Arithmetic Reasoning

Basic Math

Numbers belong to sets based on their form:

Set Definition Examples
Whole numbers all positive integers and zero 0, 1, 2, 3
Integers all positive and negative numbers and zero -2, -1, 0, 1, 2
Rational numbers any number that can be written as a fraction with integer numerator and denominator -2/5, -1/3, 0, 1, 4/3
Irrational numbers any number that is not a rational number and that cannot be written as a fraction or as a non repeating decimal Pi (3.1415…), √3 (1.7320…)
Real numbers the set of all rational and irrational numbers  -√7, -1/3, 0, 1, 3/4

Arithmetic operations include finding the sum (adding), finding the difference (subtracting), finding the product (multiplying), and finding the quotient (dividing).

Fractions are composed of a numerator (top number) and a denominator (bottom number) and can be:

  • Proper (numerator smaller than the denominator) like 1/2 and 7/8
  • Improper (numerator equal to or greater than the denominator) like 6/3 and 2/2
  • Mixed with a combination of whole and fractional numbers like 1 1/4 and 3 3/5
  • Equivalent when two fractions identify the same quantity like 1/2 and 2/4

Simplify a fraction by determining what number will divide evenly into both the numerator and the denominator.  For example, with the fraction 4/8 four divides evenly into both 4 and 8 giving 4/8 ÷ 4/4 = 1/2.

To convert an improper fraction to a mixed fraction, divide the numerator by the denominator to get a quotient of one and then write the remainder over the original denominator.  For example, 15/7 = 2 remainder 1 which is 2 1/7.

When adding and subtracting fractions the denominator must be the same for both numbers.  If you have different denominators you need to find the least common denominator by finding a number that both denominators will divide into evenly.

When multiplying and dividing fractions simply multiply the two numerators together to get the numerator of the answer and multiply the two denominators to get the denominator of the answer.  To multiply a mixed fraction, convert the mixed fraction into an improper fraction then multiply.

Always remember to simplify or convert your answer to questions involving fractions as answers on the ASVAB test will always be in simplest form.

Applied Math

Most of the questions in the arithmetic reasoning subtest involve applying math to real world problems like calculating simple interest (interest = principal x interest rate) and compound interest (calculate simple interest for each year making sure to use the new principal + accumulated interest for each year’s new interest calculation).

Series questions are also popular where you need to recognize the pattern in a series of numbers and calculate which number comes next.


The best way to prepare for the arithmetic reasoning portion of the ASVAB is to take practice tests and go through the solution to each question to make sure you understand how it is solved.  This is one area where there really is no substitute for practice.

Electronics Information

Principles of Electronics

An electrical circuit is a path through which electricity flows.  Circuits contain a variety of components and devices to which electricity is applied and flows once the circuit is closed with a complete path.  An open circuit is one where the path is incomplete or interrupted and, therefore, electricity will not flow.

Current is the flow of electrons around an electrical circuit, from a point of high potential (the electricity source) to a point of low potential (the load).  The strength of current is measured in amperes (A) and represented by the symbol I (intensity) in circuit diagrams.

There are two forms of electrical current, direct current (DC) and alternating current (AC).

Direct current is the flow of electrons in one direction, from the negative to the positive terminal of the power source (typically a battery). The higher the voltage (V) of the power source, the more electrons are available to the circuit and the greater the load that can be powered.

With alternating current the flow of electrons periodically reverses direction.  Voltage and current in an AC circuit take the form of a sine wave and is rated by the number of cycles per second (frequency) of this wave in hertz (Hz).

All materials have a certain amount of resistance to the flow of electricity.  The more resistive the material, the slower the flow of electricity will be.  Resistance is measured ohms (Ω) and equal to the voltage across the material divided by the current present in it (V/I).

conductor is a material through which electrical current can flow with little resistance.  An insulator is a material that resists electrical current and a semiconductor is a material that has conductivity partway between a conductor and an insulator with the primary advantage of showing increased conductivity with increased temperature unlike conductors where conductivity decreases with increased temperature.

Ohm’s Law

A set relationship exists between voltage (V), resistance (R), and current (I) in an electrical circuit:

  • Current is voltage ÷ resistance:  I = V / R
  • Resistance voltage ÷ current:  R = V / I
  • Voltage is resistance x current:  V = I x R

The majority of the problems on the Electronics Information subtest involve calculating current, resistance, or voltage by applying Ohm’s law.

