5.2 Work, Energy, Levers, Pulleys & Mechanical Advantage

Work, energy, levers, pulleys and mechanical advantage

Work and energy

Mechanical work occurs when a force produces displacement. For a constant force parallel to motion:

work = force × distance.

The unit is the joule, equal to one newton-metre. If force and displacement form angle θ:

work = Fd cos θ.

A person holding a stationary load may exert muscular effort, but the ideal mechanical work on the load is zero because displacement is zero.

Kinetic energy is energy of motion:

KE = 1/2 mv².

Gravitational potential energy near Earth is:

PE = mgh.

Raising a load slowly by height h requires at least mgh of ideal work, regardless of whether a ramp, lever, or pulley is used. A machine can reduce the input force by increasing the distance over which that force acts.

Power is the rate of doing work:

power = work ÷ time.

Two machines may do the same work, but the one completing it faster has greater power.

Mechanical advantage

Actual mechanical advantage is:

MA = output force ÷ input force.

Ideal mechanical advantage comes from geometry and assumes no losses. Conservation of work in an ideal machine gives:

input force × input distance = output force × output distance.

If force is multiplied by four, the effort point moves four times as far as the load. There is no free energy.

Efficiency is:

efficiency = useful output work ÷ input work × 100%.

If an ideal machine predicts 400 N output but is 80% efficient under the simplified problem model, useful output is 0.80 × 400 = 320 N.

Levers and moments

Torque or moment about a pivot is:

moment = force × perpendicular distance from pivot to line of action.

Only the perpendicular distance counts. A force applied at an angle has less turning effect than the same force perpendicular to the lever at the same point.

For balance:

clockwise moments = anticlockwise moments.

If a 600 N load acts 0.4 m from a pivot and effort is applied 1.2 m on the other side:

600 × 0.4 = effort × 1.2,

so effort = 200 N.

Lever classes

• First class: pivot between effort and load, like a seesaw.
• Second class: load between pivot and effort, like an idealised wheelbarrow.
• Third class: effort between pivot and load, like a forearm lifting a hand load.

Class alone does not give the exact advantage; distances do. A first-class lever can multiply force or speed depending on pivot placement. A second-class lever generally multiplies force. A third-class lever usually sacrifices force for speed and range of motion.

Move the pivot closer to the load or apply effort farther from the pivot to increase force advantage, assuming the geometry remains workable.

Fixed and movable pulleys

An ideal fixed pulley changes force direction but provides a mechanical advantage of 1. Pulling down can lift a load up, but the ideal effort equals the load force.

A movable pulley supported by two rope segments has ideal mechanical advantage 2. Each ideal rope segment has the same tension, so two upward tensions support the load.

For block-and-tackle diagrams, count the rope segments directly supporting the moving block. Do not count a free pulling end unless it supports the moving block, and do not count rope sections attached only to the fixed support.

If four supporting strands lift an 800 N load, the ideal effort is 800 ÷ 4 = 200 N. To lift the load 1 m, the operator pulls 4 m of rope.

Attachment-point caution

The location where the rope end is anchored can change the strand count. Trace one continuous rope from anchor through every pulley to the free end. Mark which pulleys move with the load.

Wheel and axle, ramps and screws

A wheel and axle uses different radii. Ideal force advantage is effort radius ÷ load radius when effort acts at the wheel and load at the axle. A longer handle increases torque for the same force.

An ideal ramp has mechanical advantage:

ramp length ÷ vertical rise.

A longer ramp reduces ideal force while increasing travel distance. A screw is an inclined plane wrapped around a cylinder; a finer pitch typically gives greater ideal force advantage but requires more turns.

Reasoning before arithmetic

Predict direction: longer effort arm means less effort; more supporting strands mean less ideal tension; friction means actual effort exceeds ideal effort. Then calculate. Check units: torque is newton-metres, work is joules, power is watts, and mechanical advantage has no unit.