### Potential Analysis in Classical Mechanics for CSIR NET/JRF

Potential analysis is a crucial topic in classical mechanics, particularly for exams like CSIR NET/JRF. Understanding potentials helps in analyzing forces and energy in a system, which is fundamental for solving various physical problems.

Potential Analysis , the stable equilibrium point, the unstable equilibrium point , curve plotting , transition point. IIT JAM, CSIR NET, GATE, JEST, TIFR, MSc. Entrance in Physics.

### Lecture Content Outline

#### 1. Basics of Potential Energy

• Definition: Potential energy is the energy stored in a system due to its position in a force field.
• Types of Potential Energy:
• Gravitational Potential Energy
• Elastic Potential Energy
• Electric Potential Energy
• Formula: U=−∫F⋅drU = – \int \mathbf{F} \cdot d\mathbf{r}U=−∫F⋅dr

#### 2. Gravitational Potential Energy

• Near Earth’s Surface: U=mghU = mghU=mgh
• General Case: U=−GMmrU = – \frac{G M m}{r}U=−rGMm​
• Example Problems:
• Calculate the potential energy of an object at a height hhh.
• Determine the change in potential energy when an object moves in a gravitational field.

#### 3. Elastic Potential Energy

• Hooke’s Law: F=−kxF = -kxF=−kx
• Potential Energy in a Spring: U=12kx2U = \frac{1}{2} k x^2U=21​kx2
• Example Problems:
• Find the potential energy stored in a compressed or stretched spring.
• Analyze energy conservation in a mass-spring system.

#### 4. Electric Potential Energy

• Point Charges: U=keq1q2rU = k_e \frac{q_1 q_2}{r}U=ke​rq1​q2​​
• Electric Potential: V=UqV = \frac{U}{q}V=qU​
• Example Problems:
• Calculate the potential energy between two point charges.
• Analyze the potential energy in a system of multiple charges.

#### 5. Potential Energy Curves

• Concept: Graphical representation of potential energy as a function of position.
• Stability:
• Stable Equilibrium: Local minima of potential energy curve.
• Unstable Equilibrium: Local maxima of potential energy curve.
• Neutral Equilibrium: Flat regions of the potential energy curve.
• Example Problems:
• Interpret potential energy curves to determine points of equilibrium.
• Analyze the motion of particles in different potential energy landscapes.

#### 6. Conservation of Mechanical Energy

• Total Mechanical Energy: E=K+UE = K + UE=K+U
• Principle of Conservation: ΔE=0\Delta E = 0ΔE=0
• Example Problems:
• Solve problems involving conservation of energy in gravitational fields.
• Apply conservation principles to systems with springs and electric forces.

#### 7. Potential and Force Relationship

• Gradient of Potential: F=−∇U\mathbf{F} = -\nabla UF=−∇U
• Example Problems:
• Derive force from a given potential energy function.
• Analyze the motion of a particle in a given potential field.

### Example Problems and Solutions

#### Conclusion

Potential analysis is fundamental in classical mechanics and provides a deep understanding of forces and energy conservation in physical systems. Mastering this topic is essential for excelling in exams like CSIR NET/JRF.