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Sandbox Physics

M021 · Rigid-body statics

Balance Beam & Torque Puzzle

Uniform beam, clamped loads, movable hinge and unilateral supports. Measure a sensor sweep, infer counterweight mass and release the clamp.

Interactive modelBalance Beam & Torque Puzzle
Total mass—\text{—}
Mass center along beam—\text{—}
Initial gravity torque at hinge—\text{—}
Holding clamp couple—\text{—}
Single hinge reaction—\text{—}
Actual left support reaction—\text{—}
Actual right support reaction—\text{—}
Requested left reaction—\text{—}
Requested right reaction—\text{—}
Minimum counterweight—\text{—}
Optimal counterweight position—\text{—}
Mass fitted from sensor slope—\text{—}
Equilibrium status—\text{—}
Rotation inertia about selected axis—\text{—}
Recorded observations—\text{—}
Fit residual RMS—\text{—}
Current rotation—\text{—}
Angular speed—\text{—}
Mechanical energy balance error—\text{—}
Energy absorbed at terminal stop—\text{—}

Physics tutorial

Balance requires a force and a moment budget

BackgroundA uniform beam carries two known loads and a movable counterweight.

Why it mattersOpenStax static-equilibrium conditions connect any torque origin with the same physical force balance.

Start with the essentials

Focus question
Can the smallest counterweight balance this beam?
One-sentence intuition
The longest opposing lever arm needs the least counterweight.

Core mathematical model

Force and torque balance

∑Fy=0,∑τp=0\sum F_y=0,\qquad\sum\tau_p=0

Positive moment is counterclockwise around the upward-plane normal. A hinge supplies upward force; a clamp supplies the holding couple.

Gravity and the uniform beam

τg=−g[mL(xL−p)+mR(xR−p)+mc(xc−p)−mbp]\tau_g=-g\left[m_L(x_L-p)+m_R(x_R-p)+m_c(x_c-p)-m_b p\right]

Coordinates in metres are measured from beam midpoint. The uniform beam mass acts at that midpoint.

Unilateral supports

Rb=MgxC−ab−a,Ra=Mg−Rb,Ra,Rb≥0R_b=Mg\frac{x_C-a}{b-a},\qquad R_a=Mg-R_b,\qquad R_a,R_b\geq0

In the two-support mode, a negative requested reaction means the center lies outside the support interval.

The minimum counterweight

mc,min⁡=∣mL(xL−p)+mR(xR−p)−mbp∣∣xc,opt−p∣m_{c,\min}=\frac{|m_L(x_L-p)+m_R(x_R-p)-m_b p|}{|x_{c,\mathrm{opt}}-p|}

Choose the farthest allowed position on the opposing side; the present counterweight is excluded from the target budget.

Sensor-slope mass estimate

m^c=1gdτholddxc=b−agdRbdxc\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 alternatives use different sensor units. Fit raw virtual observations with known support geometry.

Common difficulties

A constraint is a force source

Typical misconceptionA held beam with nonzero clamp torque is freely balanced.

Better mental modelRemove the clamp to see the remaining gravity moment drive rotation.

Do not forget the beam

Typical misconceptionOnly the visible blocks contribute gravity torque.

Better mental modelA shifted pivot gives uniform beam weight a nonzero lever arm.

Supports cannot pull

Typical misconceptionA negative reaction is an actual downward force from the post.

Better mental modelThe contact opens; requested reactions remain diagnostic only.

Run the experiment

  1. 01

    Solve the smallest-mass puzzle

    Move the hinge and try the automatic counterweight solution.

    What to observe: The opposing endpoint changes when the base gravity moment changes sign.
  2. 02

    Release the holding clamp

    Compare puzzle and balanced presets, then release each from rest.

    What to observe: Only an unbalanced beam rotates; the energy ledger closes at the terminal stop.
  3. 03

    Audit contact

    Select two supports and move a heavy load beyond them.

    What to observe: The requested reaction changes sign and physical acquisition is disabled.
  4. 04

    Infer an unknown mass

    Set a nonzero counterweight and scan its position, then add sensor noise.

    What to observe: The fitted slope estimates mass; moving the probe preserves the record.