Torque & Equilibrium
Balance a beam by placing weights, then read off the support reactions.
About this simulation
Load a beam on a pivot with weights on either side; the beam tilts toward the heavier torque and settles level only when clockwise and anticlockwise torques balance. Both torques and the pivot reaction are shown, making the law of the lever — double the mass, halve the distance — tangible.
How to use it
- Set the Left weight and Left distance, then the Right weight and Right distance, and watch the beam tilt towards the larger torque.
- Read τ₁ and τ₂ at the top and the verdict between them: BALANCED, TIPS LEFT or TIPS RIGHT.
- Switch Show between Both, Torques and Forces to see the turning arcs or the weights and the pivot reaction.
- Change the Beam mass and watch R on the pivot's upward arrow change while the balance does not.
- Try the presets 'Double weight, half arm' and 'Light on a long arm'.
What to notice
- The turning effect of each weight is its torque, weight × perpendicular distance from the pivot, not the weight alone.
- The beam is balanced when w₁d₁ = w₂d₂, so doubling a weight and halving its distance gives the same torque.
- The pivot must push up with a force equal to the total weight, R = w₁ + w₂ + mg, however the weights are placed.
- The beam is supported at its centre, so its own weight has no turning effect and a heavier beam only increases the reaction.
The formulas
- τ = F × d
- Torque or moment of a force F acting at a perpendicular distance d from the pivot.
- w₁d₁ = w₂d₂
- Principle of moments: anticlockwise moment equals clockwise moment when the beam is in equilibrium.
- R = w₁ + w₂ + mg
- Reaction at the pivot, balancing both weights and the beam's own weight mg.
Questions
What is the principle of moments?
The principle of moments states that when a body is in rotational equilibrium, the sum of the anticlockwise moments about any point equals the sum of the clockwise moments about the same point. For a beam on a pivot with one weight on each side, this gives w₁d₁ = w₂d₂. It is the rule behind see-saws, beam balances and levers. In the simulation, the beam reads BALANCED when the two torques match.
What are the two conditions for equilibrium of a rigid body?
First, the net force on the body must be zero, so it has no linear acceleration. Second, the net torque about any point must be zero, so it has no angular acceleration. For the beam, the first condition fixes the pivot reaction R = w₁ + w₂ + mg, and the second gives w₁d₁ = w₂d₂. In the simulation, the Forces view shows the first and the Torques view the second.
How can a small weight balance a large weight on a see-saw?
Torque depends on distance from the pivot as well as on force. A small weight placed far from the pivot can produce the same moment as a large weight placed close to it. For example, 4 N at 1.9 m and 19 N at 0.4 m both give a moment of 7.6 N·m. In the simulation, the 'Light on a long arm' preset shows exactly this balance.
What is the moment of a force and what is its SI unit?
The moment of a force, or torque, is the turning effect of a force about a pivot. It equals the force multiplied by the perpendicular distance from the pivot to the line of action of the force. Its SI unit is the newton metre (N·m). A moment can be clockwise or anticlockwise, and the simulation shows each one as a coloured arc around the pivot.