Power consumption is another important factor to consider in electrical circuits.  Power represents the overall amount of electricity that a device needs to operate and is measured in watts (w).  One watt equals one ampere multiplied by one volt or P = V x I.

Electrical Components

Batteries are the most common DC power source and the most common types of batteries are dry cell (AA, C, D, etc.), wet cell (used in cars), and storage (backup and emergency batteries).  The long bar of the battery schematic is the positive terminal and the short bar is the negative terminal.

Batteries can be connected in series to provide voltage equal to the sum of their individual voltages with a current equal to the current that a single battery produces.  Connect batteries in parallel to provide current equal to the sum of their individual currents with a voltage equal to the voltage that a single battery produces.  To increase both voltage and current, connect batteries in a series-parallel configuration.

Photovoltaic or solar cell batteries convert energy from the sun into electrical energy.  Made of semiconductors, solar cells are often used to charge other types of batteries as their output depends on how much light strikes their collection panels and that can vary greatly across the length of the day.

Electrical wiring is used to connect circuits and equipment and different electrical applications require different types and sizes of wire.  When choosing wire size, the principal factor is the amount of current that will be traveling through the wire – the higher the current, the larger gauge wire required to prevent the wire from overheating.

Often, electrical wires need to be connected (spliced) to each other and that’s most commonly done via soldering where a tin-lead mixture is applied to the wire ends and allowed to cool, producing a connection that maintains the conductivity and strength of the original wires.

Coaxial cable is another type of electrical wiring but, instead of transmitting electricity, it transmits radio frequency.  Constructed with a flexible insulating layer surrounding fine woven wire, coax is used in computer networks, antenna connections, and cable/satellite wiring.

A resistor is designed to limit the amount of current flowing through a circuit and is used to precisely control the amount of voltage and current present at various points within an electrical circuit.  When current flows through a resistor the resistor produces heat and managing heat within the circuit is a very important design consideration for engineers.

capacitor stores an electric charge using two parallel plates with a a nonconducting material (dielectric) between them.  Capacitors can provide a fixed or variable capacitance in a circuit.  Capacitance is measured in farads (F) and depends on the surface area of the plates (larger means higher capacitance), their distance apart (closer means higher capacitance), and the type of dielectric.

switch is a device for opening and closing an electric circuit – when the switch is open, no current flows.  Switches are categorized by the number of poles and throws they provide with the number of poles designating the number of electrical circuits the switch can control and the number of throws indicating the number of connections the switch can make between those circuits.

Electromagnets are artificial magnets that are made by passing electrical current through a coil of wire wrapped around an iron core.  If a conductor moves through a magnetic field, the field will induce a voltage across it.

An inductor stores energy in the form of a magnetic field and inductance is the property of a component that opposes the change in current of an AC circuit.  An inductor is often formed by wrapping a wire coil around a magnetic core and the level of inductance is directly proportional to the number of turns in the coil, its radius, and the type of material in the core.

A transformer changes AC voltage to a higher or lower potential by using an inductor. The AC voltage is connected to one coil (called the primary winding) which induces voltage across another coil (called the secondary winding) which connects to the load. The step-up or step-down in voltage depends on the ratio of the number of turns in the primary winding compared to the secondary winding and is given by the formula turn ratio = Nprimary ÷ Nsecondary

diode is a semiconductor that allows electric current to flow in only one direction and blocks current flow in the other direction.  In the symbol for a diode the direction of current flow is indicated by the point of the arrow and the vertical line indicates the direction from which current is blocked.

rectifier converts AC voltage to DC voltage while an inverter converts DC voltage to AC voltage.

thermocouple is a temperature sensitive switch that will open or close a circuit above a certain temperature.

General Science


Matter is anything that has mass and occupies space and exists in four primary states:

  • Solids have a definite mass, volume, and shape.
  • Liquids have a definite mass and volume but variable shape.
  • Gas has a definite mass but variable volume and shape.
  • Plasma is a high-energy electrically charged mix of ions and electrons that is usually in the form of a gas.

All matter is made up of atoms which are made up of the following particles:

  • Protons have a positive charge and are located in the nucleus.
  • Neutrons have a neutral charge and are located in the nucleus.
  • Electrons have a negative charge and are located in shells that orbit the nucleus.

Atoms are defined by their proton count and those with full outer electron shells don’t bond well with other atoms while atoms with relatively empty outer shells do.

