Mechanical Equilibrium - States of Equilibrium

Stable Equilibrium

What happens: A ball sits at the bottom of a bowl. If you push it slightly, it rolls back to the bottom.

Simple explanation: The object naturally returns to its original position after disturbance.

Potential energy: At its lowest point (minimum).

Mathematical sign: Second derivative of potential energy is positive (d²U/dx² > 0).

Unstable Equilibrium

What happens: A ball is balanced on top of a hill. If you push it even a little, it rolls far away and doesn't come back.

Simple explanation: The object moves away from its original position when disturbed.

Potential energy: At its highest point (maximum).

Mathematical sign: Second derivative of potential energy is negative (d²U/dx² < 0).

Neutral Equilibrium

What happens: A ball rolls on a flat, smooth surface. It stays in equilibrium no matter where you place it.

Simple explanation: The object remains in equilibrium at any position—nothing changes.

Potential energy: Same everywhere (constant).

Mathematical sign: Second derivative of potential energy is zero (d²U/dx² = 0).

• Predicts object behavior when disturbed or pushed slightly
• Determines the stability of structures like buildings, bridges, and furniture
• Safety applications – help design safe systems (like how chairs are shaped)
• Energy analysis – connects equilibrium to potential energy concepts
• Real-world design – engineers use this to make stable products
• Mathematical tool – helps solve complex physics problems using calculus
• Natural phenomena – explain why things naturally return to certain positions

Given Information:

• Wooden plank mass = 30 kg
• Boy mass = 50 kg
• Two identical vertical cables separated by 2 m
• Each cable can hold a maximum tension = 500 N
• g = 10 m/s²

Find Total Weight

Solution:

• Weight of plank = 30 × 10 = 300 N (acts at center, 1 m from either end)
• Weight of boy = 50 × 10 = 500 N (acts where the boy stands)
• Total downward force = 300 + 500 = 800 N
• T₁ + T₂ = 800 N (where T₁ and T₂ are cable tensions)
• As the boy walks toward the right cable, the right cable tension increases
• Maximum tension in cable = 500 N
• Using moment balance about the left end (point A):
  Clockwise moments: 300 × 1 + 500 × (1 + x) = 300 + 500 + 500x
  Anticlockwise moment: T₂ × 2 = 500 × 2 = 1000 N
• 300 + 500(1 + x) = 1000
• x = 0.4 m = 40 cm

The boy can walk up to 40 cm on either side of the center before a cable breaks.

Given Information:

• Each leg length = 1 m
• Angle between legs = 40°
• Person mass = 50 kg
• g = 10 m/s²
• Floor is frictionless (no horizontal reaction at the floor)

Find Vertical Reaction

Solution:

• Weight of person = W = mg = 50 × 10 = 500 N
• Two legs share the weight equally
• N = W/2 = 500/2 = 250 N (reaction at each base)
• No friction means no horizontal force at the floor
• Tension T in the cross bar acts horizontally
• Consider the left leg about its upper end
• Torque from N (clockwise) = N × L × sin(20°)
• Torque from T (anticlockwise) = T × (L/2) × cos(20°)
• (Note: 90° − 40° = 50°, so half angle = 25°, but using simplified version as sin(20°) and cos(20°))
• For equilibrium: Clockwise torque = Anticlockwise torque
• N × L × sin(20°) = T × (L/2) × cos(20°)
• T = 2N × tan(20°)
• T = 2 × 250 × 0.364
• T = 182 N

The crossbar experiences a force (tension) of 182 N to hold the ladder steady.