Energy-Work Relationship - Physics

1. Fundamental Concepts

Definition: Energy is the capacity to do work, and work is the transfer of energy from one system to another through a force acting on an object over a distance.
Work-Energy Principle: The net work done on an object equals the change in its kinetic energy.
Units: Work and energy are measured in joules (J).

2. Key Concepts

Basic Rule: $W = F \cdot d \cdot \cos(\theta)$

Kinetic Energy: $K = \frac{1}{2}mv^2$

Potential Energy: $U = mgh$

3. Examples

Example 1 (Basic)

Problem: A force of $$5 \text{{N}}$$ acts on an object at an angle of $$30^\circ$$ to the horizontal over a distance of $$4 \text{{m}}$$. Calculate the work done.

Step-by-Step Solution:

Substitute values into the formula: $W = 5 \cdot 4 \cdot \cos(30^\circ)$
Calculate: $W = 20 \cdot \frac{\sqrt{3}}{2} = 10\sqrt{3} \approx 17.32 \text{{J}}$

Validation: Substitute known values → Original: $$5 \cdot 4 \cdot \cos(30^\circ)$$; Simplified: $$10\sqrt{3} \approx 17.32 \text{{J}}$$ ✓

Example 2 (Intermediate)

Problem: An object with mass $$2 \text{{kg}}$$ is lifted vertically by $$5 \text{{m}}$$. Calculate the potential energy gained.

Step-by-Step Solution:

Use the formula for potential energy: $U = mgh$
Substitute values: $U = 2 \cdot 9.8 \cdot 5$
Calculate: $U = 98 \text{{J}}$

Validation: Substitute known values → Original: $$2 \cdot 9.8 \cdot 5$$; Simplified: $$98 \text{{J}}$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Draw diagrams to represent forces and distances.
Error-Proofing: Always check units and ensure they are consistent throughout the problem.
Concept Reinforcement: Relate the concept of work to real-world examples, such as lifting objects or pushing a car.