# Conservation of Energy ## The Principle of Conservation of Energy - The Principle of Conservation of Energy is a fundamental law in physics stating:

Energy cannot be created or destroyed, it can only be transferred from one form to another

- In any closed system, the total amount of energy remains constant - The distribution of energy across different forms may change - But the sum total never increases or decreases ## Forms of Energy - Energy exists in multiple forms that can be converted between each other - Understanding different energy forms helps track energy transfers in systems | **Form** | **Description** | |----------|----------------| | **Kinetic** | Energy possessed by a moving object | | **Gravitational Potential** | Energy an object has due to its position in a gravitational field; gained when lifted, lost when it falls | | **Elastic Potential** | Energy stored in a deformed spring or stretched elastic material (sometimes called strain energy) | | **Chemical** | Energy stored within chemical bonds in substances | | **Nuclear** | Energy contained within atomic nuclei | | **Internal** | Energy related to the temperature or state of a substance (sometimes referred to as thermal or heat energy) | ## Energy Dissipation - During energy transfers, not all energy ends up in the desired or useful form - **Dissipation** describes the process where energy spreads out into less useful forms ### Wasted Energy - Energy not transferred to useful stores is considered wasted - This energy is typically lost to the surroundings - Common forms of wasted energy include thermal (heat), light, and sound

What counts as "wasted" energy depends entirely on the system's purpose

### Examples of Useful vs Wasted Energy - **Television:** - Electrical energy → light energy + sound energy + thermal energy - Light and sound are useful outputs - Thermal energy from heated wires is wasted - **Heater:** - Electrical energy → thermal energy + sound energy - Thermal energy is the useful output - Sound from the operating mechanism is wasted ## Energy Conservation in Practice ### Common Energy Transfers - Understanding typical energy conversion scenarios helps apply conservation principles 1. **Falling object (in a vacuum):** - Gravitational potential energy → kinetic energy 2. **Battery powering a bulb:** - Chemical energy → electrical energy → light energy 3. **Horizontal mass-spring system:** - Elastic potential energy ⇌ kinetic energy 4. **Bouncing on a trampoline:** - Elastic potential energy ⇌ kinetic energy ⇌ gravitational potential energy - The jumper compresses the trampoline, converting kinetic to elastic potential - As they bounce up, elastic potential converts to kinetic then gravitational potential - At the peak, all energy is gravitational potential - Falling back down reverses the process ### Energy Conservation with Resistive Forces - In real situations, resistive forces like friction affect energy transfers - These forces cause some mechanical energy to be converted to thermal energy **Example:** Object sliding up a rough inclined plane $$\text{Loss in kinetic energy} = \text{Gain in gravitational potential energy} + \text{Work done against friction}$$ - The initial kinetic energy doesn't all become gravitational potential energy - Some is dissipated as thermal energy due to friction ### Key Energy Equations for Conservation Problems - When applying conservation of energy, these equations are frequently used: $$E_k = \frac{1}{2}mv^2$$ $$E_p = mg\Delta h$$ $$E_{\text{elastic}} = \frac{1}{2}k(\Delta L)^2$$ - Where: - $E_k$ = kinetic energy (J) - $E_p$ = gravitational potential energy (J) - $E_{\text{elastic}}$ = elastic potential energy (J) - $m$ = mass (kg) - $v$ = velocity (m s$^{-1}$) - $g$ = gravitational field strength (9.81 N kg$^{-1}$ or m s$^{-2}$) - $\Delta h$ = change in height (m) - $k$ = spring constant (N m$^{-1}$) - $\Delta L$ = extension or compression (m) ## Worked Example: Roller Coaster Energy Transfers **Problem:** A roller coaster follows path A → B → C → D. Which statement correctly describes the energy changes? - Options involve sequences of KE (kinetic energy) and GPE (gravitational potential energy) **Analysis:** - **Point A (top of initial drop):** - Roller coaster is elevated → has GPE - As it descends, GPE converts to KE - Speed increases - **Point B (bottom of loop):** - As it rises into the loop, KE converts to GPE - Speed decreases - **Point C (top of loop):** - GPE converts back to KE - Descending the loop increases speed again - **Point D (flat track):** - On level ground, only KE remains (no height change) **Answer:** D → GPE - KE - GPE - KE The energy transfers throughout this journey can be visualized as:

In energy conservation problems, identify the dominant energy form at each position by considering height (GPE) and motion (KE)

## Worked Example: Pendulum Energy Conservation **Problem:** A simple pendulum has mass 640 g and length 0.7 m. It's pulled to 20° from vertical then released. Calculate maximum speed at the vertical position, assuming negligible air resistance. **Solution:** **Step 1: Identify energy transfers** - At maximum angle: pendulum has gravitational potential energy - At vertical position: all GPE converts to kinetic energy **Step 2: Calculate height dropped** - Using trigonometry: height dropped = $0.7 - 0.7\cos(20°)$ m - $\Delta h = 0.7 - 0.7\cos(20°) = 0.7(1 - \cos(20°))$ **Step 3: Apply conservation of energy** $$\text{GPE lost} = \text{KE gained}$$ $$mg\Delta h = \frac{1}{2}mv^2$$ - Calculate GPE: - $GPE = 0.64 \times 9.81 \times (0.7 - 0.7\cos(20°))$ - $GPE = 0.265$ J **Step 4: Solve for velocity** $$\frac{1}{2}mv^2 = mg\Delta h$$ $$v^2 = 2g\Delta h$$ $$v = \sqrt{2g\Delta h} = \sqrt{\frac{2 \times 0.265}{0.64}} = 0.91 \text{ m s}^{-1}$$ The energy distribution changes as the pendulum swings:

## Worked Example: Skier on a Slope **Problem:** A skier descends 750 m down a slope at 25° to the horizontal, starting from rest. 15% of initial GPE is not transferred to KE. Calculate final speed. **Solution:** **Step 1: Identify energy transfer** - Initial: 100% GPE at top - Final: 85% converted to KE (15% lost to friction, air resistance, etc.) - Vertical height: $h = 750\sin(25°)$ **Step 2: Set up energy equation** $$E_k = 0.85 \times E_p$$ $$\frac{1}{2}mv^2 = 0.85 \times mgh$$ **Step 3: Solve for velocity** - Mass cancels: $$\frac{1}{2}v^2 = 0.85 \times gh$$ $$v^2 = 2 \times 0.85 \times gh$$ $$v = \sqrt{0.85 \times 2gh}$$ **Step 4: Calculate** $$v = \sqrt{0.85 \times 2 \times 9.81 \times 750\sin(25°)}$$ $$v = 73 \text{ m s}^{-1}$$ The energy breakdown for this problem shows how energy is distributed:

## Key Points for Problem Solving - Always identify which forms of energy are present at the start and end of a process - Remember that energy can be stored in multiple forms simultaneously - Consider whether resistive forces are present (friction, air resistance, etc.) - If present, some mechanical energy converts to thermal energy - If negligible, mechanical energy is fully conserved - The mass often cancels when using conservation equations - Familiarize yourself with common energy transfer scenarios (falling objects, pendulums, springs, projectiles) - Energy is a scalar quantity - no negative values when calculating "loss" of energy