Resultant Laboratory | Engineering Mechanics: Statics

Force calculations, step by step.

Resultant Laboratory · Engineering Mechanics: Statics

by Shihab Ibrahim (Shahab Sadeghpour)

Choose a force system, enter your known values, and follow each stage of the solution in Resultant Laboratory.

Explore the parallelogram law, resolve forces along Cartesian or oblique axes, and find coplanar or parallel force resultants. Follow the vector diagrams, equations, and explanations to understand how each answer is obtained.

Use the Statics activities to explore a concept, then work through its calculation here.

Select a Topic

Choose a statics method to begin calculation

1 Parallelogram Law

Find Resultant

Determine magnitude and angle using Law of Cosines and Sines

Open →

Resolve Resultant

Decompose resultant into two designated non-orthogonal axes

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2 Coplanar Forces

Find Resultant

Sum rectangular components (ΣFx, ΣFy) of multiple 2D forces

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Resolve Force

Project forces along Cartesian X-Y axes

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3 Parallel Forces

Find Resultant

Determine resultant magnitude and line of action position

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1. Choose 4 Given Parameters

4 / 4 selected

Force Magnitudes: F₁ (Force 1) F₂ (Force 2) R (Resultant)

Angle Directions: θ₁ (Angle of F₁) θ₂ (Angle of F₂) θ_R (Angle of R)

2. Enter Values

F₁:

kN

F₂:

kN

θ₁:

°

from

θ₂:

°

from

Force Vectors Plot

F₁ F₂ R

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Given Forces Diagram

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HINT 1: Tail-to-Tail Intersection

Before solving the question, note that in the given problem above, the two forces intersect at their tails. If you encounter a case where the head and tail intersect, use the principle of transmissibility (line of action property) of one of the forces to move it along its line of action, ensuring that the tails intersect at a common point.

Parallelogram Construction

Scale: 1.0×

Now, we're going to draw a parallelogram. A parallelogram is a four-sided shape where opposite sides are equal in length and parallel to each other.

Step 1

Draw a line parallel to one of the forces, starting from the head of the other force. The length of this line should be equal to the magnitude of the force to which it is parallel.

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Step 1 of 3

Trigonometric Vector Triangle

Step 4: Now, consider one of the two triangles created by the resultant force dividing the parallelogram. These two triangles are congruent (equal in size and shape), so either one can be used to apply trigonometric laws.

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Calculations & Trigonometric Laws

1. Interior Triangle Angles

2. Law of Cosines (Resultant Magnitude)

3. Law of Sines (Resultant Direction)

FINAL ANSWER

HINT 2: Statics Principle of 6 Variables

In solving any two-force resultant problem using the parallelogram law, there are six variables in total: three force magnitudes (F₁, F₂, R) and three angles (θ₁, θ₂, θ_R). Out of these six, exactly four must be known to uniquely determine the remaining two.

1. Force & Non-Orthogonal Axes Parameters

Arbitrary Axes (u-v)

R (Resultant Force Magnitude):

N

θ_R (Resultant Angle from +X CCW):

°

θ_u (u-axis Angle from +X CCW):

°

θ_v (v-axis Angle from +X CCW):

°

Quick Presets:

Axes & Force Vector Plot

u-axis v-axis R

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Given Axes & Force Diagram

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Step-by-Step Parallelogram Construction on u-v Axes

Step 1 of 7

Step 1: Draw the designated u-axis and v-axis oriented at their designated angles measured counterclockwise from the positive x-axis.

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Step 1 of 7

Trigonometric Vector Triangle (Law of Sines)

Force Triangle: By translating component vector $\vecec{F}_v$ to the tip of $\vecec{F}_u$, a closed vector addition triangle with resultant $\vecec{R}$ is formed. The interior angles $lpha, eta, \gamma$ are used to solve for unknown components using the Law of Sines.

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Calculations & Law of Sines Formulation

1. Force Triangle Interior Angles

2. Law of Sines & Component Magnitudes

3. Cartesian Check & Vector Equilibrium

FINAL ANSWER

1. Select Number of Concurrent Forces

System Configuration

Number of Forces:

2. Force Magnitudes & Directions

3 Forces Active

F₁ Force 1

Magnitude: N

Angle (θ₁): °

Reference:

F₂ Force 2

Magnitude: N

Angle (θ₂): °

Reference:

F₃ Force 3

Magnitude: N

Angle (θ₃): °

Reference:

Quick Presets:

Concurrent Forces Vector System

F₁ F₂ F₃ F₄

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Stage 1 of 3

1. Step-by-Step Resolution of Each Force

Step 1 of 4: Given Force

Step 1: Plot the given force vector F1 at its specified directional angle.

