Work, Energy, and Power (4A) - MCAT Chemical and Physical Foundations of Biological Systems

$f_k = \mu_k N$. Kinetic friction magnitude is proportional to normal force via the coefficient, assuming constant sliding motion.

$1\ \text{J} = 1\ \text{N}\cdot\text{m} = 1\ \text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}$. The joule measures energy transfer, derived from force times distance, equating to base units of mass, length, and time.

$\Delta U_g = mg\Delta h$. Gravitational potential energy change arises from work against gravity, proportional to mass, gravity, and height difference.

$W_{\text{net}} = \Delta K$. Net work done on an object equals its change in kinetic energy, linking force application to motion change.

$K = \frac{1}{2}mv^2$. Kinetic energy quantifies motion, scaling with mass and the square of velocity due to work-energy principles.

Work equals the area under the $F(x)$ vs. $x$ curve. The integral of force with respect to displacement geometrically represents the net energy transfer.

$W = \int F(x),dx$. Integrates force over displacement to compute total work for non-constant forces.

$\theta = 90^\circ$ so $\cos\theta = 0$. Work is zero when force is perpendicular to displacement, as no component acts along the path.

$P_{\text{avg}} = 200\ \text{W}$. Average power divides total work by time interval to determine the mean rate of energy expenditure.

$\Delta U_g = 100\ \text{J}$. Potential energy increase results from work against gravity, calculated as mass times gravitational acceleration times height change.

$W = 15\ \text{J}$. Applies the work formula with the cosine of the angle to find the effective force component along displacement.

$v = \sqrt{\frac{2K}{m}}$. Solving the kinetic energy formula for velocity yields this expression, relating energy directly to speed.

$\eta = \frac{W_{\text{out}}}{W_{\text{in}}} = \frac{E_{\text{useful}}}{E_{\text{in}}}$. Efficiency ratios useful output to total input, indicating the fraction of energy converted effectively without waste.

$W_g = -mg\Delta h$. Gravity performs negative work against upward displacement, reducing potential energy gain.

$f_s\leq \mu_s N$, with $f_{s,\max}=\mu_s N$. Static friction adjusts to prevent motion up to a maximum determined by the coefficient and normal force.

$W_f = -f_k d$. Kinetic friction opposes motion, performing negative work by dissipating energy as heat over the distance traveled.

$1\ \text{W} = 1\ \text{J}\cdot\text{s}^{-1} = 1\ \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}$. The watt quantifies power as energy per time, expressed in base units for consistency in mechanics.

$P = \vec F\cdot\vec v$. Instantaneous power equals the dot product of force and velocity, capturing the parallel component's contribution.

$P_{\text{avg}} = \frac{\Delta W}{\Delta t}$. Average power computes the mean rate of work over a finite interval.

$K_i + U_i + W_{\text{nc}} = K_f + U_f$. Nonconservative forces introduce energy dissipation or addition, modifying the total mechanical energy balance.

$W = Fd\cos\theta$. Calculates work as the component of force parallel to displacement multiplied by distance, accounting for directionality.

$K_i + U_i = K_f + U_f$. Mechanical energy remains constant in isolated systems with only conservative forces, as work done converts between kinetic and potential forms.

$W_{\text{cons}} = -\Delta U$. Conservative forces store work as potential energy, with the negative sign indicating energy conservation in closed paths.

$U_s = \frac{1}{2}kx^2$. Elastic potential energy stores deformation work in a spring, following Hooke's law with quadratic dependence on displacement.