Three interconnected performance metrics describe how well a mechanical system transmits force and motion: mechanical advantage (MA), velocity ratio (VR), and efficiency (η). Together they quantify what a system gains, what it sacrifices, and how much energy is wasted.
KEY TAKEAWAY: In a perfect (frictionless) machine, MA = VR. In reality, MA < VR because friction always causes losses. Efficiency measures the gap between ideal and real performance.
Mechanical advantage is the ratio of the output (load) force to the input (effort) force:
$$MA = \frac{F_{load}}{F_{effort}}$$
Worked example (lever):
A 2nd-class lever has a 1.5 m effort arm and a 0.5 m load arm. A load of 300 N is applied.
$$MA = \frac{1.5}{0.5} = 3 \quad \Rightarrow \quad F_{effort} = \frac{300}{3} = 100 \text{ N}$$
EXAM TIP: MA can be calculated either from forces OR from arm lengths for levers and pulleys. Both approaches must give the same answer in an ideal system.
Velocity ratio (also called speed ratio or movement ratio) is the ratio of input distance (or speed) to output distance (or speed):
$$VR = \frac{d_{effort}}{d_{load}} = \frac{v_{effort}}{v_{load}}$$
VR is determined purely by the geometry of the machine — it is independent of friction.
Velocity ratios for common mechanisms:
| Mechanism | VR formula | Notes |
|---|---|---|
| Lever | $\dfrac{\text{effort arm}}{\text{load arm}}$ | Depends on fulcrum position |
| Gear train | $\dfrac{T_{driven}}{T_{driver}}$ | More teeth on driven = higher VR |
| Pulley system | Number of supporting rope segments | Count only ropes supporting movable pulley |
| Wheel and axle | $\dfrac{r_{wheel}}{r_{axle}}$ | Larger wheel = higher VR |
| Inclined plane | $\dfrac{\text{slope length}}{\text{height}}$ | Shallower slope = higher VR |
| Screw | $\dfrac{2\pi r}{\text{pitch}}$ | Finer pitch = higher VR |
Worked example (gears):
Driver gear: 15 teeth; driven gear: 45 teeth.
$$VR = \frac{45}{15} = 3$$
For every 3 rotations of the driver, the driven gear makes 1 rotation — the output is 3× slower.
VCAA FOCUS: VCAA often asks students to calculate VR before MA. Remember: VR uses geometry (distances, teeth, arm lengths); MA uses forces measured in a real system.
Efficiency is the ratio of useful output to total input, expressed as a percentage:
$$\eta = \frac{MA}{VR} \times 100\%$$
Equivalently:
$$\eta = \frac{W_{output}}{W_{input}} \times 100\% = \frac{F_{load} \times d_{load}}{F_{effort} \times d_{effort}} \times 100\%$$
Efficiency is always less than 100% in real systems due to:
- Friction at joints, between meshing teeth, in bearings
- Elastic deformation (energy stored then lost as heat)
- Vibration and noise
Worked example:
A pulley system has VR = 4. In testing, an effort of 60 N lifts a 200 N load.
$$MA = \frac{200}{60} = 3.33$$
$$\eta = \frac{3.33}{4} \times 100\% = 83.3\%$$
16.7% of input work is lost to friction.
COMMON MISTAKE: Students sometimes calculate efficiency as $MA / VR$ and forget to multiply by 100. Efficiency is a percentage — always include the % symbol and the ×100 step.
These three quantities are linked:
$$\eta = \frac{MA}{VR} \quad \Rightarrow \quad MA = \eta \times VR$$
If any two are known, the third can be found.
Problem-solving approach:
1. Calculate VR from geometry
2. Calculate MA from measured forces
3. Calculate efficiency from MA/VR
Or, if efficiency is given:
4. $MA = \eta \times VR$ → find effort force
Engineers improve mechanical system efficiency by:
- Using roller or ball bearings instead of plain bearings
- Lubricating moving parts to reduce friction
- Selecting precision-machined gears with tight tolerances
- Minimising the number of components in the drive train
- Using appropriate materials (e.g. polymer bearings in lightly-loaded systems)
APPLICATION: A worm gear drive is self-locking (the worm can drive the wheel, but not vice versa) because its VR is very high (e.g. 40:1) and efficiency is very low (often 40–60%). This is a deliberate design feature for hoists and positioning mechanisms.
| Quantity | Symbol | Formula | Depends on |
|---|---|---|---|
| Mechanical Advantage | MA | $F_{load} / F_{effort}$ | Forces (real system) |
| Velocity Ratio | VR | $d_{effort} / d_{load}$ | Geometry only |
| Efficiency | η | $(MA / VR) \times 100\%$ | Both MA and VR |
STUDY HINT: Draw a diagram for every problem. Label effort, load, and fulcrum/pivot clearly before writing any equation. Most errors come from incorrectly identifying which force or distance is input vs. output.