Elements are substances that are made up of only one kind of atom. They’re grouped in the Periodic Table of Elements in increasing atomic number by row with elements having similar properties in the same column. Elements are divided into three types:

  • Metals are the most common elements and are typically hard, have high electrical and thermal conductivity, and a high melting point.
  • Nonmetals are typically soft, have low electrical and thermal conductivity, and lower melting and boiling points. Common nonmetals are gases.
  • Noble gases have high stability and low reaction rates with other elements.

Isotopes are atoms of the same element that have different numbers of neutrons.

A compound is created when two or more different types of elements combine to form a different material.  Elements that chemically bond by transferring electrons from one atom to another form ionic bonds while covalent bonds occur when electrons are shared between atoms.

A solution is a uniform stable mixture of a solute that’s been dissolved into a solvent. There is a limit to the amount of solute that can be dissolved by a solvent at a given temperature and this limit is called the saturation point. Solvents can be supersaturated with a solute which results in an unstable solution that will precipitate when disturbed causing the excess solute to settle out of the solution.

Acids are substances that give up an ion in a water solution. They have low pH numbers, are sour tasting, and turn blue litmus paper red. Bases (or alkalis) take on an ion in a water solution, have high pH numbers, are bitter tasting, and turn red litmus paper blue.

pH is a measure of the relative acidity or alkalinity of a solution.  Alkalinity is determined by concentration of hydrogen ions and is measured on a pH scale of values from 0 to 14.  A value of 0 is very acidic, 7 is neutral, and 14 is very alkaline.  A common neutral liquid with a pH of 7.0 is pure water.


Term Definition Measure
Work product of an object’s displacement and the force used to move the object Joule
Force push or pull that makes an object move Newton
Energy ability to do work Joule
Kinetic energy energy in motion Joule
Potential energy stored energy resulting from an object’s position Joule
Power rate at which work is done Watt
Speed distance traveled per unit of time (scalar) MpH
Velocity distance traveled per unit of time in a direction (vector) m/s
Acceleration change in velocity of an object over the amount of time required to make that change m/s2
Mass quantity of matter contained in an object (constant) lb, kb
Weight effect of gravity acting upon mass (varies with gravitational pull) Newton

Also called heat, thermal energy is created by the movement of molecules and flows from hotter to cooler substances when they come into contact.  Heat is measured in calories and can be transferred by:

  • Convection is heat transfer within a substance where hotter / less dense material displaces colder / more dense material.
  • Conduction is heat transfer between substances.
  • Radiation is heat transfer via electromagnetic waves without direct contact between substances.

Sound travels in regular waves from a source and can be both audible and inaudible.  Waves are specified by measuring the following characteristics:

  • Frequency is the number of complete wavelength cycles per second and is measured in hertz.
  • Amplitude is the maximum height of a wavelength above the average value of the wave.
  • Wavelength is the distance between corresponding points in adjacent wave cycles.

The doppler effect is generated when the source or the observer of sound is moving.  When a sound is approaching an observer its frequency appears to increase as it gets closer and change as it passes the observer and moves away.  This characteristic is what makes radar and sonar effective in indicating the size of and distance to an object via measuring reflected sound.

Light is transmitted via waves of different wavelength and frequency (perceived as color). Light is refracted when it passes from one medium into another and different wavelengths of light bend at different angles.  When light is reflected from a surface the angle at which light strikes the surface is always equal to the angle at which it is reflected.


Taxonomy is the classification of all living things according to shared characteristics, physical structures, and genetic traits.  Called the Linnaean System, taxonomy specifies a classification hierarchy from broadest (generally the same) to narrowest (completely the same) groupings:

  • Kingdom
  • Phylum
  • Class
  • Order
  • Family
  • Genus
  • Species

Organisms are recognized as belonging to one of five distinct kingdoms:

Kingdom Characteristics
Monera Single celled organisms with no true nucleus
Protista Single celled organisms with a true nucleus
Fungi Multi-celled organisms that eat by external digestion followed by absorption
Plantae Multi-celled organisms that make their own food through photosynthesis
Animalia Multi-celled organisms that find and eat their food

Carnivores are meat eaters.  Herbivores are plant eaters.  Omnivores consume both meat and plants.

Cells are the smallest unit of life and consists of cytoplasm surrounded by a plasma membrane.  The cytoplasm contain compartmented sections called organelles that do work for the cell.

