Mechanical Calculations: Levers, Gears and Pulleys

Why Calculations Matter in Mechanisms

Mechanical calculations allow engineers to predict exactly how much force a lever can generate, how fast a driven gear will spin, or how many times a pulley system multiplies an effort. In the exam, calculation questions require you to state the formula, substitute values and give a numerical answer with correct units — a formula on its own does not earn the answer mark. This lesson works through each calculation type fully before you practise, following the same method every time.

Mechanical Advantage

Mechanical advantage (MA) tells us how much a mechanism multiplies a force. A mechanical advantage greater than 1 means the mechanism amplifies the effort; less than 1 means the mechanism sacrifices force to gain speed or range.

The formula is:

$$\text{MA} = \frac{\text{load}}{\text{effort}}$$

For levers, MA can also be calculated from distances if force values are not given:

$$\text{MA} = \frac{\text{distance from fulcrum to effort}}{\text{distance from fulcrum to load}}$$

A first-order lever has its fulcrum 0.2 m from the load and 0.8 m from the effort. Calculate the mechanical advantage.

Step 1 — Write the formula:

$$\text{MA} = \frac{\text{distance from fulcrum to effort}}{\text{distance from fulcrum to load}}$$

Step 2 — Substitute values:

$$\text{MA} = \frac{0.8}{0.2}$$

Step 3 — Calculate and state answer:

$$\text{MA} = 4$$

The lever has a mechanical advantage of 4. This means an effort of 5 N would lift a load of 20 N. Note: MA has no units — it is a pure ratio.

The Principle of Moments

The principle of moments states that for a lever in equilibrium (balanced), the total clockwise moment equals the total anticlockwise moment. A moment is the turning effect of a force about a pivot.

$$\text{Moment} = \text{Force} \times \text{perpendicular distance from pivot}$$

Units: Newton-metres (Nm). Force is in newtons (N) and distance is in metres (m).

A uniform lever is balanced on a fulcrum. A 60 N load sits 0.3 m from the fulcrum on the left. What effort must be applied 0.9 m from the fulcrum on the right to keep the lever balanced?

Step 1 — Write the principle of moments:

$$\text{Clockwise moment} = \text{Anticlockwise moment}$$

Step 2 — Substitute:

$$\text{Effort} \times 0.9 = 60 \times 0.3$$

Step 3 — Solve:

$$\text{Effort} = \frac{60 \times 0.3}{0.9} = \frac{18}{0.9} = 20 \text{ N}$$

An effort of 20 N is required. The lever gives a mechanical advantage of 3 (60 ÷ 20), confirming the longer effort arm tripled the force.

Key Fact

Moment (Nm) = Force (N) × perpendicular distance from pivot (m). The longer the effort arm, the smaller the effort needed to balance a given load — this is why a long spanner undoes a tight bolt more easily than a short one.

Gear Ratio

The gear ratio compares the number of teeth on the driven gear to those on the driver gear:

$$\text{Gear ratio} = \frac{\text{teeth on driven gear}}{\text{teeth on driver gear}}$$

A driver gear has 12 teeth. The driven gear has 48 teeth. The driver turns at 200 rpm. Calculate the gear ratio and the output speed of the driven gear.

Step 1 — Gear ratio:

$$\text{Gear ratio} = \frac{48}{12} = 4:1$$

Step 2 — Output speed:

$$\text{Output speed} = \frac{\text{Input speed}}{\text{Gear ratio}} = \frac{200}{4} = 50 \text{ rpm}$$

The driven gear turns at 50 rpm — four times slower than the driver, but with four times the torque. A 4:1 ratio is described as a step-down (reduction) gear. Always include units (rpm) in your final answer.

Velocity Ratio of Pulley Systems

In a belt-and-pulley system, the velocity ratio (VR) compares pulley diameters. Because both pulleys share the same belt, the belt travels at the same linear speed at both pulleys:

$$\text{VR} = \frac{\text{diameter of driven pulley}}{\text{diameter of driver pulley}}$$

The output speed of the driven pulley is:

$$\text{Output speed} = \frac{\text{Input speed}}{\text{VR}}$$

A motor drives a 50 mm diameter pulley (driver) connected by a belt to a 200 mm diameter pulley (driven). The motor runs at 1 200 rpm. Calculate the velocity ratio and output speed.

Step 1 — Velocity ratio:

$$\text{VR} = \frac{200}{50} = 4$$

Step 2 — Output speed:

$$\text{Output speed} = \frac{1200}{4} = 300 \text{ rpm}$$

The driven pulley turns at 300 rpm. As with gears, a larger driven pulley gives a slower but more powerful (higher torque) output. Remember: for this to work, the belt must not slip — friction between belt and pulley is essential.