Exam practice

A winding road is a long, gentle inclined plane. For the same height, a longer road has a smaller slope and a larger VR (l ÷ h), so vehicles need less force to climb it with heavy loads, and it is safer to drive (and to brake) on a gentle slope than on a steep one.

A crane is made of several simple machines working together: its arm (jib) is a lever, it lifts loads with a system of pulleys, and the rope is wound on a drum, a wheel and axle. A machine made of two or more simple machines is a complex machine.

In every machine some of the input work is used to overcome friction between the moving parts (and, in many machines, to lift moving parts such as the movable pulleys). This work turns into heat and sound, not into useful output. So the output work is always less than the input work, and the efficiency is always less than 100%.

VR depends only on the distances the effort and the load move, so friction doesn’t change it. But friction (and the weight of moving parts) means a larger effort is needed than in a perfect machine. Since MA = load ÷ effort, a larger effort makes the MA smaller. So MA < VR; only in a perfect machine (no friction) would MA = VR.

A wheel and axle works like a lever: the center of the axis is the fulcrum, the effort acts at the rim of the wheel (radius R) and the load at the rim of the axle (radius r), like the two arms of a lever. Unlike an ordinary lever, it can turn round and round through 360°, with new points of the wheel and axle taking the place of the effort and load points all the time. So it is a lever that works continuously.

A simple machine is a simple tool that makes work easier, faster or more comfortable, using only human force. Its advantages:

• i.it can magnify force, so a small effort lifts or moves a large load (a lever, a jack);
• ii.it can change the direction of a force to a more convenient one (a fixed pulley);
• iii.it can increase speed or distance (a lattai, a bicycle’s gears);
• iv.it lets us work in safer or more comfortable positions.

A complex machine is a machine made of two or more simple machines working together to do a complicated job. Example: a bicycle. Its pedals and crank are a wheel and axle; the chain wheel and back sprocket, joined by the chain, are wheels and axles; the wheels turn on axles; the brake levers are levers; and its parts are held together by screws (nuts and bolts). Together they turn the rider’s push on the pedals into fast motion of the bicycle.

An inclined plane is a sloping surface used to raise a heavy load to a height with less effort than lifting it straight up. Examples: a plank used to load a drum into a truck, a ramp for wheelchairs, a staircase, a sloping path or winding road up a hill.

A wedge is a tool with a broad, flat end and a sharp edge, formed by two inclined planes meeting. It is a special form of the inclined plane. Importance: its sharp edge enters a material easily, and its sloping sides push the material apart with a force much larger than the effort, so we can split, cut and pierce things easily. Examples: axe (splitting wood), knife and chulesi (cutting food), chisel (shaping wood and stone), needle and nail (piercing), the blades of scissors and a plow.

A screw is an inclined plane wound in a spiral (the thread) round a cylinder, so it is a spiral inclined plane. Turning it moves it forward by one pitch per turn, with a large force. Uses: wood screws joining furniture, nuts and bolts fastening metal, a jackscrew lifting a vehicle, the lid of a bottle or jar, a vice holding work, a stool whose height is adjusted by turning, a corkscrew.

• i.Single fixed pulley: the pulley is fixed to a support and only turns. The load hangs on one side of the rope and the effort pulls down on the other. VR = 1; MA = 1 at best. It is used to change the direction of the force (flagpole, well).
• ii.Single movable pulley: one end of the rope is fixed above; the rope passes under the pulley, which carries the load and moves with it; the effort pulls the free end up. Two segments hold the load, so VR = 2 and the effort is about half the load.
• iii.Combined pulley (block and tackle): at least one fixed and one movable pulley, used together. VR = the number of rope segments holding the load; more pulleys give a larger MA. A fixed pulley at the top lets the effort pull downwards.

The diagrams are the ones drawn in Figure 8.2.

Screw (Figure 8.5): turn it once with the effort at a distance R from its axis (the handle). The effort moves 2πR; the screw moves forward one pitch, P. So VR = 2πR ÷ P. MA is found by measuring the load it moves (or the resistance it overcomes) and the effort: MA = load ÷ effort.
Wedge (Figure 8.4): when the wedge is driven fully in, the effort moves through its length (depth) H, and the material is pushed apart by the width L of its broad end. So VR = H ÷ L. MA = the resistance of the material (load) ÷ the effort applied.

VR = l ÷ h, so for the same height, a longer, gentler plane has a larger VR, and needs a smaller effort, so its MA is larger too. MA can also be increased by reducing friction: making the surface smooth, using rollers or wheels (a trolley instead of dragging), or greasing. Examples: winding roads on hills, long gentle ramps for wheelchairs, a longer plank to load a truck.

It means that 75% of the input work done on the machine by the effort is turned into useful output work on the load, and the remaining 25% is wasted, mostly in overcoming friction, as heat and sound. In other words, output work ÷ input work = 0.75, and MA ÷ VR = 0.75.

A perfect machine would have 100% efficiency: all the input work would become output work, and MA would equal VR. But every machine has moving parts that rub against each other, so there is always some friction, which turns part of the input work into heat; many machines must also lift some of their own parts. No surface can be made perfectly frictionless, so the output work is always less than the input work, and a perfect machine can’t exist.

In a wheel and axle (Figure 8.6), a wheel of radius R and an axle of radius r are fixed on the same axis. The effort rope is wound on the wheel and the load rope on the axle. In one complete turn, the effort rope unwinds by the circumference of the wheel, 2πR (the effort distance), and the load rope winds up by the circumference of the axle, 2πr (the load distance). So
VR = effort distance ÷ load distance = 2πR ÷ 2πr = R ÷ r.

• i.Sewing machine: lever (treadle and needle bar), wheel and axle (hand wheel and belt pulley), wedge (the needle’s point), screws.
• ii.Stapler: lever (the top arm, pressed down), wedge (the sharp points of the staple).
• iii.Bicycle: wheel and axle (pedals and crank, chain wheels, wheels, handlebar), lever (brake levers), screws (nuts and bolts).
• iv.Domestic flour mill: pulley and belt (from the motor), wheel and axle (shafts and grinding stone), screw (to set the gap between the stones), lever (the handle that adjusts it).

Load = weight of the box = mg = 90 × 10 = 900 N.
MA = load ÷ effort = 900 ÷ 300 = 3.
VR = l ÷ h = 20 ÷ 5 = 4.
Efficiency = MA ÷ VR × 100% = 3 ÷ 4 × 100% = 75%.

Four rope segments hold the load, so VR = 4.

• ii.From η = MA ÷ VR × 100%: 80 = MA ÷ 4 × 100, so MA = 3.2.
• i.MA = load ÷ effort, so effort = 1000 ÷ 3.2 = 312.5 N.
• iii.Output work = load × load distance = 1000 N × 6 m = 6000 J.

The effort distance is VR × load distance = 4 × 6 m = 24 m, so input work = effort × effort distance = 312.5 N × 24 m = 7500 J. (Check: 6000 ÷ 7500 = 80%.)

MA = load ÷ effort = 500 ÷ 250 = 2.
VR = R ÷ r = 20 ÷ 5 = 4.
Efficiency = MA ÷ VR × 100% = 2 ÷ 4 × 100% = 50%.

MA = load ÷ effort = 500 ÷ 300 = 1.67.
VR = length ÷ width = 12 ÷ 6 = 2.
Efficiency = MA ÷ VR × 100% = 1.67 ÷ 2 × 100% = 83.3%.