Equilibrium in 2D (AQA A-Level Mathematics): Revision Notes

To approach equilibrium questions in 2D Mechanics, follow these key steps:

1. Identify all forces: List all the forces acting on the object, including weight, tension, normal reaction, and friction.
2. Resolve forces into components: Split each force into horizontal and vertical components, typically using trigonometry (sine and cosine functions for angled forces).
3. Apply equilibrium conditions: For an object in equilibrium, the sum of forces in both directions must equal zero:

• $\sum F_x = 0$ (sum of horizontal forces)
• $\sum F_y = 0$ (sum of vertical forces)

4. Solve the system of equations: Use the two equations to find unknown forces or angles. This method ensures that all forces balance, confirming that the object is in equilibrium.

Example:

• Consider a scenario where a single piece of string is threaded through a particle, with the string being light and inextensible. The particle is held in equilibrium by two strings forming angles of $30^\circ$ with the horizontal. Problem:

• Given that the weight of the particle is $2$g, find the tension $T$ in the strings. Solution:

5. Resolve Forces in a Relevant Direction:

• Consider the vertical direction for resolving forces.

6. Vertical Forces:

• Since $\sin 30^\circ = \frac{1}{2}$, the equation simplifies to:

Final Answer:

• The tension in each string is $T = 2g \, \text{N}$.

Q1, (Jan 2005, Q3)

Problem Statement: A particle is in equilibrium when acted on by the forces$\left(\begin{array}{c} x \\ -7 \\ z \end{array}\right), \left(\begin{array}{c} 4 \\ y \\ -5 \end{array}\right), and \left(\begin{array}{c} 5 \\ 4 \\ -7 \end{array}\right)$, where the units are Newtons.

7. Find the values of $x$, $y$, and $z.$
8. Calculate the magnitude of $\left(\begin{array}{c} 5 \\ 4 \\ -7 \end{array}\right)$.

Problem Statement:

• A smooth bead Y is threaded on a light inextensible string.

• The string is attached to two fixed points, $X$ and $Z$, which are on the same horizontal level.

• The bead is in equilibrium, held by a horizontal force of magnitude $8 \, \text{N}$ acting parallel to $ZX$.

• The bead $Y$is vertically below $X$, and \angle $\angle XZY = 30^\circ$. Find:

• The tension in the string $T$.

• The weight of the bead $W$.

Problem Statement:

• A smooth bead $Y$ is threaded on a light inextensible string. The ends of the string are attached to two fixed points, $X$ and $Z$, on the same horizontal level.

• The bead is in equilibrium under a horizontal force of magnitude $8 \, \text{N}$ acting parallel to $ZX$.

• The bead $Y$is vertically below $X$, and $\angle ZXY = 30^\circ$. Objective:

• Find the tension $T$ in the string.

• Find the weight $W$ of the bead.