Simplified Physics

🔬 POTENTIAL ENERGY (U)

Class 12 Physics | Complete Detailed Notes

📚 1. DEFINITION & BASIC CONCEPT

U = - ∫ F · dr

"Stored energy due to position, configuration, or state of a system"

✅ Key Properties:

• Scalar quantity
• Depends on position/configuration
• Reference point arbitrary (only ΔU matters)
• Force = -∇U (Negative gradient)

🎯 2. TYPES OF POTENTIAL ENERGY

A. Gravitational Potential Energy (U_g)

Near Earth's Surface

U_g = mgh

Symbol	Meaning	SI Unit
m	Mass	kg
g	Acceleration due to gravity	9.8 m/s²
h	Height above reference	m

Universal Gravitational PE

U_g = -GMm/r

⚡ Key Points:

• Negative sign → Attractive force
• U = 0 at r = ∞
• Maximum (least negative) at infinity

Concept	Formula	Unit
Gravitational Field (g)	GM/r²	N/kg
Gravitational Potential (V)	-GM/r	J/kg
PE of mass m	mV = -GMm/r	J

B. Elastic Potential Energy (U_s)

U_s = ½kx²

📈 Graph: Parabola (U vs x)

                          U
                        ½kx²
                         │
                         └─────────→ x
                    

C. Electric Potential Energy

U = kq₁q₂/r

k = 9×10⁹ Nm²/C² | +ve (repulsive) / -ve (attractive)

🔧 3. IMPORTANT DERIVATIONS

A. Derivation of Spring PE

W = ∫₀ˣ kx dx = ½kx²

This work = Potential Energy stored

B. Relation: F = -dU/dx

System	U	F = -dU/dx
Gravity	mgh	-mg ✓
Spring	½kx²	-kx ✓

📊 4. GRAPHICAL REPRESENTATION

Potential Energy Curves

System	U vs x Graph	Force Direction
Spring	Parabola (U-shaped)	Towards equilibrium
Gravity	Straight line	Downward
Stable equilibrium	Minimum U	Restoring force
Unstable equilibrium	Maximum U	Away from point

💡 Work from PE Graph:
W_conservative = -ΔU
Area under F-x = ΔU

🌍 Gravity (Earth)

U = mgh

F = -mg

🌌 Gravity (Universal)

U = -GMm/r

F = -GMm/r²

🪢 Spring

U = ½kx²

F = -kx

⚡ Electric

U = kq₁q₂/r

F = kq₁q₂/r²

⚠️ 5. COMMON MISTAKES & TIPS

❌ WRONG	✅ CORRECT
U = mgh (absolute)	ΔU = mgΔh
PE depends on velocity	Only position
Spring PE = kx	½kx²
F = dU/dx	F = -dU/dx

🎯 6. NEET/JEE LEVEL QUESTIONS

Q1: Work done by gravity when body falls from height h?
Ans: W_g = -ΔU = mgh ✓

Q2: Spring constant if 2 kg mass compresses spring by 10 cm?
Ans: k = mg/x = 196 N/m

✨ Ready for NEET/JEE! | Class 12 Physics Chapter 6 ✨