Conservation of Energy and Mechanical Advantage (4A) - MCAT Chemical and Physical Foundations of Biological Systems

$MA = 1$. A fixed pulley changes force direction but not magnitude, resulting in no net force amplification ideally.

$W_{in} = W_{out}$. In ideal machines without friction, energy conservation implies equal input and output work despite force-distance trade-offs.

$W_{net} = \Delta K$. The theorem states that net work on an object equals its change in kinetic energy, derived from Newton's second law integrated over displacement.

$W = Fd\cos\theta$. Work measures energy transfer by force along displacement, with $\cos\theta$ accounting for the component parallel to motion.

$P = \frac{W}{\Delta t}$. Power represents the rate of work done, averaging energy transfer over time for non-instantaneous processes.

$P = \vec{F}\cdot\vec{v}$. Instantaneous power is the dot product capturing the rate of energy transfer at a specific moment for varying forces or velocities.

Negative work. Kinetic friction opposes motion, so the angle between force and displacement is 180°, yielding a negative cosine and energy dissipation.

$f_k = \mu_k N$. Kinetic friction is proportional to the normal force, with $\mu_k$ as the coefficient depending on surface properties.

$W_{nc} = \Delta E_{mech}$. Nonconservative forces like friction change mechanical energy through path-dependent work, equaling the net gain or loss in $E_{mech}$.

$K_i + U_i + W_{nc} = K_f + U_f$. This equation accounts for energy input or dissipation by nonconservative forces, balancing initial and final mechanical energies.

$U_s = \frac{1}{2}kx^2$. Elastic potential energy derives from Hooke's law, integrating force over displacement to store energy quadratically with deformation.

$IMA = \frac{d_{in}}{d_{out}}$. Ideal mechanical advantage assumes no energy losses, equating to the ratio of input to output distances based on geometry.

Only conservative forces do work (no nonconservative work). Conservation holds when work is path-independent, as conservative forces like gravity store energy as potential without dissipation.

$U_g = mgh$. This formula quantifies the energy stored due to an object's position in a gravitational field, derived from the work done against gravity over height $h$.

$K = \frac{1}{2}mv^2$. This equation calculates the energy of motion, arising from integrating force over distance for constant acceleration.

$E_{mech} = K + U$. Mechanical energy sums kinetic and potential energies, representing the total energy convertible between motion and position without losses.

$\eta = \frac{W_{out}}{W_{in}}$. Efficiency quantifies the fraction of input work converted to useful output, less than 1 due to losses like friction.

$\eta = \frac{AMA}{IMA}$. This relates actual performance (AMA) to theoretical maximum (IMA), highlighting losses in real machines.

$MA = 2$. The movable pulley halves the required input force by distributing the load across two rope segments ideally.

$MA = n$. In a block-and-tackle system, mechanical advantage equals the number of load-supporting ropes, amplifying force proportionally.

$v = \sqrt{2gh}$. Conservation of energy converts gravitational potential to kinetic, yielding speed independent of mass for free fall near Earth.

$W_f = -\mu_k mgd$. Friction work is negative as it opposes motion, dissipating energy proportional to coefficient, weight, and distance on a flat surface.

$F_{in} = \frac{F_{out}}{4}$. For ideal machines, mechanical advantage inversely relates input and output forces, requiring one-fourth force to lift with MA=4.

$x = v\sqrt{\frac{m}{k}}$. Equating initial kinetic energy to stored elastic potential gives the deformation where the spring constant balances mass and velocity.