Organelle Function
Nucleus Controls cell functions, contains chromosomes
Endoplasmic reticulum Manufactures proteins and lipids
Golgi apparatus Transports proteins
Lysosomes Recycles materials
Ribosomes Synthesizes proteins
Mitochondria Extracts energy from carbohydrates
Vacuole Stores waste and other materials (this is where food is digested and then absorbed into the protoplasm)
Chloroplast (plant cell) Performs photosynthesis (converts solar energy into chemical energy)
Cell walls (plant cell) Supports and protects the cell

Photosynthesis is the process by which plants create their own food via chloroplasts which contain a green light-absorbing pigment called chlorophyll.  Solar energy from light is then used to convert carbon dioxide and water into carbohydrates and glucose which the plant uses for food.

Cells reproduction is made possible via chromosomes which contain deoxyribonucleic acid (DNA) strands which provide a set of instructions for all cell function. When the cell divides, DNA is replicated allowing the new cell to immediately exhibit all the characteristics of the original cell.

Cells replicate by mitosis in which a cell duplicates its chromosomes and then splits in two resulting in both cells receiving a set of chromosomes that perfectly matches the parent cell or through meiosis which reduces the number of chromosomes in the parent cell by half and produces four gamete cells. This process is required to produce egg and sperm cells for sexual reproduction.

Anatomy deals with the study of the structure of an organism and physiology focuses on the functions and processes of an organism’s bodily systems:

System Function
Musculoskeletal provides strength, protects internal organs, produces blood cells
Nervous neurons receive stimuli from their surroundings and transmit those stimuli to the brain via electrical impulses
Endocrine controls the body through the release of hormones which influence the function of specific organs and processes
Respiratory intakes and filters air, extracts oxygen, then transfers it to the bloodstream for use by the body
Circulatory transports oxygen and nutrients throughout the body and removes waste
Lymphatic returns fluids to the blood and helps fight infections
Digestive ingests and breaks down food into nutrients that are then absorbed into the body
Excretory disposes of bodily waste
Reproductive produce and transport sperm (male) or ovum (female) used in reproduction

Planetary Science

Ecology is the study of interactions between organisms and their environment. Regions that share a similar climate and soil type (and, therefore, similar animal and plant life) are called biomes and house a community of species that each belong to a population.  This combination of biome and community make up an ecosystem that consists of predators that utilize other living organisms as food, prey that becomes food to predators, and parasites that lives in or on another organism and takes nutrients from its host.

Symbiosis is a mutually beneficial relationship between two or more species.

The biosphere of the Earth is divided into seven biomes:

Biome Characteristics
Tundra northern location, permanently frozen soil, short rooted plants and shrubs, vast plains, short summers, long winters
Taiga south of tundra, evergreen trees, long and severe winters
Deciduous forest south of taiga, leafy trees, milder climate with four distinct seasons
Savannah central grasslands, few trees, sporadic rainfall
Tropical rainforest high humidity, high rainfall, high temperature, dense vegetation
Desert extremely low humidity and precipitation, scarce plant life, large daily temperature swings, scarce water supply
Marine covered in water, diverse animal and plant species

The food chain describes the interdependency of the various organisms within a community.  The food chain is made up of producers that make their own food and consumers that cannot produce their own food and feed on producers.  Primary consumers feed directly on producers, secondary consumers feed on primary consumers and producers, and tertiary consumers feed on all organisms lower in the food chain.  Also in the food chain are decomposers like bacteria and fungi which break down all members of the food chain to recycle their organic compounds.

Geology is devoted to the study of the Earth which consists of an outermost layer called the crust made of rock, a mantle directly below the crust which varies from rigid rock near the crust to molten rock near the core which consists of a fluid outer core and a solid inner core.  The core is responsible for generating the Earth’s magnetic field.

