🔋 Work-Energy Theorem for Variable Forces
1. Work Done by a Variable Force
When a force changes with distance (e.g., the woman pushing the trunk 🧳), work is calculated by finding the area under the force vs. displacement graph:
- For a linearly decreasing force (like in Example 5.5):
\( W = \text{Area of rectangle} + \text{Area of trapezium} \)
Example: \( W_F = 100 \, \text{N} \times 10 \, \text{m} + \frac{1}{2}(100 + 50) \, \text{N} \times 10 \, \text{m} = 1750 \, \text{J} \) - For constant forces (like friction 🔻):
\( W_f = \text{Force} \times \text{Total displacement} \)
Example: \( W_f = (-50 \, \text{N}) \times 20 \, \text{m} = -1000 \, \text{J} \)
2. Work-Energy Theorem 🔄
Change in kinetic energy = Total work done:
- For variable forces:
\( K_f – K_i = \int_{x_i}^{x_f} F(x) \, dx \) - Example 5.6 (block on rough patch):
Retarding force \( F_r = -\frac{k}{x} \), work done:
\( \int_{0.1}^{2.01} \frac{-k}{x} dx = -k \ln\left(\frac{2.01}{0.1}\right) \)
Final kinetic energy: \( K_f = 2 \, \text{J} – 0.5 \ln(20.1) \approx 0.5 \, \text{J} \)
Final speed: \( v_f = \sqrt{\frac{2K_f}{m}} = 1 \, \text{m/s} \)
3. Potential Energy 🌄
Energy stored due to position/configuration (e.g., gravitational potential energy):
- \( V(h) = mgh \) (for small heights near Earth’s surface)
- Force = Negative derivative of potential energy:
\( F = -\frac{d}{dh} V(h) = -mg \) - When released, potential energy converts to kinetic energy:
\( \frac{1}{2}mv^2 = mgh \)
🌟 High-Yield NEET Concepts
- Work-energy theorem: \( \Delta K = W_{\text{net}} \) (applies to variable forces too!).
- Calculating work via integration (e.g., for \( F(x) = -k/x \)).
- Potential energy and force relationship: \( F = -\frac{dV}{dx} \).
- Kinetic to potential energy conversion (e.g., falling objects).
💡 Remember: Work done by friction is always negative (opposes motion)! For variable forces, draw the graph and find the area or integrate. Practice logarithmic integrals for forces like \( 1/x \).
