Understanding Force Balance and Composition

Balance

A body is in equilibrium when at rest or when moving with uniform rectilinear motion. Some conditions are:

• When a body acts on a single force, it cannot be in balance.
• Two equal and opposite forces acting on a body produce equilibrium.
• The total strength of various forces must be zero for a body to be in balance.

Resultant Force

The resultant force (R) is the force that substitutes various forces, and its effect is the same as all the initial forces together. The calculation of the force resulting from a group of them acting on a body is called the composition of forces.

Resultant Force of Forces Applied in the Same Direction and Sense

• The point of application, direction, and sense will be the same as those of the component forces.
• The module is calculated by summing the magnitudes of the component forces.

Resultant Force of Forces Applied in the Same Direction and Opposite Sense

• The point of application and direction are the same as those of the component forces.
• The sense is that of the component of greater force.
• The module is calculated by subtracting the modules of the component forces.

Resultant Force of Forces Applied in Different Directions - Parallelogram Rule

To combine forces with different directions, the parallelogram rule is applied (to be added the x components of each force and the y components of each force). The resultant force of forces applied in different directions is defined as:

• The point of application is the same as that of the component forces.
• The direction, sense, and modulus are determined by the diagonal of the parallelogram formed.

If the forces are perpendicular, the module is calculated using the Pythagorean theorem: R = √(F₁² + F₂²).

The other force will have the same point of application, module, and direction as the resultant, so it will be the balancing force.

Composition of Parallel Forces

Mostly, forces acting on a body do not have the same point of application.

Parallel Forces in the Same Direction

The point of application is graphically placed by drawing F₁, F₂, and F₂ in the opposite direction to F₁. The ends are joined by a line, and the point where both lines intersect is the point of application.

The resultant force of parallel forces applied in the same direction is defined as:

• The point of application is calculated using the inverse proportionality rule: F₁ / F₂ = d₂ / d₁ -> F₁ · d₁ = F₂ · d₂.
• The direction and sense are the same as those of the component forces.
• The module is the sum of the modules of the components.

Parallel Forces in Different Senses

The force resulting from forces applied in different senses is defined as:

• The point of application is calculated using the inverse rule.
• The direction is the same as that of the component forces.
• The sense is the same as that of the component force with the greater module.
• The module is calculated as the difference of the magnitudes of the components.

Decomposition of Forces

To decompose a force into its components is the reverse problem; starting from a resultant force, the component forces are obtained. A force can be easily decomposed in two directions. Two lines are selected and plotted from the origin. From the end of the force, we draw parallels to those two directions, and the points of intersection on both directions are the extremes of the component forces.