Summary of Work, Energy, and Power

## Introduction Work and Energy are central concepts in physics that explain how forces move objects and how energy is stored, transferred, and converted. This material presents clear definitions, formulas, worked examples, and practice ideas suitable for a student studying independently. > **Definition:** Energy is the ability to do work. Work is the energy transferred to or from an object by a force acting over a displacement. ## Key concepts overview - **Energy** is a scalar quantity and is conserved in isolated systems (total energy constant). - **Work** relates a force applied over a displacement to energy transferred. - **Power** measures how quickly work (or energy transfer) happens. ## 1. Types of energy ### Kinetic energy > **Definition:** Kinetic energy is the energy associated with motion. The kinetic energy of a particle of mass $m$ moving at speed $v$ is $$K = \frac{1}{2} m v^{2}$$ ### Potential energy > **Definition:** Potential energy is stored energy due to position or configuration. Common forms: - Gravitational potential near Earth's surface: $$U_{g} = m g h$$ - Elastic (spring) potential: $$U_{s} = \frac{1}{2} k x^{2}$$ ### Mechanical energy > **Definition:** Mechanical energy is the sum of kinetic and potential energies: $$E = K + U$$ Fun fact: Energy takes many forms such as thermal, chemical, nuclear, and electromagnetic, but mechanical energy (kinetic + potential) is often enough to analyze many everyday problems. ## 2. Work: definition and formulas > **Definition:** Work $W$ done by a constant force $\mathbf{F}$ acting through a displacement $\mathbf{d}$ is the dot product of force and displacement. For a constant force at angle $\theta$ to the displacement: $$W = F d \cos\theta$$ In one dimension, if the force component along displacement is $F_{x}$ and displacement is $\Delta x$: $$W = F_{x} \Delta x$$ Units: joule (J), where $1\ \text{J} = 1\ \text{N}\cdot\text{m}$. ### Positive and negative work - Work is positive when the force has a component in the direction of motion (energy is added to the object). - Work is negative when the force opposes motion (energy is removed from the object). ## 3. Work done by variable forces For variable forces (e.g., springs), work equals the area under the force versus displacement curve. For a spring force $F(x) = -kx$, the work done by the spring when moving from $x_{1}$ to $x_{2}$ is $$W = \int_{x_{1}}^{x_{2}} (-k x)\, dx$$ which yields the change in elastic potential energy. ## 4. Work–Energy Principle > **Statement:** The net work done on an object equals its change in kinetic energy. $$W_{\text{net}} = \Delta K = K_{2} - K_{1}$$ This is a powerful tool to relate forces and motion without directly solving Newton's second law differential equations. ### Extended form including potential energy If conservative forces are present, their work can be expressed via potential energy: $$W_{C} = -\Delta U$$ For nonconservative forces $W_{NC}$, the general relation becomes: $$W_{NC} = \Delta K + \Delta U$$ If no nonconservative forces do work ($W_{NC}=0$), mechanical energy is conserved: $$\Delta K + \Delta U = 0 \quad\Rightarrow\quad E_{1} = E_{2}$$ ## 5. Conservation of mechanical energy When only conservative forces (e.g., gravity, ideal springs) act: $$K_{1} + U_{1} = K_{2} + U_{2}$$ Applications: - Free fall and pendulum motion. - Roller-coaster speed estimates using height changes. - Mass-spring conversions between $\frac{1}{2}kx^{2}$ and $\frac{1}{2}mv^{2}$. Example (roller coaster): starting from rest at height $h$, speed at bottom $y=0$ is given by $$\frac{1}{2} m v^{2} = m g h \;\Rightarrow\; v = \sqrt{2 g h}$$ Did you know that the maximum speed of a frictionless roller coaster starting from rest at height $h$ depends only on $h$ and $g$, not on the shape of the track? ## 6. Power > **Definition:** Power $P$ is the rate of doing work or the rate of energy transfer: $$P = \frac{W}{t}$$ Average power is $\overline{P} = W/\Delta