### 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=21kx2**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=kerq1q2**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.