Module 1/5 · Weeks 1–3 · 30 h

Mechanics and the four forces

DRT 112 Applied Physics for Aircraft Technology

About 80 minDraft, awaiting reviewLast updated 26 September 2026

By the end of this module you will be able to

Explain Newton's three laws of motion with examples from drone flight
Draw free-body diagrams of a multirotor and a fixed-wing aircraft
Calculate net force, acceleration and thrust-to-weight ratio
Resolve tilted thrust into components and find horizontal acceleration
Use force and moment balance to analyse an off-centre payload

Prerequisites: DRT 111 Modules 1–3 (units, trigonometry and vectors)

Why a drone can hover

A hovering drone looks as if nothing is happening, but several forces act at once and cancel exactly. Change any one slightly and the drone accelerates, drops or tilts at once. Understanding forces is therefore the basis of everything in this programme, from choosing motors and mounting payloads to tuning the control system.

This module uses the vectors from DRT 111 to describe the forces on an aircraft, starting with Newton’s laws.

Newton’s laws of motion

First law (inertia). An object stays at rest or moves at constant velocity if the net force on it is zero. A hovering drone and a drone flying straight at constant speed both have zero net force.

Second law (force and acceleration). Net force equals mass times acceleration:

\sum F = m a

Third law (action and reaction). When one object pushes another, the second pushes back with an equal and opposite force. A propeller pushes air down, so the air pushes the propeller up. That is where thrust comes from.

Mass and weight

Mass ( $m$ , in kg) is the amount of matter and does not change with location. Weight ( $W$ , in N) is the force of gravity on that mass:

W = m g, g \approx 9.81 m/s^{2}

We often say “the drone weighs 2 kilograms”, meaning its mass. Its actual weight is $2 \times 9.81 = 19.62$ N. Motor makers usually quote thrust in “grams” (gram-force); convert to newtons by multiplying by $9.81 \times 1 0^{- 3}$ . For example, 1000 gf is about 9.81 N.

Free-body diagrams

A free-body diagram isolates an object and shows only the external forces acting on it, with their directions. It is the first step in every mechanics problem.

Side view of a multirotor with thrust T1 and T2 pointing up at each rotor and weight W pointing down at the centre of gravity — Figure 1. Free-body diagram of a hovering multirotor

A fixed-wing aircraft has four main forces: lift from the wing, weight, thrust from the propeller or engine, and drag from the air.

Fixed-wing aircraft with lift pointing up, weight down, thrust forward and drag backward — Figure 2. The four forces on a fixed-wing aircraft

In level flight at constant speed the net force is zero, so $L = W$ and $T = D$ . A multirotor is different: its rotors provide both lift and thrust at the same time.

Net force and acceleration

Example 1. Climbing

A 2 kg drone’s rotors produce 25 N of total upward thrust. What is its acceleration?

Weight $W = 2 \times 9.81 = 19.62$ N
Net upward force $\sum F = 25 - 19.62 = 5.38$ N
Acceleration $a = \frac{5.38}{2} = 2.69$ m/s² upward

Thrust-to-weight ratio

The thrust-to-weight ratio (TWR) is maximum thrust divided by weight. At TWR = 1 the drone can just hover, with nothing left for climbing, fighting wind or correcting attitude. In practice, general-purpose multirotors are designed for a TWR of about 2 or more, hovering at around half throttle; racing drones can reach 5–10.

Example 2. Sizing motors

A 1.5 kg quadcopter needs TWR = 2. What maximum thrust must each motor give?

Total thrust needed $2 \times 1.5 \times 9.81 = 29.43$ N
Per motor $29.43 \div 4 \approx 7.36$ N, about 750 gf

Tilting to move

A multirotor has no forward-facing propeller. It moves horizontally by tilting, which tilts its thrust. The thrust then has two components.

Multirotor tilted at angle theta; thrust T tilts with it and splits into T cos theta vertical and T sin theta horizontal — Figure 3. Tilted thrust splits into vertical and horizontal components

The vertical component $T cos θ$ supports the weight.
The horizontal component $T sin θ$ accelerates the drone forward.

To hold height, the vertical forces must balance, $T cos θ = W$ , so

T = \frac{W}{cos θ}, a_{x} = \frac{T sin θ}{m} = g tan θ

Example 3. Tilting 20 degrees

Thrust needed $T = W / cos 2 0^{\circ} = 1.064 W$ , 6.4% more than in hover
Horizontal acceleration $a_{x} = 9.81 \times tan 2 0^{\circ} \approx 3.57$ m/s²

This is why a drone uses more energy flying fast or into wind: it must tilt further and raise total thrust.

