M021 · IN02 · Precision weighing

Precision Balance & Buoyancy

Why can the same object give a different mass reading in air? Air pushes upward on both the sample and the reference weights. Different densities mean different displaced volumes, so the indicated mass changes; correcting the buoyancy recovers the mass when those densities are known.

Move the pan suspension, calibrate zero and sensitivity, and compare direct with substitution weighing. Record the detector signal and correct the air buoyancy. The original torque workshop remains available.

Indicated mass $99.8950\,\mathrm{g}$

Buoyancy-corrected mass $100.0000\,\mathrm{g}$

Air correction $105\,\mathrm{mg}$

Recorded detector signal $-15.4\,\mathrm{div}$

What did the air lift?

Drag the sample-pan suspension or the gold reference stack. Keep a weighing before changing the apparatus.

InstrumentWeighing methodSample mass

m_x/\mathrm{g}

100

Reference mass

m_r/\mathrm{g}

100

Air density

\rho_a/(\mathrm{kg}\,\mathrm{m}^{-3})

1.2

Sample density

\rho_x/(\mathrm{kg}\,\mathrm{m}^{-3})

1000

Geometry, drift and noise

Left suspension position

x/\mathrm{mm}

0

Pivot offset

p/\mathrm{mm}

0

Zero offset

z/\mathrm{mg}

0.8

Zero drift per second

\dot z/(\mathrm{mg}\,\mathrm{s}^{-1})

0.08

Noise bound

\epsilon/\mathrm{mg}

0.03

Seed

s

175

Moving the suspension changes the lever arm; moving a sample inside an ideal freely hanging pan would not. Recalibrate after changing geometry or instrument.

Model assumptions

Quasi-static teaching apparatus with rigid arms, a prescribed linear restoring stiffness and known material densities. The indicated mass compares the sample with true-mass standards of density eight thousand kilograms per cubic metre in the current air. Air density is prescribed, not calculated from weather readings. The electromagnetic comparison assumes a fixed ten-newton-per-ampere force constant and ideal feedback. Friction, hysteresis, convection and detailed servo dynamics are omitted. The beam stops at its stated angle; the weighing signal is the force comparison, not the clipped scene angle.

Detector observations

Horizontal: observation time. Vertical: recorded signal; each dot is an acquired reading.

Time	Left load	Raw signal	Counterpoise

Calibration, model and sources

F_g-F_b=mg\left(1-\frac{\rho_a}{\rho}\right)

Air pushes upward by the weight of the displaced air. The reference and sample can displace different volumes.

m_x=m_{\rm ind}\frac{1-\rho_a/\rho_r}{1-\rho_a/\rho_x}

Mass indication is a comparison with true-mass reference weights in the current air. Sample and reference densities and air density must be known for this correction.

The arm length is measured from the pivot to the suspension. Positive beam rotation lowers the left pan. Pointer stiffness is prescribed; equilibrium outside the allowed angle rests at a stop.
The EMFR illustration replaces the right reference with a fixed-force-constant coil. An ideal servo holds the beam level; the measured current supplies the balancing moment.
Same-pan substitution cancels the fixed counterpoise and arm asymmetry. Averaging the two sample readings cancels a linear drift across the symmetric sequence.
Calibration stores an empty-pan reading and a 10 g standard reading. Repeatability noise is bounded and seeded; zero and sensitivity observations are idealized short measurements.

NIST SOP 2 (2019), equations 3 and 5, section 5.1 · METTLER TOLEDO, External Influences (2001), pp. 1–2

Start with the essentials

Focus question: Why can the same object give a different mass reading in air?
One-sentence intuition: A balance compares downward forces; different displaced volumes require a buoyancy correction.

Core mathematical model

Weight after air buoyancy

F=mg(1-\rho_a/\rho)

Mass is in kilograms, density in kilograms per cubic metre, and force in newtons. Air contributes an upward force equal to the weight of the displaced air.

Mass from the indicated comparison

m_x=m_{\rm ind}\frac{1-\rho_a/\rho_r}{1-\rho_a/\rho_x}

The indication compares the sample with true-mass standards of reference density in the current air. Sample and reference densities must be known.

Symmetric same-pan substitution

\Delta s=\frac{s_{A,0}+s_{A,10}}{2}-s_{B,5}

The two sample readings surround the reference reading at equal intervals. Averaging cancels a linear zero drift. The stored calibration sensitivity turns their difference into a mass difference.

Common difficulties

Mass and indication

Typical misconceptionThe mass of the sample changed when the air density changed.

Better mental modelThe sample displaces a different weight of air. That changes the downward force on the pan, and therefore the indication.

