Equilibrium in 1D

• Vector Summation: Since forces are vectors, direction is indicated by sign. In a 1D system (e.g., along the x-axis), equilibrium is mathematically expressed as $\sum F_x = 0$.

• Weight and Gravity: Weight is the force exerted by gravity on a mass ($m$) and always acts vertically downwards. It is calculated using $W = mg$, where $g$ is the acceleration due to gravity.

• Normal Reaction: When an object is in contact with a surface, the surface exerts a Normal Reaction Force ($R$ or $N$) perpendicular to the surface. In a simple vertical 1D equilibrium on a horizontal floor, $R = mg$.

• Step 1: Draw a Force Diagram: Represent the object as a dot and draw arrows for every force acting on it. Ensure arrows point in the correct direction (e.g., tension pulls away, thrust pushes toward).

• Step 2: Define a Positive Direction: Choose one direction (e.g., Up or Right) as positive. Forces in the opposite direction are then treated as negative values in your equation.

• Step 3: Formulate the Equilibrium Equation: Set the sum of all forces equal to zero. For example, in a vertical system: $T - W = 0$, where $T$ is tension and $W$ is weight.

• Step 4: Solve for Unknowns: Use algebraic manipulation to find missing magnitudes or masses. If the result for a force magnitude is negative, it indicates the force actually acts in the opposite direction to what was assumed.

Static vs. Dynamic Equilibrium

• Static Equilibrium: The object is at rest ($v = 0$). All forces cancel out, and the object remains stationary.
• Dynamic Equilibrium: The object moves at a constant velocity ($v \neq 0, a = 0$). Even though it is moving, the resultant force is still zero because there is no acceleration.

Tension vs. Thrust

• Rounding and $g$: Always check the specified value for $g$. If $g = 9.8 \text{ m s}^{-2}$ is used, provide your final answer to two significant figures. If $g = 10 \text{ m s}^{-2}$ is used, round to one significant figure.

• The 'Hidden' Weight: Never forget to include the weight ($mg$) of the particle in vertical problems, even if the question doesn't explicitly mention it.

• Sanity Check: If a particle is in equilibrium, ensure your calculated forces 'balance' the diagram. If one side has $50 \text{ N}$ and the other has $10 \text{ N}$, the particle cannot be in equilibrium.

• Units: Ensure all masses are in kilograms (kg) and all forces are in Newtons (N) before performing calculations.

• Equilibrium $\neq$ Rest: A common mistake is assuming equilibrium only applies to stationary objects. Remember that an object moving at $50 \text{ m/s}$ in a straight line with no change in speed is also in equilibrium.

• Confusing Mass and Weight: Mass ($m$) is the amount of matter in kg; Weight ($W$) is a force in Newtons. Using $m$ instead of $mg$ in a force equation is a frequent error.

• Sign Errors: Forgetting that forces in opposite directions must have opposite signs in the summation $\sum F = 0$ will lead to incorrect totals.