MCAT Physics and Math

Free Body Diagrams

Representations of the forces acting on an object.

Translational Equilibrium

Occurs in the absence of any net forces acting on an object.

Rotational Equilibrium

Occurs in the absence of any net torques acting on an object, where the center of mass is the most commonly used pivot point.

Vectors

Physical quantities that have both magnitude and direction, such as displacement, velocity, acceleration, and force.

Scalars

Quantities without direction, which may be the magnitude of vectors (like speed) or dimensionless (like coefficients of friction).

Dot Product

$A \cdot B = |A||B| \cos(\theta)$, resulting in a scalar quantity.

Cross Product

$A \times B = |A||B| \sin(\theta)$, resulting in a new vector with a direction found using the right-hand rule.

Displacement

A vector representation of a change in position that is path independent.

Distance

A scalar quantity that reflects the path traveled.

Velocity

A vector representation of the change in displacement with respect to time.

Average Speed

$\text{Total distance traveled} / \text{total time}$.

Instantaneous Velocity

The change in displacement over time as the time approaches $0$.

Force

Any push or pull that has the potential to result in an acceleration.

Gravity

The attractive force between two objects as a result of their masses.

Friction

A force that opposes motion as a function of electrostatic interactions at the surfaces between two objects.

Mass

A measure of the inertia of an object; its amount of material.

Weight

The force experienced by a given mass due to the gravitational attraction to the Earth.

Acceleration

The vector representation of the change in velocity over time.

Torque

A twisting force that causes rotation, calculated as $\tau = r F \sin(\theta)$, where counterclockwise is positive and clockwise is negative.

Newton’s First Law

An object will remain at rest or move with a constant velocity if there is no net force on the object ($F_{net} = m a = 0$).

Newton’s Second Law

Any acceleration is the result of a net force greater than $0$ ($F_{net} = m a$).

Newton’s Third Law

Any two objects interacting experience equal and opposite forces ($F_{AB} = -F_{BA}$).

Linear Motion

Includes free fall and motion in which the velocity and acceleration vectors are parallel or antiparallel.

Projectile Motion

Contains both an x- and y-component where, assuming negligible air resistance, the only force acting is gravity and X velocity is constant.

Centripetal Force

A force vector pointing radially inward in circular motion, calculated as $F_c = \frac{m v^2}{r}$.

Energy

The property of a system that enables it to do something or make something happen, including the capacity to do work; SI units are joules ($J = kg \cdot m^2 / s^2$).

Kinetic Energy

Energy associated with the movement of objects, depending on mass and speed squared ($KE = \frac{1}{2} m v^2$).

Gravitational Potential Energy

Related to the mass of an object and its height above a zero point ($U = m g h$).

Elastic Potential Energy

Related to the spring constant and the degree of stretch or compression squared ($U = \frac{1}{2} k x^2$).

Conservative Forces

Path independent forces that do not dissipate mechanical energy, such as gravity and electrostatic forces.

Nonconservative Forces

Path dependent forces that cause dissipation of mechanical energy, such as friction, air resistance, and viscous drag.

Work

The process by which energy is transferred from one system to another ($W = F d = F d \cos(\theta)$).

Power

The rate at which work is done or energy is transferred, measured in watts ($W = \frac{J}{s}$).

Work-Energy Theorem

When net work is done on or by a system, the system’s kinetic energy will change by the same amount ($W_{net} = \Delta K = K_f - K_i$).

Mechanical Advantage

The factor by which a simple machine multiplies the input force to accomplish work.

Efficiency

The ratio of a machine's work output to work input when nonconservative forces are taken into account.

Thermal Equilibrium

Occurs when systems have the same average KE and temperature, resulting in no heat transfer.

Temperature

The average kinetic energy of the particles that make up a substance.

Thermal Expansion

How a substance changes in length ($\Delta L = \alpha L \Delta T$) or volume ($\Delta V = \beta V \Delta T$) based on temperature change.

First Law of Thermodynamics

A statement of conservation of energy where $\Delta U = Q - W$.

Heat

The energy transfer between two objects at different temperatures until they reach thermal equilibrium ($q = m c \Delta T$).

Specific Heat

The amount of energy necessary to raise one gram of a substance by $1^{\circ} C$ or $1\,K$ (H2O = $4.184\,J/g \cdot K$).

Heat of Transformation

The energy required for a phase change where temperature does not change ($q = m L$).

Isobaric Process

A thermodynamic process where pressure is constant ($\Delta P = 0$).

Isothermal Process

A thermodynamic process where temperature is constant ($\Delta U = 0$).

Adiabatic Process

A thermodynamic process where no heat is exchanged ($Q = 0$).

Isovolumetric Process

Also called isochoric; volume is constant ($\Delta V = 0$), so work is zero.

Entropy

A measure of how much energy has spread out or how spread out energy has become.

Second Law of Thermodynamics

In a closed system, energy will spontaneously and irreversibly go from being localized to being spread out.

Isolated System

A system that does not exchange matter or energy with its surroundings.

State Functions

Pathway independent variables including pressure, density, temperature, volume, enthalpy, internal energy, Gibbs free energy, and entropy.

