Mechanical Calculations: Gears, Pulleys and Mechanical Advantage
Why Use Gears and Pulleys?
Gears and pulleys are power-transmission components — they carry force and motion from one shaft or point to another. They are also force multipliers: by choosing the right gear ratio or pulley arrangement, engineers can increase or reduce force, speed and torque to match what the application needs. A bicycle has a small front chainring and a large rear sprocket in low gear so the rider can pedal easily uphill; in high gear the ratio reverses to allow high speed on a flat road.
Gear Trains and Gear Ratio
A gear train consists of two or more gears with interlocking teeth. The driver gear is connected to the motor or input shaft. The driven gear is connected to the output shaft. When the driver turns, its teeth push the driven gear’s teeth in the opposite direction.
The gear ratio tells you how many times the driver must rotate to turn the driven gear once:
$$\text{Gear ratio} = \frac{\text{teeth on driven gear}}{\text{teeth on driver gear}}$$
Example: a driver gear has 20 teeth and a driven gear has 60 teeth. $$\text{Gear ratio} = \frac{60}{20} = 3:1$$ This means the driver must rotate 3 times for the driven gear to complete one full turn. The driven gear turns at one-third of the speed but with three times the torque. This is a speed reducer — used when high torque is needed at low speed (e.g. a winch, a robot arm).
Gear ratio = driven teeth ÷ driver teeth. A ratio greater than 1 (more teeth on driven than driver) means the output turns slower but with greater torque. A ratio less than 1 means the output turns faster but with less torque. Always state which gear is the driver in your working.
A compound gear train uses intermediate shafts that each carry two gears. This lets engineers achieve very large overall gear ratios without making gears unreasonably large. A clock uses a compound gear train to step down the motor speed to one revolution per minute at the minute hand.
A chain and sprocket drive works on the same principle as a gear train but the two sprockets can be far apart. The chain transmits force without slipping (unlike a belt drive). Bicycles, motorcycles and industrial conveyors use chain-and-sprocket drives. Gear ratio is still calculated using tooth counts: driven teeth ÷ driver teeth.
Pulleys and Velocity Ratio
A pulley changes the direction of a force or transmits rotary motion between shafts using a belt. A single fixed pulley only changes direction — you pull down on the rope and the load goes up, but you still need a force equal to the load’s weight. A block and tackle uses multiple pulleys to reduce the effort needed.
For a pulley system lifting a load, the velocity ratio (VR) equals the number of rope sections supporting the movable block: $$\text{Velocity ratio} = \text{number of rope sections supporting the load}$$ With 4 rope sections, VR = 4: the effort moves 4 metres for every 1 metre the load rises. This also means the effort force needed is approximately one-quarter of the load (ignoring friction).
Mechanical Advantage
Mechanical advantage (MA) tells you how much a mechanism multiplies the input force: $$\text{MA} = \frac{\text{load}}{\text{effort}}$$ If a pulley system requires an effort of 200 N to lift a load of 800 N: $$\text{MA} = \frac{800}{200} = 4$$ An MA of 4 means the machine multiplies the input force by 4. A higher MA means less effort is needed, but the effort must be applied over a greater distance. Real machines always have an MA slightly less than the VR because friction wastes some energy.
Mechanical advantage (MA) = load ÷ effort. An MA greater than 1 means the machine multiplies your force — it takes less effort to move a large load. Always include units: load and effort are both in newtons (N). MA itself has no units because it is a ratio.
A bearing supports a rotating shaft and reduces friction. The two main types you need to know are plain (journal) bearings and ball (rolling) bearings. A plain bearing is a sleeve of soft metal or polymer that the shaft slides against; it relies on an oil film. A ball bearing uses small steel balls running in a hardened steel race — the rolling contact dramatically reduces friction compared with sliding contact.
In the examination, “spins” alone is insufficient to describe what a ball bearing does. You must say that it reduces friction between rotating parts, allowing the shaft to turn with less energy loss and less heat generation. Ball bearings are used in electric motors, wheel hubs, gearboxes and any mechanism where a shaft must rotate at speed for extended periods.
Worked Calculation: Gear Ratio
Bramford Drives Ltd makes conveyor gearboxes for a bottling plant. The motor pinion has 18 teeth; the output gear has 72 teeth. Calculate the gear ratio.
$$\text{Gear ratio} = \frac{\text{teeth on driven}}{\text{teeth on driver}} = \frac{72}{18} = 4:1$$ The output shaft turns at one-quarter of the motor speed. If the motor runs at 1440 rpm, the conveyor shaft turns at 360 rpm with four times the torque.
Worked Calculation: Mechanical Advantage
A technician uses a block-and-tackle with 3 rope sections to lift a 600 N engine component. What effort is required (assume ideal conditions)?
Velocity ratio = 3 (three rope sections). In ideal conditions MA = VR = 3. $$\text{Effort} = \frac{\text{load}}{\text{MA}} = \frac{600}{3} = 200\text{ N}$$ In practice, friction in the pulleys means the actual effort needed would be higher than 200 N, giving a real MA slightly less than 3.