2 Forces and Particle Equilibrium

Representing a force

A force is a vector with both magnitude and direction. In SI units, force is measured in newtons, written $\mathrm{N}$. It may be represented by an arrow: the arrow's length represents magnitude, its orientation gives direction, and its line of action shows where the force acts.

In a two-dimensional Cartesian coordinate system, a force can be written as

where $F_x$ and $F_y$ are signed components along the $x$- and $y$-axes, and $\mathbf i$ and $\mathbf j$ are unit vectors. In three dimensions, the representation is

The magnitude of a two-dimensional force is

and its direction angle, measured counterclockwise from the positive $x$-axis, can be found using $\theta=\operatorname{atan2}(F_y,F_x)$. The signs of the components identify the force's quadrant.

Force components and resolution

Force components are perpendicular vectors whose sum equals the original force. If a force of magnitude $F$ makes an angle $\theta$ from the positive $x$-axis, its components are

The formulas depend on how the angle is measured. If the angle is measured from the vertical, sine and cosine switch roles. Assign signs according to the actual directions of the components. Component form makes vector sums straightforward: add corresponding components.

For example, a $100\,\mathrm{N}$ force acting $30^\circ$ above the positive $x$-axis has components

Thus, $\mathbf F=(86.6\mathbf i+50.0\mathbf j)\,\mathrm{N}$. The component values are not separate additional forces; together they represent the original force.

Concurrent forces and diagrams

A concurrent force system has all its forces' lines of action passing through one common point. When an object's size and rotation are not relevant, it can be modeled as a particle, with all forces drawn as acting at that point.

A free-body diagram isolates the particle and shows every external force acting on it, such as applied forces, cable tensions, and weight. Do not include forces the particle exerts on other objects.

The resultant force is the vector sum of the forces:

For example, two concurrent forces on a particle—$40\,\mathrm{N}$ to the right and $30\,\mathrm{N}$ upward—have resultant

It points $\tan^{-1}(30/40)=36.9^\circ$ above the positive $x$-axis.

Particle equilibrium

Particle equilibrium occurs when the resultant force is zero. In two dimensions, the component sums must each be zero:

In three dimensions, the third condition is also required:

For the example with forces of $40\,\mathrm{N}$ rightward and $30\,\mathrm{N}$ upward, a third force must be equal and opposite to their resultant:

This force has magnitude $50\,\mathrm{N}$ and acts down and left. The component checks are $40-40=0\,\mathrm{N}$ horizontally and $30-30=0\,\mathrm{N}$ vertically.

Setting up an equilibrium problem

Use this sequence to set up a particle-equilibrium problem:

1. Isolate the particle and sketch its free-body diagram.

2. Choose coordinate axes; align an axis with a known force when convenient.

3. Label each force with its magnitude, direction, and component signs. For an unknown force, assume a direction and keep the signs consistent. A negative solution means the actual direction is opposite to the assumed direction.

4. Resolve angled forces into components and apply $\sum F_x=0$, $\sum F_y=0$, and, if needed, $\sum F_z=0$.

5. Check that units are consistent and that the final vector directions make physical sense.

These equations apply to particle equilibrium and concurrent forces. For an extended rigid body, forces may also cause rotation, so force balance alone is not generally sufficient; moment equilibrium must also be considered.