The Geometry of Power – Advanced Simple Machines (Advanced)

How machines trade force for distance: the wheel and axle, mechanical advantage and efficiency, gear trains, tension in strings and two masses over a pulley.

Key ideas

A wheel and axle turn together; a small effort on the large wheel gives a large force at the small axle.
A machine multiplies force or speed but never energy: work out is never more than work in.
Real machines lose some work to friction, so their efficiency is always below 100%.
In a gear train the output of one gear drives the next, so one source can turn several hands at different speeds.
Tension pulls along a string, away from the body; a hanging body at rest has tension equal to its weight.
Two masses over a frictionless pulley move together with one acceleration, towards the heavier side.

Remember

Wheel and axle	MA = radius of wheel ÷ radius of axle; effort = load ÷ MA
Efficiency	η = useful output work ÷ input work × 100%
Tension	at rest T = mg; lifted T = m(g + a); lowered T = m(g − a)
Two masses over a pulley	a = (m₁ − m₂)g ÷ (m₁ + m₂); T = 2m₁m₂g ÷ (m₁ + m₂)

Watch out

Dividing the axle radius by the wheel radius: MA is bigger than 1.
Mixing cm and mm when comparing the two radii.
Dividing by m₁ alone instead of by the total mass m₁ + m₂.

Worked example

A wheel of radius 72 cm and an axle of radius 6 cm lift a 300 N load with an effort of 31.25 N. Find the efficiency of the machine.

Working: per turn: output 300 × 2π × 6, input 31.25 × 2π × 72; η = (300 × 6) ÷ (31.25 × 72) × 100

Answer: 80%