The crust contains three major types of rocks which are classified based on how they’re formed:

  • Igneous rock is formed by cooling lava
  • Sedimentary rock is formed by the compacting and cementing together of sediment (small rocks, minerals, and organics)
  • Metamorphic rock is formed by extreme heat, pressure, or chemical action on existing rock

By analyzing rock deposits and fossils, geologists have been able to break down Earth’s history and development into a timeline of major geological eras:

Era Timeline Major Features
Precambrian 4.6 billion to 570 million years ago violent volcanic activity, primitive life forms
Paleozoic 570 million to 245 million years ago continents formed, aquatic animals and land plants emerged
Mesozoic 245 million to 66 million years ago dinosaurs lived then became extinct
Cenozoic 66 million years ago to today mammals developed and flourish

Meteorology is the study of climate and weather and the Earth’s atmosphere.  The atmosphere is composed of four layers:

  • Troposphere which extends from the surface to 10 miles above the surface and contains most of the Earth’s clouds and weather. As altitude increases, temperature and pressure decrease.
  • Stratosphere from 10 to 30 miles above the surface and contains the ozone layer. Has very little water and while pressure continues to decrease with altitude, temperature increases.
  • Mesosphere from 30 to 50 miles above the surface where both temperature and pressure again decrease with altitude.
  • Thermosphere from 50 to 300 miles above the surface where temperature increases dramatically and pressure decreases with altitude. Contains the ionosphere.

Clouds are formed when water vapor in warm air currents cools as the currents rise.  When the temperature of the vapor reaches the dew point, it condenses into water droplets or ice crystals and forms clouds.  When the droplets or crystals become heavy enough, they precipitate our of the cloud as rain or snow.

Clouds are divided into three basic types.  Stratus clouds are flat and layered, cumulus are clumped clouds with elongated tall shapes, and cirrus clouds are striped and wispy high-altitude clouds.  Clouds can be a combination of these types like cumulonimbus (thunderstorm) clouds.

When two air masses of differing temperature and/or humidity collide, the line that separates them is called a front.  A warm front is formed when the leading edge of the colliding air mass is warmer than the air it is displacing and a cold front when the leading edge is cooler.  Front movement is characterized by wind and meteorologists use an anemometer to measure wind speed and a barometer to measure air pressure when forecasting the weather.

Astronomy is the study of space and the universe, starting with our solar system which contains the following planetary bodies in order from closest to farthest away from the sun:

  • Mercury
  • Venus
  • Earth
  • Mars
  • Jupiter
  • Uranus
  • Neptune
  • Pluto

star is a large gaseous body and planets and other material orbit stars and other massive objects with the shape of an orbit determined by the strength of the gravitational pull of the object that is being orbited.  A galaxy is a large cluster of stars held together by gravity where distance is measured in multiples of a light year which is the distance light travels in one year and is approximately 5.9 trillion miles.

Math Knowledge

Math Operations

All whole numbers can be classified as either prime or composite numbers.  Composite numbers can be expressed as the product of two or more whole number factors (excluding the number itself and one) while prime numbers have only two factors, the number itself and one.

The greatest common factor of two numbers is the largest number that can divide equally into both numbers.  For example, the factors of 18 are {1, 2, 3, 6, 9, 18} and the factors of 15 are {1, 3, 5} so the greatest common factor of 18 and 15 is 3.

The multiples of a number are the set of numbers formed by multiplying a number by other numbers.  For example, the first five multiples of 3 are {3, 6, 9, 12, 15}.  The least common multiple of two numbers is the smallest number that both numbers factor into evenly.  The first five multiples of of 4 are {4, 8, 12, 16, 20} making the least common multiple of 3 and 4 equal to 12.

Denoted by a number followed by an exclamation point, a factorial is the product of all whole numbers from 1 through that number.  For example, 4! is 4 x 3 x 2 x 1 = 24.

A number in superscript next to a base (for example, 23) is called an exponent and represents the number of times (3) that the base (2) will be multiplied by itself.  Another way of expressing a number with an exponent of two is squared and with an exponent of three, cubed so 23 is two cubed and 42 is four squared.

The inverse of an exponent is the root which is indicated by a radical sign √.  A root consists of a radicand which is the number for which you’re finding the root and an index indicating which root you’re finding.  For example, in 3√8 the number 8 is the radicand and the number 3 is the root.  (A radical sign with no specified index is assumed to have an index of 2.)

When adding or subtracting signed numbers with the same sign, the answer will always have the sign of the two numbers.  When the signs are different, subtract the smaller number from the larger number and give the answer the sign of the larger number.  When multiplying and dividing signed numbers, the answer will be negative if one of the numbers is negative and positive if the signs are the same.

The order of operations defines in what order you perform operations in a complex equation to ensure that you get the correct result.  From first to last, the order of operations is:

  • Operations inside parentheses and brackets
  • Exponents and roots
  • Multiplication
  • Division
  • Addition
  • Subtraction


Algebraic equations contain a mix of constants (numbers), variables (letters that represent unknown numbers), and coefficients (a constant multiplied by a variable).  If a term uses one or more variables it is called a polynomial.

