---
title: "Balance Beam & Torque Puzzle Interactive Lab | Sandbox Physics"
description: "Can the smallest counterweight balance a loaded beam? Yes, if it hangs on the longest lever arm on the opposite side."
canonical_url: "https://sandboxphysics.com/labs/lever-torque/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/lever-torque/"
last_modified: "2026-11-03"
---

M021 · Rigid-body statics

# Balance Beam & Torque Puzzle

**Can the smallest counterweight balance a loaded beam?** Yes, if it hangs on the longest lever arm on the opposite side. Count the beam's own weight in the torque balance, and reject any arrangement where a support would have to pull down instead of push up.

Drag loads along a beam and move the hinge. Find the least counterweight, watch a support lift off, then use a virtual sensor sweep to estimate an unknown mass.

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

Model assumptions

The beam is slender and the loads are point masses; block dimensions and hardware are schematic. On a single hinge, the beam's rotation ends at ideal stops. After one of two supports lifts, a guide keeps rotation about the remaining contact. Ideal stops end the trial at twenty-five degrees and absorb its remaining kinetic energy.

Physics tutorial

## A small weight can balance a large load

**Background** The two loads pull on opposite sides of a beam. Their distances from the hinge decide which side drops.

**Why it matters** A crane uses a counterweight for the same reason: moving a weight farther from the pivot increases its turning effect. The beam's own weight matters when the hinge is off center.

Start with the essentials

- **Focus question**: Can the smallest counterweight balance a loaded beam?
- **One-sentence intuition**: The lightest counterweight sits at the farthest available point on the side opposing the other loads.

### Core mathematical model

#### A resting beam needs zero force and zero moment

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

Take counterclockwise rotation as positive. A clamp can supply a holding couple; an ideal hinge supplies force and allows rotation after the clamp is removed.

#### Gravity turns each load about the hinge

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

Positions are measured from the middle of the four-metre uniform beam. Its weight acts at the middle, so shifting the hinge gives the beam itself a lever arm.

#### The longest opposing lever arm uses the least mass

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

Leave the present counterweight out of the moment to be balanced, then put the new one at the opposite end of the allowed placement range. The automatic solution keeps full numerical precision.

#### Two supports divide the total weight

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

Both reactions can be positive only when the mass center lies between the supports. A negative required value signals lift-off; the trial then rotates about the remaining contact.

#### A moving hinge reaction follows the mass center

$$
R_x=M a_{C,x},\qquad R_y=M(g+a_{C,y}),\qquad I_p\ddot\phi=\tau_g(\phi)
$$

After release, the support force changes with the center's acceleration. The guided pivot fixes one rotation axis; the ideal stops absorb the remaining kinetic energy at twenty-five degrees.

#### A sensor sweep measures the counterweight

$$
\widehat m_c=\frac1g\frac{\mathrm d\tau_{\mathrm{hold}}}{\mathrm d x_c}=\frac{b-a}{g}\frac{\mathrm dR_b}{\mathrm d x_c}
$$

Clamp the beam for the sensor trial. A known change in counterweight position changes the holding couple or right reaction in proportion to that mass. The orange observations and fitted line are kept separate from the ideal reference.

### Common difficulties

#### Equal weights need not balance

**Typical misconception** Matching the two masses is enough.

**Better mental model** Moment also depends on distance. A smaller mass farther from the hinge can balance a larger mass nearby.

#### A hinge and a clamp do different jobs

**Typical misconception** The beam stays still, so its weights must already balance.

**Better mental model** A clamp can hold an unbalanced beam. Release it to see whether gravity starts a rotation.

#### A negative post reaction means lift-off

**Typical misconception** A post can exert the negative force given by the equilibrium equations.

**Better mental model** These posts push upward. When one would need to pull downward, that contact opens and the remaining support becomes the trial's pivot.

### Run the experiment

1. 01

   #### Watch the heavier moment win

   Let Torque puzzle run, then pause it.

   **What to observe:** The right side drops even though its mass is only moderately larger; the turning effects combine with their distances from the hinge.

2. 02

   #### Move a load directly

   Drag either colored load along the beam, or focus its handle and press an arrow key.

   **What to observe:** The initial moment changes continuously. Moving the hinge also changes the beam's own contribution.

3. 03

   #### Find the lightest balance

   Press Place minimum counterweight, then release the beam again.

   **What to observe:** The gold weight appears at the farthest opposing position and the free beam stays level. At settings that need more than the allowed mass, the solution button is disabled.

4. 04

   #### Make a support lift

   Compare Two supports with Support lift-off. Open Forces, energy and diagnostics.

   **What to observe:** The first case has two upward reactions. The second has an impossible downward requirement at one post and rotates about the other.

5. 05

   #### Measure a known mass

   Set a nonzero Counterweight mass, open Clamp and measure a counterweight, and press Scan 19 positions.

   **What to observe:** The clamp holds the beam while the virtual sensor readings give a slope. The fitted mass uses those readings; the teal curve is a separate ideal reference.

6. 06

   #### Test the estimate

   Change the sensor noise in Geometry, release and force display, scan again, then export the observations.

   **What to observe:** Noise scatters the readings and changes the fitted mass. The export contains position and sensor signal, so the fit can be repeated independently.

Physical conclusion

Balance every external force and moment, including the beam's weight. A post loses contact when keeping the beam level would require it to pull downward.

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)