Force Diagrams and Resolution

Overview

Force Diagrams and Resolution is one of the most important problem-solving skills in mechanics. Many errors in Forces, Dynamics, and circular motion arise not from algebra, but from drawing the forces incorrectly.

This page focuses on:

• choosing the system correctly
• identifying external forces
• drawing accurate free-body diagrams
• resolving forces into components
• 1D and 2D sign conventions
• resultant force methods
• worked examples
• exam pitfalls

Why It Matters

Good mechanics work usually starts with the correct diagram, not the correct equation.

Definition

A free-body diagram isolates one chosen body and shows all the external forces acting on it. Resolution replaces angled vectors by perpendicular components.

Key Representations

$$\sum \vec{F}=m\vec{a}$$ $$\sum F_x=m{a}_x$$ $$\sum F_y=m{a}_y$$ $$F_x=F\cos \theta ,F_y=F\sin \theta$$

Why Free-Body Diagrams Matter

A free-body diagram (FBD) isolates one object and shows all external forces acting on it.

It helps you apply:

$$\sum \vec{F}=m\vec{a}$$

without confusion.

Typical uses:

• blocks on rough surfaces
• inclined planes
• hanging masses
• pulley systems
• accelerating lifts
• equilibrium problems

Choosing the System

Before drawing forces, decide:

Which object am I analysing?

Examples:

• the block only
• the hanging mass only
• the car only
• the combined two-block system

Different choices give different valid equations.

External vs Internal Forces

External Forces

Forces acting on the chosen system from outside.

Examples:

• weight
• normal reaction
• tension from rope outside system
• friction from ground

Internal Forces

Forces between parts within the chosen system.

If analysing a combined system, internal forces are not shown.

Example:

Two blocks tied together. If both blocks are chosen as one system, tension between them is internal.

Common Forces to Identify

Weight

$$\vec{W}=m\vec{g}$$

Always vertically downward.

Normal Contact Force

$N$

Perpendicular to contact surface.

Tension

$T$

Along string, away from object.

Friction

$f$

Parallel to surface, opposing relative motion or attempted motion.

Applied Force

Push / pull by external agent.

Drag / Resistive Force

Opposes motion through fluid.

Free-Body Diagram Rules

Good Method

1. Draw object as box / dot / block.
2. Remove surroundings.
3. Draw all external forces from object.
4. Label each force.
5. Choose axes.

Avoid

• drawing velocity as a force
• drawing acceleration as a force
• including reaction pair acting on other object
• forgetting weight
• wrong direction for friction

Example Shapes

Block on Horizontal Floor

For stationary block:

• $W$ downward
• $N$ upward

If pushed right:

• push right
• friction left

Hanging Mass

• tension upward
• weight downward

Block on Inclined Plane

• weight vertically downward
• normal perpendicular to plane
• friction along plane if present

Axes Choice

Choose axes that simplify equations.

Horizontal Surface

Use:

• x horizontal
• y vertical

Inclined Plane

Usually use:

• x parallel to slope
• y perpendicular to slope

This avoids unnecessary trig.

1D Signed Convention

Force is a vector, but in one dimension we often use signed scalar components.

Choose rightward as positive:

• $+T$
• $-f$

Then:

$$\sum F=ma$$

is the component form of:

$$\sum \vec{F}=m\vec{a}$$

Negative result means opposite to chosen positive direction.

Resolving a Force into Components

Suppose force $F$ makes angle $\theta$ above horizontal.

Horizontal component:

$$F_x=F\cos \theta$$

Vertical component:

$$F_y=F\sin \theta$$

Then:

$$\vec{F}=\overrightarrow{F_x}+\overrightarrow{F_y}$$

Always check where the angle is measured from.

Inclined Plane Weight Components

For slope angle $\theta$:

Weight remains vertical downward.

Resolve into:

• parallel to plane:

$$mg\sin \theta$$

• perpendicular to plane:

$$mg\cos \theta$$

Hence:

$$N=mg\cos \theta$$

if no other perpendicular forces.

Resultant Force

The resultant force is:

$${\vec{F}}_{\text{net}}=\sum \vec{F}$$

In Components

$$\sum F_x=m{a}_x$$ $$\sum F_y=m{a}_y$$

Use separate equations for perpendicular directions.

Vector Triangle Method (Equilibrium)

If three forces keep a body in equilibrium:

$$\sum \vec{F}=0$$

They may be represented by a closed triangle drawn head-to-tail.

Useful in tension and statics problems.

Worked Examples

Example 1: Horizontal Pull

A $4.0\text{kg}$ block is pulled right by $12\text{N}$ on smooth floor.

Choose right positive.

$$\sum F=ma$$ $$12=4a$$ $$a=3.0{\text{m s}}^{-2}$$

Example 2: With Friction

Same block with friction $5\text{N}$ left.

$$\sum F=12-5=7$$ $$7=4a$$ $$a=1.75{\text{m s}}^{-2}$$

Example 3: Hanging Mass

Mass $2.0\text{kg}$ hangs at rest.

Then:

$$T=mg=19.6\text{N}$$

Example 4: Inclined Plane

A block rests on smooth $30^{\circ }$ slope.

Parallel component of weight:

$$mg\sin 30^{\circ }=0.5mg$$

Perpendicular component:

$$mg\cos 30^{\circ }=0.866mg$$

Common Exam Pitfalls

1. Drawing weight perpendicular to slope

Wrong. Weight is always vertical downward.

2. Friction always opposite motion

More precisely, friction opposes relative motion or tendency.

3. Wrong trig choice

Check whether angle is with horizontal, vertical, or slope.

4. Using one equation only in 2D

Use separate component equations.

5. Including action-reaction pair on same diagram

They act on different bodies.

6. Forgetting sign convention

State positive direction first.

7. Drawing acceleration as force

Acceleration is not a force.

Summary

Core Idea

Use a clean diagram first, then equations.

Process

1. choose object
2. identify external forces
3. choose axes
4. resolve angled forces
5. apply:

$$\sum \vec{F}=m\vec{a}$$

or component forms.

Most Important Habit

A correct free-body diagram often solves half the question.

Related Links

• Forces
• Dynamics
• Kinematics
• Vectors
• Fluid Forces and Resistive Motion
• Equilibrium, Moments, and Couples

Links

• Forces
• Dynamics
• Kinematics
• Vectors