Algebraic properties describe mathematical relationships that can help you solve equations:

  • Distributive property: when you have an equation that contains both multiplication and addition you can distribute the multipliers.  For example, (2+4)(3+5) = 2(3+5) + 4(3+5).
  • Commutative property: you can add or multiply numbers in any order.  For example, 2 x 4 x 7 = 7 x 2 x 4.
  • Associative property:  when adding or multiplying you can group numbers in any order.  For example, (3+5) + 7 = 3 + (5+7)
  • Property of inequality:  if two numbers are equal you can add or multiply those numbers by the same value to get the same result.  For example, if 4 = 2y then 4(7) = 2y(7).
  • Additive identity:  a number or variable added to zero equals that number or variable.  For example, 7y + 0 = 7y.
  • Multiplicative identity:  a number or variable multiplied by one equals that number or variable.  For example, 7y x 1 = 7y.
  • Additive inverse:  a number or variable added to its inverse equals zero.  For example, 7y + (-7y) = 0.
  • Multiplicative inverse:  a number or variable multiplied by its reciprocal equals one.  For example, (7y)(1/7y) = 1.

When solving equations with inequalities like <, >, ≤, and ≥ make sure to reverse the direction of the inequality when you multiply or divide by both sides of the inequality.  If you add or subtract the inequality symbol remains the same.


line is a connection of points that go in a straight path in only two directions.  A ray is a line that has one defined endpoint but other end goes infinitely in a straight path.  Parallel lines are equidistant lines that never cross and perpendicular lines are lines that intersect at a right angle (90°).

An angle is the point at which two rays intersect:

Angle Degrees Example
Line 180°  
Circle 360° Screen Shot 2016-07-10 at 9.01.48 AM
 Right 90° Screen Shot 2016-07-10 at 9.01.08 AM
Acute < 90°  Screen Shot 2016-07-10 at 9.01.15 AM
Obtuse > 90° but less than 180° Screen Shot 2016-07-10 at 9.00.56 AM
Reflex > 180° but less than 360° Screen Shot 2016-07-10 at 9.01.56 AM

Two angles are said to be complementary when their sum equals 90° and supplementary when their sum equals 180°.

Transversals are lines that intersect two or more lines.  Knowing that the sum of all angles for a given line equals 360°, if the transversed lines are parallel then the angles of each transversal are equivalent:

Screen Shot 2016-07-10 at 9.32.12 AM

Corresponding angles (1 and 3, 2 and 4) are equal.  Alternate interior angles (3 and 5) are equal.  Alternate exterior angles (2 and 8) are equal.

Triangles are formed by the intersection of three line segments and have three side and three angles.  The sum of the interior angles of a triangle always equal 180°.  Triangles are equilateral when all three sides have the same length, isosceles when two sides have equal length, and right when one of the angles is 90°.

The perimeter of a triangle can be found by adding the lengths of each side and the area can be calculated as 1/2(base)(height).

Quadrilaterals are four-sided geometric shapes whose four angles sum 360°:

Type Angles Area Example
Square All right angles (side)2  
Rectangle All right angles (length) x (width)
Parallelogram Opposite angles are equal (base) x (height)
Trapezoid Sum of angles is 360° 1/2(base1 + base2) x height  

Circles have a radius which measures the length from the center of the circle to any point on its curve.  The diameter is the length, through the center, from one point on the curve to another and equals radius x 2.  The circumference is the length around the curve and is equal to 2 x pi x radius (2πr) or pi times diameter (πd).  The area of a circle is pi times radius squared (πr2).

Volume measures the three dimensional area of an object.  Volume for a rectangle is found by multiplying length x width x height which, for a cube with equal side lengths, can be simplified to side3.  Volume for a cylinder can be found by multiplying the two dimensional area (πr2) by the height to get πr2h.

Mechanical Comprehension

Motion and Force

The First Law of Motion states that an object at rest tends to remain at rest unless acted upon by an outside force and that an object in motion tends to remain in motion at the same speed and at the same direction unless acted upon by an outside force.

The Second Law of Motion states that acceleration imparted upon an object is directly related to the strength of the force and inversely related to the object’s mass and always occurs in the direction of the force.