Where the force acts

Typical misconceptionSliding an object inside a freely hanging pan changes its beam lever arm.

Better mental modelThe pan transmits its load at the suspension. This experiment moves that suspension, so it really does change the arm.

Run the experiment

01
Compare densities
Compare Air lifts the sample with Equal densities.
What to observe: Equal sample and reference densities remove the buoyancy correction.
02
Compare weighing methods
Compare Unequal arms and Replace on the same pan, then keep both weighings.
What to observe: Same-pan substitution cancels the fixed counterpoise contribution despite unequal arms.
03
Retain a measurement
Keep weighing, move the sample suspension, then select the old weighing.
What to observe: The apparatus preview changes while the stored detector observations and calibration stay fixed.
04
Replace weights with a coil
Select Balance with a current.
What to observe: Ideal feedback holds the beam horizontal; the measured coil current carries the force information.

Start with the essentials

Focus question: Can the smallest counterweight balance a loaded beam?
One-sentence intuition: The counterweight needs the least mass when it sits on the opposing side with the longest possible lever arm.

Core mathematical model

Force and torque balance

\sum F_y=0,\qquad\sum\tau_p=0

Counterclockwise moments about the normal to the beam's plane are positive. The hinge pushes up on the beam, and the clamp supplies the couple that holds it still.

Gravity torque, including the beam

\tau_g=-g\left[m_L(x_L-p)+m_R(x_R-p)+m_c(x_c-p)-m_b p\right]

Positions are in metres, measured from the middle of the beam. The weight of the uniform beam acts at that midpoint.

Supports that can only push

R_b=Mg\frac{x_C-a}{b-a},\qquad R_a=Mg-R_b,\qquad R_a,R_b\geq0

With two supports, a negative required reaction means the center of mass lies outside the supports.

The minimum counterweight

m_{c,\min}=\frac{|m_L(x_L-p)+m_R(x_R-p)-m_b p|}{|x_{c,\mathrm{opt}}-p|}

Put the counterweight as far out as allowed on the opposing side. The counterweight already on the beam is left out of the torque it has to balance.

Mass from the slope of a sensor reading

\widehat m_c=\frac{1}{g}\frac{\mathrm d\tau_{\mathrm{hold}}}{\mathrm d x_c}=\frac{b-a}{g}\frac{\mathrm dR_b}{\mathrm d x_c}

The two forms read different sensors, so their units differ. Fit the raw virtual readings using the known support geometry.

Common difficulties

A clamp can hide an imbalance

Typical misconceptionA beam held still by a clamp with nonzero torque is balanced on its own.

Better mental modelRemove the clamp, and the leftover gravity torque starts the beam rotating.

Do not forget the beam

Typical misconceptionOnly the visible blocks produce gravity torque.

Better mental modelOnce the pivot moves off the middle, the uniform beam's own weight has a lever arm too.

Supports cannot pull

Typical misconceptionA negative reaction is a real downward pull from the post.

Better mental modelThe beam lifts off that support instead. The required reaction is only a warning sign.

Run the experiment

01
Solve the smallest-mass puzzle
Move the hinge and try Place minimum counterweight.
What to observe: When the gravity torque from the loads and beam changes sign, the best spot moves to the other end.
02
Release the holding clamp
Compare the Torque puzzle and Balanced beam presets, then release each from rest.
What to observe: Only the unbalanced beam rotates, and its energy is conserved all the way to the end stop.
03
Check the supports
Select Two supports and move a heavy load outside them.
What to observe: The required reaction changes sign, and the sensor stops recording physical readings.
04
Infer an unknown mass
Set a nonzero counterweight, scan its position, then add sensor noise.
What to observe: The slope of the fit gives the mass. Moving the counterweight keeps the readings already recorded.

Precision Balance & Buoyancy

Detector observations

Air changes the force a pan receives

Core mathematical model

Weight after air buoyancy

Mass from the indicated comparison

Symmetric same-pan substitution

Common difficulties

Mass and indication

Where the force acts

Run the experiment

Compare densities

Compare weighing methods

Retain a measurement

Replace weights with a coil

Balance requires a force and a moment budget

Core mathematical model

Force and torque balance

Gravity torque, including the beam

Supports that can only push

The minimum counterweight

Mass from the slope of a sensor reading

Common difficulties

A clamp can hide an imbalance

Do not forget the beam

Supports cannot pull

Run the experiment

Solve the smallest-mass puzzle

Release the holding clamp

Check the supports

Infer an unknown mass

Related experiments and articles

How physics instruments measure