Force Composition Calculator

Calculate the resultant of two forces using the parallelogram law

Inputs

F_1

Force

F_2

Force

Formula Explanation

Fig 1 — Parallelogram of Forces

Formula ① — Magnitude (Right Angle)

F = \sqrt{F_1^2 + F_2^2}

When the two forces are perpendicular, the resultant equals the hypotenuse of the right triangle formed by $F_1$ and $F_2$ .

Formula ② — Direction (Right Angle)

\tan\varphi = \dfrac{F_2}{F_1}

The angle $\varphi$ is measured from $F_1$ to the resultant $F$ .

Formula ③ — Magnitude (General Angle)

F = \sqrt{F_1^2 + F_2^2 + 2F_1 F_2\cos\alpha}

The law of cosines generalises the right-angle case: when $\alpha$ = 90°, cos 90° = 0 and formula ③ reduces to formula ①.

Formula ④ — Direction (General Angle)

\tan\varphi = \dfrac{F_2\sin\alpha}{F_1 + F_2\cos\alpha}

When $\alpha$ = 90°, sin 90° = 1 and formula ④ reduces to formula ②.

Key Concepts

Resultant Force

When several forces act on an object, there exists a single equivalent force that produces the same effect. This equivalent force is called the resultant.

Parallelogram Law

Two forces acting at a point can be combined by the parallelogram rule: draw both vectors from the same origin; the resultant is the diagonal of the parallelogram they form.

Special Case: α = 90°

When the two forces are mutually perpendicular, the resultant simplifies to the Pythagorean theorem.

Force Decomposition

The reverse process — splitting one force into two perpendicular components $F_x$ and $F_y$ — is called orthogonal decomposition. It is widely used in structural analysis.

Worked Example

Two forces act on a bracket: $F_1 = 30\,\text{N}$ and $F_2 = 40\,\text{N}$ , with a right angle between them. Find the magnitude and direction of their resultant.

Step 1 — Magnitude (apply formula ①)

F = \sqrt{30^2 + 40^2} = \sqrt{900+1600} = \sqrt{2500} = 50\text{ (N)}

Step 2 — Direction (apply formula ②)

\tan\varphi = \dfrac{40}{30} = 1.33 \quad\Rightarrow\quad \varphi \approx 53.1°

The resultant has a magnitude of $50\,\text{N}$ and acts at $53.1°$ from the 30 N force.

Force Composition Calculator

Calculate the resultant of two forces using the parallelogram law

Inputs

F_1

Force

F_2

Force

Formula Explanation

Fig 1 — Parallelogram of Forces

Formula ① — Magnitude (Right Angle)

F = \sqrt{F_1^2 + F_2^2}

When the two forces are perpendicular, the resultant equals the hypotenuse of the right triangle formed by $F_1$ and $F_2$ .

Formula ② — Direction (Right Angle)

\tan\varphi = \dfrac{F_2}{F_1}

The angle $\varphi$ is measured from $F_1$ to the resultant $F$ .

Formula ③ — Magnitude (General Angle)

F = \sqrt{F_1^2 + F_2^2 + 2F_1 F_2\cos\alpha}

The law of cosines generalises the right-angle case: when $\alpha$ = 90°, cos 90° = 0 and formula ③ reduces to formula ①.

Formula ④ — Direction (General Angle)

\tan\varphi = \dfrac{F_2\sin\alpha}{F_1 + F_2\cos\alpha}

When $\alpha$ = 90°, sin 90° = 1 and formula ④ reduces to formula ②.

Key Concepts

Resultant Force

When several forces act on an object, there exists a single equivalent force that produces the same effect. This equivalent force is called the resultant.

Parallelogram Law

Two forces acting at a point can be combined by the parallelogram rule: draw both vectors from the same origin; the resultant is the diagonal of the parallelogram they form.

Special Case: α = 90°

When the two forces are mutually perpendicular, the resultant simplifies to the Pythagorean theorem.

Force Decomposition

The reverse process — splitting one force into two perpendicular components $F_x$ and $F_y$ — is called orthogonal decomposition. It is widely used in structural analysis.

Worked Example

Two forces act on a bracket: $F_1 = 30\,\text{N}$ and $F_2 = 40\,\text{N}$ , with a right angle between them. Find the magnitude and direction of their resultant.

Step 1 — Magnitude (apply formula ①)

F = \sqrt{30^2 + 40^2} = \sqrt{900+1600} = \sqrt{2500} = 50\text{ (N)}

Step 2 — Direction (apply formula ②)

\tan\varphi = \dfrac{40}{30} = 1.33 \quad\Rightarrow\quad \varphi \approx 53.1°

The resultant has a magnitude of $50\,\text{N}$ and acts at $53.1°$ from the 30 N force.

Force Composition Calculator

Inputs

Formula Explanation

Key Concepts

Worked Example

Further Reading

Force Composition Calculator

Inputs

Formula Explanation

Key Concepts

Worked Example

Further Reading