Both start with 600 mL on the left and 450 mL on the right and the beam clamped by its arrest. A: clamped with the right end low. B: clamped with the left end low. Press "Start balancing" to release it.
Drag to orbit, wheel / pinch to zoom, right-drag to pan. Keyboard: arrow keys orbit when the scene has focus.
The science
Torque balance. A beam balance is in equilibrium when the torques about the pivot cancel: sum(tau) = 0, tau = r F = r m g. With equal arms a, the net torque from the pans is (m_R - m_L) g a cos(theta). Water mass is m = rho V with rho = 1000 kg/m3, so 1 mL of water adds 1 g.
Why the beam tilts only part of the way. The beam's own centre of mass sits a distance h below the pivot, giving a restoring torque M g h sin(theta). At rest: tan(theta) = (m_R - m_L) a / (M h). A mechanical stop at 12 degrees limits the swing, so a 150 g imbalance simply lies on the stop. The beam is damped, so it overshoots and settles.
Archimedes. A submerged body feels an upward force equal to the weight of displaced fluid: B = rho_w g V_sub. Hung from the beam, the block's pull on the left side is T = (rho_s - rho_w V_sub/V) V g. The water level in the dish rises by V_sub / A_dish.
Honest simplifications. Rigid massless hangers and chains, point masses at the beam ends, a single damping coefficient (zeta = 0.35), no air buoyancy, no surface tension or meniscus, water at a fixed density (no temperature dependence). "Balanced" means the torque difference at the level position is within 0.01 N m (about 3.4 g). All masses, lengths and starting volumes are model values, not measurements of a real instrument.
Two pictures, one sim. The two reference pictures show the same bench with the beam clamped in two different positions; one of them shows the beam tilting toward the lighter vessel, which physics does not allow once the beam is released. Both starting states are available above, and the simulation always releases to the physically correct side.
Check. node scripts/check.mjs re-derives the torque difference (0.44145 N m for 600 vs 450 mL), the rest angle by bisection against the closed form, a time integration, and the Archimedes relations by an independent quadrature. Virtual lab only: no real-world instructions.