The double-link balanced scale shown by Hiscox is a weighing mechanism. Two equal parallel links guide the pan, while a counterweight and radial index plate rotate with the upper link. Increasing the load lowers the pan and changes the equilibrium angle.
Double Link Balanced Scale Interactive Calculator
Model the weighing scale shown by Hiscox: parallel links keep the pan level while a pendulum counterweight and index sector rotate with the upper link. Adjust load, counterweight and dimensions to calculate ideal static equilibrium.
Equation Used
- Equal links and equal vertical pivot spacing keep the pan level.
- Counterweight radius and mass define a concentrated gravitational load.
- Pan, links and sector self-weight are neglected.
- The 135° arm offset is illustrative, not a measured historical dimension.
- No friction, stop, collision, damping or capacity model is included.
Ideal weighing-scale model. The counterweight offset is an explicit illustrative choice; clearances, stops and structural capacity are not calculated.
A level pan on two parallel links
The two frame pivots have the same vertical separation as the two pivots on the pan carrier. Equal connecting links form a parallelogram, keeping the carrier upright and its pan horizontal through the motion.
The upper link is rigidly joined to the radial index sector. A counterweight moves with that assembly. Its changing horizontal distance from the fixed pivot creates a balancing moment against the pan load. A fixed pointer reads the moving sector.
The example places the counterweight arm 135° clockwise from the upper link. This illustrative angle sets the balance equation; Hiscox does not supply a dimension for it. The animation applies and removes load through a succession of static equilibria.
A load-position model
The calculator estimates the equilibrium angle, pan drop and opposing moments for the specified load, link length, counterweight and radius. It also reports the load that makes the links horizontal for this geometry.
The pan load is balanced by the counterweight. Their opposing gravitational moments determine the link angle and pan position.
Balance of gravitational moments
Let m be the pan load, M the counterweight mass, L the link length, R the counterweight radius, and θ the upper-link angle above horizontal. For the chosen counterweight offset α=−135°, moment balance is mL cosθ + MR cos(θ+α)=0.
The stable equilibrium is θ=atan(1−√2mL/(MR)). The pan is unloaded at θ=45°. Its drop from that position is L(sin45°−sinθ).
Each opposing moment has magnitude mgL cosθ, with L expressed in metres for N·m. The mass giving horizontal links is MR/(√2L). This is a geometric calibration point, not a rated maximum load.
Links, pan and index sector are treated as weightless. The counterweight is concentrated at the entered radius. Pin friction, travel stops, clearances and dynamic response are not included.
A sample equilibrium
With a 2 kg pan load, 150 mm links, a 2 kg counterweight and 250 mm counterweight radius, the calculated links lie about 8.6° above horizontal. The pan drops about 83.6 mm from the unloaded 45° position.
The load and counterweight then produce equal opposing moments of about 2.91 N·m. A pan load of about 2.36 kg makes these links horizontal.
Changing either mass, the link length or the counterweight radius changes the equilibrium. The displayed animation keeps the upper and lower links equal and parallel at every pose.
Interpreting the result
The original source is a small mechanism diagram and supplies no dimensions, certified accuracy or load rating. This page explains its kinematic layout using an explicit ideal counterbalance model.
The model does not confirm that a constructed scale has enough travel or clearance for every entered combination. It does not model mass of the pan, links or sector, friction, damping or mechanical stops. Those are needed to design or calibrate a real instrument.
The index marks show an illustrative angular sector. The numeric readout comes from the entered dimensions and masses; no universal linear mass scale is implied.
Double-link scale questions
Why does the pan remain level?
The parallel links and equal pivot spacing form a parallelogram that keeps the pan carrier upright.
What determines the pan position?
The pan moves until the load and counterweight produce equal opposing gravitational moments in the ideal model.
Is the horizontal-link load its capacity?
No. It is just the calculated mass at which the links are horizontal for the chosen counterweight geometry.
Does the animation show real settling time?
No. It shows successive static equilibria as load is applied and removed. Dynamic motion is not calculated.
What does zero load show?
The chosen geometry rests at a 45° link angle with its counterweight directly below the upper pivot.
References
Hiscox, Mechanical Movements, Powers, Devices and Appliances, mechanism 161, printed page 52 (PDF page 62), shows the double-link balanced scale and states that its upper link is fixed to the radial index plate.
OpenStax, University Physics Volume 1, section 12.1, explains the force and moment equilibrium used in the ideal model.
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