Force

A force is a push or pull with magnitude, direction and point of application. Learn force as a vector, units, resolving into components, adding forces to find a resultant, and solving concurrent force systems with Lami's theorem.

Everything a mechanical engineer designs — a bracket, a bridge, a robot arm, an engine — exists to carry, transmit or produce forces. Force is the most basic quantity in mechanics: the interaction that changes an object's motion or shape.

Because force has both size and direction, it is a vector. Most early mechanics problems come down to breaking forces into components, adding them up, and making sure they balance.

An analogy

Two people pulling a boat with ropes from different angles: the boat doesn't move toward either person, but along the combined (resultant) direction of both pulls.

What a force is

A force is an interaction — contact (pushing, pulling, friction, pressure) or at a distance (gravity, magnetism). 1 N accelerates 1 kg at 1 m/s². A 1 kg mass weighs about 9.81 N on Earth; engineers often use kN (1000 N) for structures.

Force as a vector

A force is fully described by magnitude, direction (line of action) and point of application. Moving a force along its line of action doesn't change its effect on a rigid body (principle of transmissibility), but moving it sideways does — it creates a moment.

Resolving and adding forces

The easiest way to add forces at different angles is to resolve each into x and y components, sum each direction separately, then recombine. The parallelogram and triangle laws are the graphical version of the same idea.

Force systems

Forces are classified by how their lines of action relate.

• Collinear: all on one line — simple addition.
• Concurrent: lines meet at one point — resultant through that point.
• Parallel: e.g. loads on a beam — resultant plus position.
• General (non-concurrent): resultant force plus a moment.

3D forces and unit vectors

In 3D, a force is F = F·û, where the unit vector û = (x, y, z)/√(x² + y² + z²) points along a cable or member. Direction cosines cos θx, cos θy, cos θz give components. This is how cable-stayed structures, cranes and truss analysis software handle forces.

Formulas

• Components: F_x = F cos θ, F_y = F sin θ
• Resultant: R = √(ΣF_x² + ΣF_y²), θ_R = atan2(ΣF_y, ΣF_x)
• Weight: W = m · g (g = 9.81 m/s²)
• Lami's theorem (3 concurrent forces): F₁/sin α = F₂/sin β = F₃/sin γ

Worked example: Resultant of two forces

Given: F₁ = 100 N at 0°; F₂ = 80 N at 60°

Find: Magnitude and direction of the resultant.

• ΣF_x = 100 + 80 cos 60° = 140 N
• ΣF_y = 0 + 80 sin 60° = 69.28 N
• R = √(140² + 69.28²) = 156.2 N
• θ = atan(69.28/140) = 26.3°

Answer: R = 156.2 N at 26.3°

Worked example: Resolving a force

Given: F = 500 N at 30° above horizontal

Find: Horizontal and vertical components.

• F_x = 500 cos 30° = 433 N
• F_y = 500 sin 30° = 250 N

Answer: 433 N horizontal, 250 N vertical

Common mistakes

• Wrong: Adding force magnitudes directly when they act at angles. — Right: Resolve into components first, add components, then recombine.
• Wrong: Using mass (kg) as if it were a force. — Right: Convert with W = mg: a 100 kg load weighs 981 N.

Applications

• Cranes and rigging: Sling angles change cable tension dramatically — at 30° from horizontal each leg carries the full load.
• Truss design: Bridges and roof trusses are analysed as networks of concurrent forces at each joint.
• Vehicle dynamics: Tyre forces are resolved into longitudinal (traction/braking) and lateral (cornering) components.

Key takeaways

• Force is a vector: magnitude, direction, point of application.
• Resolve → sum components → recombine for the resultant.
• Weight = mg; use newtons, not kilograms, for forces.
• Lami's theorem solves three concurrent forces in equilibrium.

Going further

• Dot and cross products: The dot product F·û projects a force onto a direction (e.g. the component along a member); the cross product r × F gives the moment vector — the core of 3D statics.
• Distributed forces: Pressure, weight and wind are distributed; they are replaced by equivalent resultants acting at the centroid of the load distribution.
• Numerical statics: Finite-element and multibody software assemble thousands of force-balance equations into matrices and solve them simultaneously — the same ΣF = 0 at every node.