–test interactions are summed. The real minute rotation is retained; the drawn optical path is compressed. Time runs sixty times faster than the laboratory clock. Moving light markers show propagation direction; their size and speed are schematic.

Start with the essentials

Focus question: How can a slowly twisting wire measure the attraction between two small spheres?
One-sentence intuition: The optical displacement measures an angle; an independently timed free oscillation supplies the force scale.

Core mathematical model

Each source attracts both test spheres

\tau_g=-G\sum_{i=1}^{2}\sum_{j=1}^{2}m_iM_j\frac{(\mathbf r_i\times(\mathbf R_j-\mathbf r_i))_y}{|\mathbf R_j-\mathbf r_i|^3}

Positions are measured from the suspension axis in the horizontal plane. The positive rotation convention is from positive x toward positive z; the displayed scalar torque uses that convention.

A measured free oscillation sets the ribbon stiffness

\kappa=I\left[\left(\frac{2\pi}{T_d}\right)^2+\gamma^2\right]

The damped period and exponential decay rate are fitted from stored source-free spot readings. The inertia includes the two spheres and their own spin about the suspension axis.

The reflected spot doubles the mirror angle

x-x_0=L\tan(2\theta)

The optical zero comes from a separate locked-rotor reference. Both source positions retain their own timestamps and readings.

Common difficulties

The far source sphere also pulls

Typical misconceptionEach test sphere only feels the large sphere nearest to it.

Better mental modelBoth sources attract both test spheres. The farther attraction partly cancels the near-pair torque and becomes important when the source is moved away.

Run the experiment

01
Move the source mass
Drag a turquoise sphere or change Closest center spacing, then use Swap source sides.
What to observe: The two source spheres move together. Their torque changes sign when they pass to the other side of the test masses.
02
Record both equilibria
Use Record both positions and calibrate. Inspect the gold mean markers near the end of each trace.
What to observe: The light spot settles on opposite sides of the calibrated zero. Their separation supplies a measured angle change.
03
Let the detector drift
Choose A drifting zero and compare the recovered gravitational constant with the default record.
What to observe: A changing detector zero moves both late-time means and biases their difference. Repeating a precise reading does not remove this systematic effect.

Cavendish Gravitational Balance

Recorded spot after the source changes sides

Move the source away and time the ribbon

Turn a tiny attraction into a readable displacement

Core mathematical model

Each source attracts both test spheres

A measured free oscillation sets the ribbon stiffness

The reflected spot doubles the mirror angle

Common difficulties

The far source sphere also pulls

Run the experiment

Move the source mass

Record both equilibria

Let the detector drift

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