Principle of moments and centre of gravity

An ideal lever is a light rod pivoted at the fulcrum, like a see-saw or the beam of a balance. With load F₁ at distance d₁ and effort F₂ at d₂ from the fulcrum, the reaction there is R = F₁ + F₂.

Principle of moments: d₁F₁ = d₂F₂, load arm × load = effort arm × effort. Anticlockwise moments are usually taken positive. It also holds when the parallel forces are not perpendicular to the lever.

Mechanical advantage M.A. = F₁/F₂ = d₂/d₁. When the effort arm is longer than the load arm, M.A. > 1 and a small effort lifts a large load.

The centre of gravity is the point about which the total gravitational torque on the body is zero; a cardboard balances horizontally on a pencil tip placed there.

In uniform gravity Σmᵢrᵢ × g = 0 gives Σmᵢrᵢ = 0, so the centre of gravity coincides with the centre of mass. For a body so large that g varies across it, they differ; the centre of mass depends only on how the mass is distributed.

A plate hung freely from a point settles with its centre of gravity on the vertical through that point. Two or three such verticals from different points cross at the centre of gravity.

A 70 cm, 4.00 kg uniform bar on knife-edges 10 cm from each end carries 6.00 kg at 30 cm from one end. With g = 9.8 m/s², forces give R₁ + R₂ = 98.00 N and moments about the centre give R₁ − R₂ = 11.76 N: R₁ = 54.88 N and R₂ = 43.12 N, about 55 N and 43 N.

A 3 m, 20 kg ladder rests against a frictionless wall with its foot 1 m out. Moments about the foot give the wall's push F₁ = W/(4√2) = 34.6 N; the floor supplies N = 196.0 N up and friction 34.6 N, a resultant of 199.0 N at about 80° to the horizontal.