Process Functions

Variables that describe the pathway from one equilibrium state to another, including work and heat.

Density

Mass per unit volume of a substance ($\rho = \frac{m}{V}$).

Pressure

A scalar measure of force per unit area ($P = \frac{F}{A}$) which is always perpendicular to container walls for a gas.

Absolute Pressure

The sum of all pressures at a point within a fluid ($P_{total} = P_0 + \rho g h$).

Gauge Pressure

The difference between absolute pressure and atmospheric pressure ($P_{gauge} = P - P_{atm}$).

Viscosity

A measure of a fluid's internal friction; it generates a nonconservative force called viscous drag.

Laminar Flow

Smooth and orderly fluid flow.

Poiseuille’s Law

Determines the rate of laminar flow with an inverse exponential relationship between radius and pressure gradient to the fourth power.

Continuity Equation

States that fluids flow more quickly through narrow passages and slowly through wide ones ($Q = v_1 A_1 = v_2 A_2$).

Bernoulli’s Equation

The sum of static pressure and dynamic pressure is constant between two points in a closed system.

Venturi Effect

The velocity of a fluid in a constricted area increases while its static pressure decreases.

Pascal’s Principle

Pressure applied to an incompressible fluid is distributed undiminished throughout the entire volume.

Archimedes’ Principle

An object in a fluid experiences a buoyant force equal to the weight of the fluid it displaces ($F_b = \rho V g$).

Specific Gravity

The ratio of the density of an object to the density of water.

Cohesive Forces

Forces experienced between molecules of the same fluid, giving rise to surface tension.

Adhesive Forces

Forces experienced between fluid molecules and other materials.

Fundamental Unit of Charge

$e = 1.60 \times 10^{-19}\,C$, possessed by both protons and electrons.

Conductors

Materials that allow the free and uniform passage of electrons when charged.

Insulators

Materials that resist the movement of charge, resulting in localized areas of charge.

Coulomb’s Law

Calculates the magnitude of the electrostatic force between two charges ($F = k \frac{|q_1 q_2|}{r^2}$).

Electric Field

Generated by every charge; the force exerted on a test charge divided by the magnitude of that charge ($E = \frac{F_e}{q} = k \frac{Q}{r^2}$).

Electrical Potential Energy

The work required to bring a test charge from infinitely far away to a given position ($U = k \frac{Q q}{r}$).

Electrical Potential

The electrical potential energy per unit charge ($V = \frac{U}{q}$), measured in volts ($1\,V = 1\,J/C$).

Voltage

The potential difference ($\Delta V$) that accompanies the movement of a test charge between positions.

Equipotential Lines

Lines where the potential is the same at every point, always perpendicular to electric field lines.

Electric Dipole

Two charges of opposite sign separated by a fixed distance $d$; experiences net torque until aligned with an external electric field.

Dipole Moment

The product of charge and separation distance ($p = q d$).

Magnetic Field

Field created by magnets and moving charges, measured in tesla ($1\,T = 10,000\text{ gauss}$).

Diamagnetic Materials

Materials with no unpaired electrons that are slightly repelled by magnets.

Paramagnetic Materials

Materials with some unpaired electrons that become weakly magnetic in an external field.

Ferromagnetic Materials

Materials with some unpaired electrons that become strongly magnetic in an external field.

Lorentz Force

The sum of the electrostatic and magnetic forces acting on a body.

Current

The movement of charge between points with different electrical potentials, defined by convention as the movement of positive charge ($I = \frac{Q}{\Delta t}$).

Kirchhoff’s Junction Rule

The sum of currents flowing into a junction equals the sum of currents flowing out.

Kirchhoff’s Loop Rule

In a closed loop, the sum of voltage sources equals the sum of voltage drops.

Resistance

The opposition to the flow of electrons, calculated as $R = \rho \frac{L}{A}$.

Ohm’s Law

$V = I R$; current magnitude is proportional to the voltage drop for a given resistance.

Capacitance

The ability to store and discharge electrical potential energy, determined by plate area and distance ($C = \frac{Q}{V}$).

Dielectric Materials

Insulators that increase a capacitor's capacitance by a factor of the dielectric constant ($k$).

Ammeters

Meters inserted in series to measure current with negligible resistance.

Voltmeters

Meters inserted in parallel to measure voltage drop with very large resistance.

Ohmmeters

Self-powered meters inserted around a resistive element to measure resistance with negligible resistance.

Transverse Waves

Waves with oscillations perpendicular to direction of propagation, such as light.

Longitudinal Waves

Waves with oscillations parallel to direction of propagation, such as sound.

Amplitude

The magnitude of maximal displacement in a wave.

Frequency

The number of cycles a wave makes per second, expressed in Hz.

Constructive Interference

Occurs when waves are in phase, resulting in an amplitude equal to the sum of individual amplitudes.

Destructive Interference

Occurs when waves are out of phase, resulting in an amplitude equal to the difference between individual amplitudes.

Standing Waves

Produced by interference of two waves of same frequency traveling in opposite directions.