The Third Law of Motion states that for every action there is an equal and opposite reaction.

Recognize that a force (F) is any effort that can push, pull, or rotate an object to cause movement.  The direction of the movement is in the direction of the force and the amount of movement depends on the strength of the force and on its distance from the object:  torque = force x distance.

vector is a force on an object that is represented by an arrow with the length of the arrow representing the magnitude of the force and the head pointing in the direction that the object is moving.  Vector quantities include:

  • Displacement
  • Force
  • Momentum
  • Spin
  • Velocity

Scalar quantities are forces that have magnitude but no direction.  Scalar quantities include:

  • Area
  • Energy
  • Length
  • Mass
  • Speed
  • Work

Mechanical Principles

Speed is the rate (distance traveled per amount of time) of motion.  Instantaneous speed is the speed of a moving object at a particular instant along its motion while average speed is overall distance ÷ overall time to travel that distance.

Velocity is speed in a particular direction and acceleration is change in velocity per unit of time.

Momentum is the product of an object’s mass and velocity and can be either linear (from straight line movement) or angular (from rotation).

Equilibrium occurs when the forces acting upon an object are balanced.  There still may be forces acting upon the object but their effects cancel each other out.

Power is the rate of doing work over time: P = W ÷ time and horsepower is a common measure of power for machines.  One horsepower is 550 ft. lb. of work done in one second.

Machines create mechanical advantage and let you multiply input force to overcome more resistance than you cold without use of the machine.  Mechanical advantage (MA) = resistance force ÷ effort force and tells you the number of times that a machine multiplies effort force.

Friction reduces mechanical advantage as some portion of effort force must be used to overcome it. Also called drag, friction can be calculated through the ratio of work output to work input and is given by efficiency which equals (work output / work input) x 100%.

Torque is a measure of moments of rotational force around a specific point.  When equilibrium exists, these moments of force in clockwise and counterclockwise directions are equal.

Forces are are said to be collinear if they act along the same line of action, concurrent if they pass through a common point, and coplanar if they act in a common plane.

Simple Machines

A machine is any mechanism that makes work easier.  Machines can multiply applied effort, speed, or distance or change the direction of the effort without multiplying it.  Machines are simple if they multiply effort in a single way and are compound if they multiply effort in many ways through the use of more than one simple machine.

lever is a rigid beam resting on a pivot point (fulcrum).  The mechanical advantage of a lever is calculated by:

  • dividing the length of the effort arm by the length of the resistance arm
  • dividing the distance the effort moves by the distance the resistance moves
  • dividing the resistance force by the effort force

Levers are classified by where the effort and resistance forces are exerted in relation to the fulcrum.

First-class levers have the effort and the load on opposite sides of the fulcrum.  The mechanical advantage is (effort)(distance) = (load)(distance). If the distances are equal, then no mechanical advantage exists. If the effort arm is longer than the load arm then the lever will multiply the effort force and if the effort arm is shorter the lever will multiply the effort speed but not the effort force.

Second-class levers place the load in-between the fulcrum and the effort force.  As with first-class levers, the distance between the load and the fulcrum determines the mechanical advantage.  The closer the load is to the fulcrum, the greater the mechanical advantage.

Third-class levers have the fulcrum at one end, the load at the other, and the effort force in-between.  Third-class levers always multiply distance and speed without changing the direction of the effort and, because effort is greater than the resistance of the load, the mechanical advantage is always less than one.

wheel and axle can be thought of as a flexible lever that rotates around a center point that acts as the fulcrum. The arm lengths are represented by the radius of the wheel upon which the load force is applied and the radius upon which the load is applied. Depending upon which wheels the load and resistance forces are applied, the wheel and axle can multiply force or multiply speed.

pulley consists of a belt wrapped around a grooved wheel that reverses the direction of the effort and, depending on how it is arranged, may or may not multiply the effort.

fixed pulley does not move with the resistance, does not multiply effort, but does change the direction of the effort.  A fixed pulley has a mechanical advantage of one.




movable pulley attaches to the resistance and moves with it.  It multiples the effort but does not change the direction of the effort. Mechanical advantage is calculated by dividing effort distance by load distance.



block and tackle is a combination of fixed and movable pulleys which combines the advantages of both.  The movable pulleys multiply the force while the fixed pulleys change the direction of the effort. Mechanical advantage is equal to the number of times the force changes direction through the pulley system.

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