Moments and equilibrium

A force acting away from a pivot can make an object rotate. This turning effect is a moment (torque), equal to force times perpendicular distance from the pivot:

τ = F d

An object is in equilibrium when both net force and net moment are zero, $\sum F = 0$ and $\sum τ = 0$ . For a drone, the key pivot is the centre of gravity (CG).

Example 4. Camera mounted forward

A 0.3 kg camera is mounted 0.10 m ahead of the original CG. The front and rear motors are 0.20 m from the centre.

Nose-down moment from the camera $τ = 0.3 \times 9.81 \times 0.10 \approx 0.294$ N·m
To balance it, the front motors must give $0.294 \div 0.20 \approx 1.47$ N more total thrust than the rear

The front motors work harder, run hotter and drain the battery faster. Keeping the CG at the centre of the frame matters a great deal; DRT 332 returns to this in detail.

Momentum and impulse

Momentum is mass times velocity, $p = m v$ . When velocity changes in a short time, large forces arise, as described by impulse:

F Δ t = Δ p = m Δ v

Example 5. A hard landing

A 2 kg drone hits the ground at 3 m/s and stops in 0.05 s. The average force is $F = \frac{2 \times 3}{0.05} = 120$ N, about six times its weight. Flexible landing gear lengthens $Δ t$ and so reduces the force.

Module lab: measuring thrust with a scale

In class

Mount a motor and propeller upside down on a test stand placed on a digital scale, so the propeller blows air upward.
Set throttle to 25%, 50% and 75%; record the scale reading (gf) and current (A).
Convert to newtons and work out what mass of drone would have TWR = 2 with four of these motors.

Safety. Remove propellers whenever the battery is connected for setup. Always stand to the side of the propeller plane, wear eye protection and keep a fire-safe container for batteries.

Common mistakes

Watch out

Confusing mass and weight. Multiply by $g$ before comparing with thrust in newtons.
Forgetting gf units. Makers quote thrust in grams; convert before calculating.
Thinking hover means no forces. Forces are present but cancel exactly.
Assuming thrust stays the same when tilting. Thrust must rise to hold height.

Summary

Newton’s three laws explain why a drone hovers, accelerates and produces thrust.
Weight $W = m g$ is in newtons; makers quote thrust in gf.
Free-body diagrams come first; a fixed-wing aircraft has four main forces.
A TWR of about 2 or more gives control margin.
Tilting to move requires thrust $W / cos θ$ and gives acceleration $g tan θ$ .
Moment $τ = F d$ ; equilibrium needs $\sum F = 0$ and $\sum τ = 0$ , and an off-centre CG makes some motors work harder.

Check your understanding

What is the weight of a 3 kg drone in newtons?
A 1.2 kg drone has 15 N of total thrust. What is its vertical acceleration?
A 2.5 kg drone needs TWR = 2 with six motors. What thrust in gf must each give?
Tilting at $3 0^{\circ}$ while holding height, by what percentage must thrust rise, and what is the horizontal acceleration?
A 0.5 kg battery is mounted 0.04 m behind the CG. What moment does it create?

Answers

$3 \times 9.81 = 29.43$ N
$(15 - 11.772) /1.2 \approx 2.69$ m/s² upward
$2 \times 2.5 = 5$ kgf total, divided by 6 gives about 833 gf each
$1/ cos 3 0^{\circ} = 1.155$ , about 15.5% more, and $a_{x} = 9.81 tan 3 0^{\circ} \approx 5.66$ m/s²
$0.5 \times 9.81 \times 0.04 \approx 0.196$ N·m (nose up)

Key formulas

Newton's second law	$\sum F = m a$
Weight	$W = m g, g \approx 9.81 m/s^{2}$
Thrust-to-weight ratio	$TWR = \frac{T _{m a x}}{W}$
Thrust when tilted at constant height	$T = \frac{W}{cos θ}, a_{x} = g tan θ$
Moment	$τ = F d$
Equilibrium	$\sum F = 0, \sum τ = 0$
Impulse	$F Δ t = Δ p = m Δ v$

Key references

Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers (10th ed.). Cengage.
NASA Glenn Research Center. (2022). Beginner's guide to aeronautics. link
Quan, Q. (2017). Introduction to multicopter design and control. Springer.
Anderson, J. D. (2016). Introduction to flight (8th ed.). McGraw-Hill.

In class / field

Lab or field practice from worksheets with a safety checklist

Learning evidence: Checked worksheets and quiz results

Module quiz

This is a formative self-check, not a graded exam

Knowledge domain: Mathematics, physics and statistics · Aircraft, structures and design · Mission planning, flight and simulation

Mechanics and the four forces

By the end of this module you will be able to

Why a drone can hover