---
title: "Precision Balance and Air Buoyancy Interactive Lab | Sandbox Physics"
description: "Why can the same object give a different mass reading in air? Air pushes upward on both the sample and the reference weights."
canonical_url: "https://sandboxphysics.com/labs/balance-beam/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/balance-beam/"
last_modified: "2026-10-28"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

[Precision weighing](https://sandboxphysics.com/labs/balance-beam/?) [Original torque workshop](https://sandboxphysics.com/labs/balance-beam/?balance-mode=mechanics)

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](https://www.nist.gov/system/files/documents/2019/05/13/sop-2-applying-air-buoyancy-20190506.pdf) · [METTLER TOLEDO, External Influences (2001), pp. 1–2](https://www.mt.com/mt_ext_files/Editorial/Generic/2/Weigh_Uncertain_Number1_0x0003d6750003db6700091746_files/Adverse_Influences.pdf)

Physics tutorial

## Air changes the force a pan receives

**Background** A 100 gram sample occupies more space than its steel reference. Air lifts both, but lifts the larger one more.

**Why it matters** Small differences in buoyancy can move the final digits of a precise weighing. Unequal arms can produce an additional error.

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 misconception** The mass of the sample changed when the air density changed.

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

#### Where the force acts

**Typical misconception** Sliding an object inside a freely hanging pan changes its beam lever arm.

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

### Run the experiment

1. 01

   #### Compare densities

   Compare Air lifts the sample with Equal densities.

   **What to observe:** Equal sample and reference densities remove the buoyancy correction.

2. 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.

3. 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.

4. 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.

Physical conclusion

Use the same pan to compare sample and standard, measure the zero and sensitivity, and apply the correction using the known densities.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Air helps hold up the object on a balance**](https://sandboxphysics.com/stories/instruments/precision-balance/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)