Equilibrium in 2D

• Newton's First Law in 2D: An object remains in its state of motion unless acted upon by an unbalanced force. In 2D, this implies that if the object is not accelerating, the net force in every possible direction must be zero.

• Independence of Components: A crucial principle is that forces acting in perpendicular directions (like $x$ and $y$) do not affect each other. Therefore, equilibrium can be treated as two simultaneous 1D equilibrium problems.

• Resultant Force Calculation: The total force $\mathbf{F}_{net}$ is the vector sum $\sum \mathbf{F}_i$. For equilibrium, $\sum F_x = 0$ and $\sum F_y = 0$ must both be true.

Resolution of Forces

• Step 1: Choose Axes: Select two perpendicular directions. Usually, horizontal and vertical are used, but for inclined planes, it is often easier to use directions parallel and perpendicular to the surface.

• Step 2: Resolve Vectors: Use trigonometry to find the components of any diagonal forces. A force $F$ at an angle $\theta$ to the $x$-axis has components $F_x = F \cos(\theta)$ and $F_y = F \sin(\theta)$.

• Step 3: Set Up Equations: Create two independent algebraic equations: one for the sum of all 'left' forces equaling 'right' forces, and one for 'up' forces equaling 'down' forces.

Vector Addition (Nose-to-Tail)

• If three forces are in equilibrium, they can be drawn as a closed triangle. If there are more than three, they form a closed polygon. The fact that the shape closes indicates the resultant displacement (force) is zero.

• Always Draw a Diagram: Even if one is provided, sketch a clear force diagram (Free Body Diagram) showing all forces acting on the single point representing the particle.

• Check for Hidden Forces: Ensure weight ($W = mg$) is always included acting vertically downwards, and normal reactions ($R$) are always perpendicular to the contact surface.

• Coordinate Choice: If a problem involves an inclined plane, always resolve parallel and perpendicular to the plane. This usually prevents having to resolve the normal reaction and friction, simplifying the algebra.

• Sanity Check: If you calculate a negative magnitude for a force, it usually means the direction you assumed in your diagram was opposite to the actual direction.

• Mixing Sine and Cosine: A common error is using $\sin$ when $\cos$ is required. Always check if the component is adjacent to the angle (use $\cos$) or opposite to the angle (use $\sin$).

• Ignoring the Zero Vector: In vector notation, students sometimes set the sum of vectors equal to a scalar zero. Remember that $\mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 = \mathbf{0}$ (the zero vector).

• Incomplete Equilibrium: Assuming equilibrium in the $x$-direction implies equilibrium in the $y$-direction. Both must be verified independently.