Equilibrium in 1D (AQA A-Level Mathematics): Revision Notes

1. Definition of Equilibrium in 1D

• Equilibrium: An object is in equilibrium when the net (resultant) force acting on it is zero. In one dimension, this means the forces acting in opposite directions are equal in magnitude but opposite in direction.

• Mathematically:

• For an object in equilibrium in 1D, the sum of all forces acting on the object along the line of action (e.g., the $x$-axis or $y$-axis) must be zero.

• Formula: $∑F=0$ $∑F=0\sum F = 0$

• This implies that: $Fright=Fleft(or)Fup=Fdown$ $Fright=Fleft(or)Fup=FdownF_{\text{right}} = F_{\text{left}} \quad \text{(or)} \quad F_{\text{up}} = F_{\text{down}}$

• If the object is stationary or moving at a constant velocity, the forces acting in one direction are exactly balanced by the forces acting in the opposite direction.

Example

1. Horizontal Equilibrium:

• Consider an object placed on a frictionless surface with two forces acting on it: a $10$ $N$ force to the right and a $10$ $N$ force to the left.
• Condition: The object is in equilibrium because the forces are equal and opposite.
• Equation: $10N(right)−10N(left)=0N.$ $10 N (right)−10 N (left)=0 N10 \, \text{N} \, \text{(right)} - 10 \, \text{N} \, \text{(left)} = 0 \, \text{N}$

2. Vertical Equilibrium:

• A book rests on a table. The weight of the book (force due to gravity) acts downward, and the normal force from the table acts upward.
• Condition: The book is in equilibrium because the downward gravitational force is balanced by the upward normal force.
• Equation: $W=N$, where W is the weight of the book, and $N$ is the normal force. $W=NW = N$

$WW$

$NN$

• Step 1: Identify All Forces:
• Draw a diagram (Free Body Diagram) showing all the forces acting on the object. Label their magnitudes and directions.
• Step 2: Resolve Forces Along the Line of Action:
• Ensure all forces are resolved along the same line (horizontal or vertical). If dealing with angled forces in 1D problems, resolve them into their components along the relevant axis.
• Step 3: Apply the Equilibrium Condition:
• Set up the equilibrium equation by summing the forces and setting the sum equal to zero.
• Solve for the unknowns (e.g., force magnitude, tension in a string, etc.).

Problem: A $5$ $kg$ block is being pulled to the right with a force of $30$ $N$, and it experiences a frictional force of $10$ $N$ to the left. Is the block in equilibrium?

• Predicting Motion: Understanding equilibrium helps predict whether an object will remain stationary, move at a constant velocity, or accelerate.
• Practical Applications: Equilibrium concepts are used in designing structures, analysing forces in cables and strings, and understanding statics problems in engineering.

In one-dimensional mechanics, equilibrium occurs when all forces acting along a single line cancel each other out, resulting in no net force. This concept is fundamental for analysing the motion of objects and ensuring stability in physical systems. When forces are balanced, the object remains in equilibrium; otherwise, it will accelerate in the direction of the net force.