Chapter 5 / 5.1 Introduction Work, Energy and Power

Chapter 5: Work, Energy, and Power

Key Definitions

• Work: Done when a force causes displacement. Mathematically, work \( W = \mathbf{F} \cdot \mathbf{d} = Fd \cos \theta \), where \( \theta \) is the angle between force (\( \mathbf{F} \)) and displacement (\( \mathbf{d} \)).
• Energy: The capacity to do work. Types include kinetic (energy of motion) and potential (stored energy).
• Power: Rate of doing work. \( P = \frac{\text{Work}}{\text{Time}} \).

Scalar (Dot) Product

• For vectors \( \mathbf{A} \) and \( \mathbf{B} \), the dot product is: \[ \mathbf{A} \cdot \mathbf{B} = AB \cos \theta \] It can also be calculated using components: \[ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z \]
• Properties:
  • Commutative: \( \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} \)
  • Distributive: \( \mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C} \)
  • For unit vectors: \( \hat{i} \cdot \hat{i} = 1 \), \( \hat{i} \cdot \hat{j} = 0 \), etc.

Work-Energy Theorem

• The work done by a force changes the kinetic energy of an object: \[ W = \Delta KE = \frac{1}{2}mv^2 – \frac{1}{2}mu^2 \]

Example Calculation

Problem: Find the angle between force \( \mathbf{F} = 3\hat{i} + 4\hat{j} – 5\hat{k} \) and displacement \( \mathbf{d} = 5\hat{i} + 4\hat{j} + 3\hat{k} \).

Solution:

• Dot product: \( \mathbf{F} \cdot \mathbf{d} = (3)(5) + (4)(4) + (-5)(3) = 16 \).
• Magnitudes: \[ F = \sqrt{3^2 + 4^2 + (-5)^2} = \sqrt{50}, \quad d = \sqrt{5^2 + 4^2 + 3^2} = \sqrt{50} \]
• Angle: \[ \cos \theta = \frac{16}{\sqrt{50} \cdot \sqrt{50}} = 0.32 \quad \Rightarrow \quad \theta = \cos^{-1}(0.32) \]

Important Concepts for NEET

• Work-energy theorem and its application to kinetic energy changes.
• Conservation of mechanical energy (kinetic + potential energy).
• Calculating work using the scalar product \( \mathbf{F} \cdot \mathbf{d} \).
• Potential energy of a spring: \( PE_{\text{spring}} = \frac{1}{2}kx^2 \).
• Power calculations involving force and velocity: \( P = \mathbf{F} \cdot \mathbf{v} \).