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.

Initial weight moment $-11.8\,\mathrm{N\,m}$

Minimum counterweight $0.667\,\mathrm{kg}$

Equilibrium status $\text{Rotating}$

Current rotation $-1.97\,\mathrm{{}^\circ}$

Place the loads and test the balance

Can the smallest counterweight balance a loaded beam?

Gravity and support forcesOne fixed force scale per trialLeft mass

m_L

4\,\mathrm{kg}

Left position

x_L

-1.2\,\mathrm{m}

Right mass

m_R

6\,\mathrm{kg}

Right position

x_R

1\,\mathrm{m}

Counterweight mass

m_c

0\,\mathrm{kg}

Counterweight position

x_c

-1.8\,\mathrm{m}

Uniform beam mass

m_b

2\,\mathrm{kg}

Single pivot position

p

0\,\mathrm{m}

Supports: single or two

s

0

Left support

a

-1.2\,\mathrm{m}

Right support

b

1.2\,\mathrm{m}

Sensor noise bound in selected mode

\epsilon

0\,\mathrm{N\,m}

A single hinge supplies force; a separate clamp supplies the couple for a held sensor trial. Free releases remove the clamp. Two posts can push upward only. If holding the beam level would require one post to pull down, that contact opens and the trial rotates about the other post. A guide prevents slip and sideways motion about the selected axis. Ideal stops at twenty-five degrees absorb the remaining kinetic energy; rebound and later contact changes are omitted.

The four-metre beam is uniform and slender. Loads are ideal points rigidly attached to it; block dimensions and hardware are schematic. The minimum-mass challenge includes the beam weight and excludes the present counterweight. The automatic solution keeps full precision and is disabled if it exceeds the thirty-kilogram limit. Sensor observations are simulated from separate held configurations. The fitted slope estimates the counterweight mass; noise is bounded in newton-metres for the clamp or newtons for the right post. Errors repeat at the same station. Physical changes clear the observations; moving the counterweight preserves them. Releases use fixed-step RK4 at 240 steps per model second and observe at most eight seconds.

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.

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 misconceptionMatching the two masses is enough.

Better mental modelMoment 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 misconceptionThe beam stays still, so its weights must already balance.

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

A negative post reaction means lift-off

Typical misconceptionA post can exert the negative force given by the equilibrium equations.

Better mental modelThese 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

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.
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.
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.
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.
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.
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.

Core mathematical model

A resting beam needs zero force and zero moment

Gravity turns each load about the hinge

The longest opposing lever arm uses the least mass

Two supports divide the total weight

A moving hinge reaction follows the mass center

A sensor sweep measures the counterweight

Common difficulties

Equal weights need not balance

A hinge and a clamp do different jobs

A negative post reaction means lift-off

Run the experiment

Watch the heavier moment win

Move a load directly

Find the lightest balance

Make a support lift

Measure a known mass

Test the estimate