What is Equilibrium of a Particle?

A particle is in equilibrium when the net external force acting on it is zero. This means:

  • It is either at rest or moving with constant velocity (no acceleration).

Conditions for Equilibrium

  • Two forces: They must be equal in magnitude and opposite in direction.
    Equation: \( F_1 = -F_2 \)
  • Three or more forces: The vector sum of all forces must be zero.
    Equation: \( F_1 + F_2 + F_3 + \dots = 0 \)
    For 3 forces, this means the forces form a closed triangle when drawn head-to-tail.

Breaking Forces into Components

For solving problems, resolve forces into x, y, and z components. In equilibrium:

  • Sum of x-components = 0: \( F_{1x} + F_{2x} + F_{3x} = 0 \)
  • Sum of y-components = 0: \( F_{1y} + F_{2y} + F_{3y} = 0 \)
  • Sum of z-components = 0: \( F_{1z} + F_{2z} + F_{3z} = 0 \)

Example Problem Solved

Situation: A 6 kg mass hangs from a rope. A 50 N horizontal force is applied at the midpoint. Find the angle θ between the rope and the vertical.

Steps:

  1. Weight of the mass: \( T_2 = 6 \times 10 = 60 \, \text{N} \).
  2. At point P, forces are:
    • Tension \( T_1 \) in the rope (at angle θ).
    • Horizontal force: 50 N.
    • Vertical tension: \( T_2 = 60 \, \text{N} \).
  3. Balance components:
    Vertical: \( T_1 \cos\theta = 60 \, \text{N} \)
    Horizontal: \( T_1 \sin\theta = 50 \, \text{N} \)
  4. Divide equations: \( \tan\theta = \frac{50}{60} \)
    Result: \( \theta = \tan^{-1}\left(\frac{5}{6}\right) \approx 40^\circ \)

Common Forces in Mechanics

  • Gravitational force: Acts on all objects (e.g., weight).
  • Contact forces: Require physical contact:
    • Normal reaction (perpendicular to surfaces).
    • Friction (parallel to surfaces).
    • Tension (in strings/ropes).
    • Spring force: \( F = -kx \) (k = spring constant, x = displacement).

Important for NEET Exams

  • Equilibrium conditions for particles (ΣF = 0).
  • Resolving forces into components to solve equilibrium problems.
  • Understanding tension, normal force, and friction as contact forces.
  • Application of \( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \) in force balance.
  • Spring force equation \( F = -kx \).