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Step 1 of 4

Stage 2 of 3

2. Force System & Cartesian Component Projections

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Calculations & Rectangular Vector Summation

1. Component Resolution Summary Table

2. Summation of X-Components (ΣFx)

3. Summation of Y-Components (ΣFy)

4. Resultant Magnitude & Direction Angle

Stage 3 of 3

3. Resultant Vector Construction from ΣFx and ΣFy

Step 1 of 6 (Draw Rx)

Step 1: Plot the total horizontal component Rx = ΣFx starting from the origin along the Cartesian X-axis.

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Step 1 of 5

FINAL RESULTANT ANSWER

1. Force & Orientation Parameters

Orthogonal Components

F (Force Magnitude):

kN

θ (Angle from Axis):

°

Direction / Reference:

Quick Presets:

Force Vector Plot

F

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Given Vector Diagram

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Cartesian Decomposition Construction

Step 1 of 4

Step 1: Drop perpendicular projection lines from the head of force vector F onto the x-axis and y-axis to form a right-angled triangle.

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Step 1 of 4

Calculations & Right-Triangle Trigonometry

1. Trigonometric Formulation

2. Component Magnitudes & Directions

3. Vector Form & Verification

FINAL ANSWER

Textbook Presets:

1. Number of Parallel Forces

System Configuration

Number of Forces:

2. Forces & Positions along Member

3 Forces Active

F₁ Force 1 (Reference Point A, x = 0)

Magnitude: kN

Direction:

Position x₁ from A: m

F₂ Force 2

Magnitude: kN

Direction:

Position x₂ from A: m

F₃ Force 3

Magnitude: kN

Direction:

Position x₃ from A: m

Live Interactive System Diagram

Live Vector Model

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Stage 1: System Analysis & Moment Equilibrium Calculations

Algebraic Method

Parallel Force System Summary Table

1. Resultant Force Magnitude (ΣFy)

2. Moment Summation about Point A (ΣMA)

3. Resultant Line of Action Position (d)

Stage 2: Step-by-Step Equivalent Resultant Construction

Step 1 of 4 (Applied Force System)

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Step 1 of 4

FINAL RESULTANT ANSWER

Force calculations, step by step.

Select a Topic

1 Parallelogram Law

Find Resultant

Resolve Resultant

2 Coplanar Forces

Find Resultant

Resolve Force

3 Parallel Forces

Find Resultant

Parallelogram Law

Parallelogram Law Methodology & Formulas

1. Choose 4 Given Parameters

2. Enter Values

Force Vectors Plot

Parallelogram Law

Determine Resultant Magnitude and Direction

Given Forces Diagram

HINT 1: Tail-to-Tail Intersection

Parallelogram Construction

Trigonometric Vector Triangle

Calculations & Trigonometric Laws

1. Interior Triangle Angles

2. Law of Cosines (Resultant Magnitude)

3. Law of Sines (Resultant Direction)

HINT 2: Statics Principle of 6 Variables

Parallelogram Law

1. Force & Non-Orthogonal Axes Parameters

Axes & Force Vector Plot

Parallelogram Law

Decompose Resultant into u-v Components

Given Axes & Force Diagram

Step-by-Step Parallelogram Construction on u-v Axes

Trigonometric Vector Triangle (Law of Sines)

Calculations & Law of Sines Formulation

1. Force Triangle Interior Angles

2. Law of Sines & Component Magnitudes

3. Cartesian Check & Vector Equilibrium

Coplanar Forces

1. Select Number of Concurrent Forces

2. Force Magnitudes & Directions

Concurrent Forces Vector System

Coplanar Forces

Addition of a System of Coplanar Forces

1. Step-by-Step Resolution of Each Force

2. Force System & Cartesian Component Projections

Calculations & Rectangular Vector Summation

1. Component Resolution Summary Table

2. Summation of X-Components (ΣFx)

3. Summation of Y-Components (ΣFy)

4. Resultant Magnitude & Direction Angle

3. Resultant Vector Construction from ΣFx and ΣFy

Coplanar Forces

1. Force & Orientation Parameters

Force Vector Plot

Coplanar Forces

Decompose Force into Cartesian Components

Given Vector Diagram

Cartesian Decomposition Construction

Calculations & Right-Triangle Trigonometry

1. Trigonometric Formulation

2. Component Magnitudes & Directions

3. Vector Form & Verification

Parallel Forces

1. Number of Parallel Forces

2. Forces & Positions along Member

Live Interactive System Diagram

Resultant of Parallel Force System

Stage 1: System Analysis & Moment Equilibrium Calculations

Parallel Force System Summary Table

1. Resultant Force Magnitude (ΣFy)

2. Moment Summation about Point A (ΣMA)

3. Resultant Line of Action Position (d)

Stage 2: Step-by-Step Equivalent Resultant Construction

Module Title