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.

The state of equilibrium in which the potential energy of the system is at its local minimum is called stable equilibrium.

OR

The equilibrium in which the potential energy at the equilibrium position is minimum as compared to its neighbouring points, where \[\frac {d^2U}{dr^2}\]​ is positive, is called Stable Equilibrium.

The state of equilibrium in which the potential energy of the system is at its local maximum is called unstable equilibrium.

OR

The equilibrium in which the potential energy at the equilibrium position is maximum as compared to other positions, where \[\frac {d^2U}{dr^2}\] is negative, is called Unstable Equilibrium.

The state of equilibrium in which the potential energy of the system is constant over a plane and remains same at any position is called neutral equilibrium.

OR

The equilibrium in which the potential energy remains constant even if the body moves to neighbouring points, where \[\frac {d^2U}{dr^2}\] = 0, is called Neutral Equilibrium.