Lever Lab

What is modelled

A lever is a stiff beam that turns about a fulcrum. It balances when the torques of the load, the beam and the effort about the fulcrum add up to zero. The class depends only on the order of the three points: fulcrum between load and effort (class 1), load between fulcrum and effort (class 2), effort between fulcrum and load (class 3).

Equations

• Torque balance F (xe − xf) = m g (xl − xf) + mb g (L/2 − xf), with the effort pointing the way that opposes the load torque.
• Mechanical advantage MA = (xe − xf) / (xl − xf), the ratio of the arms.
• Fulcrum reaction R = (m + mb) g − F (signed).
• Dynamics I θ″ = Στ cos θ − c θ′ with I = m (xl − xf)2 + mb (L2/12 + (L/2 − xf)2).

Checks run in the page

• The torques about the left end of the beam also add up to zero at the balancing effort.
• The torque of 4000 slices of the beam against the weight at the centre.
• Virtual work: the work done by the effort equals the work gained by the load.
• The first instant of the beam motion against the exact angular acceleration.

Honest simplifications

The beam is rigid and uniform, the forces stay vertical as the beam tilts, the tilt is limited to about 20 degrees by stops, and the pivot friction is a simple viscous damping. Use it for the trends, not for designing a lifting tool.

• Press Class 2: wheelbarrow, then Class 3: tweezers, and compare the mechanical advantage with 1.
• Press Out of balance, then Set the balancing effort and see the beam settle.
• Move the fulcrum toward the load and watch the balancing effort fall.