This section elucidates this relationship, explicating the principles of energy conservation, particularly in scenarios like non-linear motion, and provides an overall perspective on energy considerations in various contexts.

Energy and Work: Concepts and Applications

Energy Changes

• Definition: Energy changes involve the transformation of energy from one form to another or its movement into/out of a system.

Work Done by External Forces

• Concept: When an external force works on a system, it changes the system's energy.
• Quantification: This energy change is measurable.

Work Done Formula

• Equation: $W = F \cdot d \cdot \cos(\theta)$
• Variables: $F$ = force, $d$ = displacement, $\theta$ = angle between force and displacement.

Conservation of Energy Principle

• Principle: In an isolated system, total energy remains constant but can change forms.
• Application in Non-linear Motions: For an object sliding down a curve, gravitational potential energy converts to kinetic energy, simplifying motion analysis.

Energy Considerations in Complex Systems

Approach: Focusing on energy changes simplifies understanding complex systems where direct force analysis is hard.

Example Problems

Example 1: Sliding Object Problem

Problem: Calculate speed of a 5 kg object sliding down a 10m high slide (no friction).

Solution:

• Initial Potential Energy (PE): $PE = mgh = 5 \times 9.8 \times 10 = 490$ joules.
• Final Kinetic Energy (KE) = Initial PE.
• Speed at bottom: $v = \sqrt{\frac{980}{5}} \approx 9.9 \text{ m/s}$.

The illustration shows a simple slide with a 10-meter height and base. The red dot represents the object (5 kg) at the bottom of the slide.

Example 2: Power Usage Problem

Problem: Find average power used by a bulb consuming 3600 joules in 1 hour.

Solution:

• Power: $P = \frac{\text{Energy}}{\text{Time}} = \frac{3600 \text{ joules}}{3600 \text{ seconds}} = 1 \text{ watt}$.

The graph illustrates the energy consumption of a light bulb over time. The green line shows the energy consumed over an hour, with the red dot indicating the total energy consumption of 3600 joules by the light bulb after 1 hour.