Newtonian Mechanics

Cheatsheet Content

### Work, Energy & Power (Chapter 4) #### Work ($W$) - **Definition:** Work is done when a force causes a displacement of an object. It is a scalar quantity. - **Formula for Constant Force:** $W = Fd \cos\theta$ - Where: - $F$ = magnitude of the applied force (N) - $d$ = magnitude of the displacement (m) - $\theta$ = angle between the force vector ($\vec{F}$) and the displacement vector ($\vec{d}$) - **Units:** Joules (J). $1 \text{ J} = 1 \text{ N} \cdot \text{m}$. - **Types of Work:** - **Positive Work:** When $0^\circ \le \theta *Conceptual diagram for Work: A force $\vec{F}$ pulls an object, causing a displacement $\vec{d}$. The work done depends on the component of the force parallel to the displacement, given by $F \cos\theta$.* #### Energy ($E$) - **Definition:** The ability to do work. Scalar quantity. Units: Joules (J). - **Kinetic Energy ($KE$):** Energy possessed by an object due to its motion. - $KE = \frac{1}{2}mv^2$ - Where: $m$ = mass (kg), $v$ = speed (m/s). - **Potential Energy ($PE$):** Stored energy due to an object's position or state. - **Gravitational Potential Energy ($GPE$):** Energy stored due to an object's height in a gravitational field. - $GPE = mgh$ - Where: $m$ = mass (kg), $g$ = acceleration due to gravity ($9.8 \text{ m/s}^2$), $h$ = height above a reference level (m). - **Elastic Potential Energy ($EPE$):** Energy stored in an elastic object (e.g., spring) when stretched or compressed. - $EPE = \frac{1}{2}kx^2$ - Where: $k$ = spring constant (N/m), $x$ = displacement from equilibrium position (m). - **Conservation of Mechanical Energy:** In the absence of non-conservative forces (like friction, air resistance), the total mechanical energy ($ME = KE + PE$) of a system remains constant. - $KE_i + PE_i = KE_f + PE_f$ - **Conservation of Total Energy:** Energy cannot be created or destroyed, only transformed from one form to another. Total energy of an isolated system is always conserved. #### Power ($P$) - **Definition:** The rate at which work is done or energy is transferred. Scalar quantity. - **Formula:** $P = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$ - Where: $W$ = work done (J), $\Delta E$ = change in energy (J), $\Delta t$ = time interval (s). - **Alternative Formula (Constant Force & Velocity):** $P = Fv \cos\theta$ - Where: $F$ = force (N), $v$ = velocity (m/s), $\theta$ = angle between force and velocity vectors. - **Units:** Watts (W). $1 \text{ W} = 1 \text{ J/s}$. Another common unit is horsepower ($1 \text{ hp} \approx 746 \text{ W}$).