Work and Kinetic Energy Summary

A system consists of objects whose interactions are analyzed, while external objects constitute the environment.
Energy can be transformed within a system without loss or transferred between system and environment via work ( $W$ ) or heat.
Primary forms of energy include kinetic energy ( $K$ ), potential energy ( $U$ ), and thermal energy ( $E_{\text{th}}$ ), measured in joules ( $1\,\text{J} = 1\,\text{kg}\cdot\text{m}^2/\text{s}^2$ ).
The energy principle states that a system's total energy changes by the amount of net external work performed on it: $\Delta E_{\text{sys}} = W_{\text{ext}}$
An isolated system has no interaction with its environment, maintaining constant total energy.

Kinetic energy represents energy of motion: $K = \frac{1}{2} m v^2$
Work ( $W$ ) is the process of transferring energy to or from a system via mechanical forces.
For a single particle, work changes kinetic energy according to the energy principle: $\Delta K = K_f - K_i = W$
Directional characteristics of work:
- $W > 0$ : Force acts in the direction of displacement (speeds up the particle, adding energy).
- $W < 0$ : Force acts opposite to displacement (slows down the particle, removing energy).
- $W = 0$ : Particle has no displacement, or force acts perpendicular to displacement.
For systems with multiple forces, total work is additive: $W_{\text{tot}} = \sum_i W_i$
For multiparticle systems, total kinetic energy is the sum of individual kinetic energies ( $K_{\text{tot}} = \sum K_i$ ), yielding: $\Delta K_{\text{tot}} = W_{\text{tot}}$

Work done by a variable force along a path: $W = \int_{s_i}^{s_f} F_s\,ds$
Graphically, work corresponds to the area under the force-versus-position curve.
Work done by a constant force at an angle $\theta$ relative to displacement $\Delta r$ : $W = F \Delta r \cos(\theta) = \mathbf{F} \cdot \mathbf{\Delta r}$
The dot product (scalar product) of vectors $\mathbf{A}$ and $\mathbf{B}$ separated by angle $\alpha$ : $\mathbf{A} \cdot \mathbf{B} = A B \cos(\alpha) = A_x B_x + A_y B_y$
Zero-work situations include zero displacement ( $\Delta s = 0$ ), forces perpendicular to motion (e.g., uniform circular motion), and forces applied at stationary contact points (e.g., pushing off a wall).

A restoring force acts to return a system to its equilibrium position.
Hooke's law models ideal restoring forces for springs with spring constant $k$ (measured in $\text{N/m}$ ): $(F_{\text{sp}})_s = -k \Delta s$
Spring displacement is defined relative to equilibrium position $s_{\text{eq}}$ : $\Delta s = s - s_{\text{eq}}$ .
Work done by an ideal spring on an object: $W = -\left(\frac{1}{2} k (\Delta s_f)^2 - \frac{1}{2} k (\Delta s_i)^2\right)$

Thermal energy ( $E_{\text{th}}$ ) is microscopic energy comprising atomic kinetic energy ( $K_{\text{micro}}$ ) and molecular bond potential energy ( $U_{\text{micro}}$ ): $E_{\text{th}} = K_{\text{micro}} + U_{\text{micro}}$
Extended energy principle including thermal energy: $\Delta E_{\text{sys}} = \Delta K + \Delta E_{\text{th}} = W_{\text{ext}}$
Dissipative forces (such as kinetic friction $f_k$ or drag) convert macroscopic kinetic energy into thermal energy: $\Delta E_{\text{th}} = f_k \Delta s$

Power ( $P$ ) is the rate at which energy is transferred or transformed: $P = \frac{dE_{\text{sys}}}{dt}$
Power as the rate of doing work: $P = \frac{dW}{dt}$
Power delivered to a particle by force $\mathbf{F}$ moving at velocity $\mathbf{v}$ : $P = \mathbf{F} \cdot \mathbf{v} = F v \cos(\theta)$
Standard units of power: watt ( $1\,\text{W} = 1\,\text{J/s}$ ) and horsepower ( $1\,\text{hp} = 746\,\text{W}$ ).