Force, Torque, Speed, and Power in Mechanical Systems

Overview

Four fundamental physical quantities describe the behaviour of mechanical systems: force, torque, speed, and power. Mastery of their definitions, units, interrelationships, and calculations is essential for analysing and designing mechanical systems in VCE Systems Engineering.

KEY TAKEAWAY: Force is the push/pull on an object; torque is the rotational equivalent of force; speed describes how fast motion occurs; power is the rate at which work (energy) is transferred.

Force

Force (\(F\)) is a push or pull that causes, or tends to cause, a change in motion.

In static mechanical systems (no acceleration), the forces are in equilibrium:
\$\(\sum F = 0\)\$

Worked example:
A 20 kg object rests on a surface. What force must a lever exert to lift it?
\$\(F_{load} = mg = 20 \times 9.8 = 196 \text{ N}\)\$

EXAM TIP: Always use \(g = 9.8\) m/s² (or 10 m/s² if the question specifies) when converting mass to weight force.

Torque

Torque (\(\tau\)) is the rotational equivalent of force — it is the turning effect of a force about a pivot point.

where:
- \(\tau\) = torque (N·m)
- \(F\) = force applied (N)
- \(d\) = perpendicular distance from the pivot to the line of action of the force (m)

Principle of moments (equilibrium):
\$\(\tau_{clockwise} = \tau_{anticlockwise}\)\$

Worked example:
A spanner applies 25 N at 0.3 m from the bolt centre.
\$\(\tau = 25 \times 0.3 = 7.5 \text{ N·m}\)\$

If a second force of 15 N acts on the other side at distance \(d\):
\(\$15 \times d = 7.5 \quad \Rightarrow \quad d = 0.5 \text{ m}\)\$

VCAA FOCUS: Torque in gear systems: when a gear reduces speed, it multiplies torque proportionally (and vice versa). This is directly related to the gear ratio.

Torque in gear trains:
If gear ratio \(GR = T_{driven}/T_{driver}\):
\$\(\tau_{driven} = \tau_{driver} \times GR \quad (\text{ignoring losses})\)\$

A 3:1 reduction gear triples the output torque.

Speed

In mechanical systems, two types of speed are used:

Linear speed (\(v\)): Distance per unit time
\$\(v = \frac{d}{t} \quad \text{(m/s or m/min)}\)\$

Rotational speed (\(N\)): Revolutions per minute (rpm) or radians per second

Relationship between linear and rotational speed:
\$\(v = \omega r = \frac{2\pi N r}{60}\)\$

where \(r\) = radius (m), \(N\) = speed in rpm, \(\omega\) = angular velocity (rad/s).

Worked example:
A pulley of diameter 0.2 m rotates at 300 rpm. Find the belt speed.
\$\(r = 0.1 \text{ m}, \quad v = \frac{2\pi \times 300 \times 0.1}{60} = \frac{60\pi}{60} = \pi \approx 3.14 \text{ m/s}\)\$

Speed in gear trains:
\$\(\frac{N_{driver}}{N_{driven}} = \frac{T_{driven}}{T_{driver}}\)\$

A step-up gear train increases output speed; a step-down increases output torque.

REMEMBER: Speed and torque trade off against each other in a gear train. The product (power) is approximately constant (minus losses).

Power

Power (\(P\)) is the rate at which work is done or energy is transferred:

For rotational systems:
\$\(P = \tau \times \omega = \frac{2\pi N \tau}{60}\)\$

Worked example:
A motor produces 8 N·m of torque at 1500 rpm. Find the power output.
\$\(\omega = \frac{2\pi \times 1500}{60} = 50\pi \approx 157 \text{ rad/s}\)\$
\$\(P = 8 \times 157 = 1257 \text{ W} \approx 1.26 \text{ kW}\)\$

COMMON MISTAKE: Forgetting to convert rpm to rad/s before using \(P = \tau\omega\). Always convert: \(\omega = 2\pi N/60\).

Interrelationships Summary

These relationships mean that when a gear train reduces speed (smaller \(\omega\)), torque (\(\tau\)) increases proportionally to maintain constant power (minus losses).

Application: Gear Train Design

A machine requires 50 N·m torque at 200 rpm. The motor provides 10 N·m at 1000 rpm.

Required gear ratio:
\$\(GR = \frac{1000}{200} = 5:1 \text{ reduction}\)\$

Torque check (ideal):
\$\(\tau_{out} = 10 \times 5 = 50 \text{ N·m} \checkmark\)\$

Power check:
\$\(P_{motor} = \frac{2\pi \times 1000 \times 10}{60} = 1047 \text{ W}\)\$
\$\(P_{output} = \frac{2\pi \times 200 \times 50}{60} = 1047 \text{ W} \checkmark \text{ (ideal)}\)\$

APPLICATION: When selecting a motor for a system, calculate the required torque and speed at the output, then work backwards through the drive train using gear ratios to determine what the motor must provide.

Units Quick Reference