Forces, machines, and how mechanical things work. Each lesson below is also in the interactive study guide, where you can take a quiz to check yourself.
Lesson 1 Forces and Motion
Every question on the Mechanical Comprehension section comes back to one big idea: forces make things move, stop, or change direction. If you understand pushes, pulls, friction, and gravity, you can reason your way through gears, levers, and pulleys later on. This chapter gives you that foundation in plain language, with the kind of examples the ASVAB loves to ask about.
What a Force Is, and Net Force
A force is simply a push or a pull, and it is measured in newtons (N). Most objects have several forces acting on them at once, like when you push a crate forward while friction drags it backward. The net force is what you get when you combine all of those forces, keeping track of direction. Forces in the same direction add together, and forces in opposite directions subtract. If the net force is zero, the object is in equilibrium: it either stays still or keeps moving at the same steady speed. If the net force is not zero, the object speeds up, slows down, or changes direction.
Newton's Three Laws in Everyday Terms
Newton's first law says objects keep doing what they are doing unless a net force changes it, which is why you lurch forward when a vehicle brakes hard. This tendency to resist changes in motion is called inertia, and objects with more mass have more of it. Newton's second law connects force, mass, and acceleration with the formula force equals mass times acceleration (F = m x a); the same push accelerates a light object more than a heavy one. Newton's third law says every action has an equal and opposite reaction, which is how a rifle kicks back against your shoulder when it fires. These three laws explain almost every motion question the ASVAB can throw at you.
Weight vs Mass
Mass is the amount of matter in an object, measured in kilograms, and it never changes no matter where you go. Weight is the force of gravity pulling on that mass, measured in newtons, and it depends on location. On Earth, weight equals mass times about 9.8 meters per second squared, so a 60 kilogram person weighs about 588 newtons. Take that same person to the Moon and their mass is still 60 kilograms, but their weight drops to about one sixth of the Earth value because the Moon's gravity is weaker. If a test question asks what stays the same in space, the answer is mass.
Friction: Static vs Kinetic, Helpful vs Harmful
Friction is the force that resists sliding between two surfaces in contact. Static friction acts on objects that are not yet moving, and it is what you have to overcome to get a heavy box sliding. Kinetic friction acts on objects that are already sliding, and it is almost always weaker than static friction, which is why the box gets easier to push once it starts moving. Friction can be helpful: it lets your boots grip the ground, lets tires steer a truck, and lets brakes stop a vehicle. It can also be harmful, wearing out engine parts and wasting energy as heat, which is why machines use oil, grease, and bearings to reduce it.
Gravity and Falling Objects
Gravity pulls every object toward the center of the Earth, and near the surface it accelerates falling objects at about 9.8 meters per second squared. Ignoring air resistance, all objects fall at the same rate, so a heavy wrench and a light bolt dropped together hit the ground at the same time. In the real world, air resistance pushes back against falling objects, which is why a feather drifts down slowly while a rock plummets. A falling object speeds up until air resistance grows large enough to balance its weight; after that it falls at a constant terminal velocity. Remember that gravity acts on an object whether it is falling, sitting on a table, or flying through the air.
Work, Energy, and Power
In physics, work has a specific meaning: work equals force times the distance moved in the direction of the force, measured in joules. Holding a heavy pack perfectly still does no work in the physics sense, because nothing moves. Energy is the ability to do work, and it comes in two main mechanical forms: kinetic energy is the energy of motion, and potential energy is stored energy, like a raised weight or a stretched spring. Kinetic energy grows with the square of speed, so doubling an object's speed gives it four times the kinetic energy. Power is how fast work gets done: power equals work divided by time, measured in watts, so a more powerful motor does the same job in less time.
Worked example
A worker pushes a 10 kilogram cart with a force of 50 newtons while friction pushes back with 20 newtons. What is the cart's acceleration?
- Find the net force first. The push and the friction act in opposite directions, so subtract: 50 N minus 20 N equals 30 N of net force in the direction of the push.
- Apply Newton's second law, F = m x a, rearranged to a = F divided by m.
- Divide the net force by the mass: 30 N divided by 10 kg equals 3 meters per second squared.
- The cart accelerates at 3 meters per second squared in the direction the worker is pushing.
Tip: Always find the NET force before using F = m x a. A common trap answer uses the full 50 N push and gets 5 meters per second squared, which ignores friction.
Worked example
Two soldiers pull on a stalled truck with ropes. One pulls east with 300 newtons and the other pulls west with 300 newtons. What happens to the truck?
- Identify the forces: 300 N east and 300 N west, which are equal in size and opposite in direction.
- Combine them to find the net force: 300 N minus 300 N equals 0 N.
- A net force of zero means the truck is in equilibrium.
- By Newton's first law, an object with zero net force does not change its motion, so the stationary truck stays exactly where it is.
Tip: Balanced forces do not mean no forces are acting; they mean the forces cancel out. Equilibrium can also apply to something already moving at a constant speed in a straight line.
Worked example
A mechanic lifts a toolbox with a steady force of 40 newtons through a height of 5 meters, and the lift takes 4 seconds. How much work is done, and how much power is used?
- Use the work formula: work equals force times distance.
- Multiply: 40 N x 5 m equals 200 joules of work.
- Use the power formula: power equals work divided by time.
- Divide: 200 joules divided by 4 seconds equals 50 watts of power.
Tip: Keep the two formulas straight: work cares about distance, power cares about time. If two people do the same work but one finishes faster, the faster one used more power, not more work.
Key takeaway: Forces are pushes and pulls, and it is the net force that decides whether an object stays in equilibrium or accelerates according to F = m x a. Mass measures matter and never changes, while weight is gravity's pull and depends on location; friction and gravity are the everyday forces you will see most on the test. Work is force times distance, energy is stored either in motion (kinetic) or position (potential), and power measures how quickly that work gets done.
Lesson 2 Levers and Simple Machines
Almost every mechanical device you will ever touch - from a crowbar to a truck jack to a pair of pliers - is built from a handful of basic building blocks called simple machines. The ASVAB loves these because they test whether you can reason about force, not just memorize facts. The good news: once you learn one golden rule (trade distance for force) and one simple formula (mechanical advantage), most of these questions become quick arithmetic. In this chapter you will learn the six simple machines, break down the three classes of levers with examples you already use in daily life, and practice the classic seesaw balance problems that show up on nearly every version of the test.
The Six Simple Machines
Engineers group nearly all basic mechanical devices into six simple machines: the lever, the inclined plane, the wedge, the screw, the wheel and axle, and the pulley. Each one does the same fundamental job - it changes the size or direction of a force to make work easier. The key idea is the trade-off: a simple machine can multiply your force, but only by making you apply that force over a longer distance. You never get free work out of a machine. If a ramp lets you push with one quarter of the force, you must push four times as far. That trade-off number is called mechanical advantage (MA). MA equals the load force divided by the effort force, or equivalently the distance the effort moves divided by the distance the load moves. An MA of 3 means the machine multiplies your force by 3.
Lever Basics: Effort, Load, and Fulcrum
A lever is a rigid bar that pivots on a fixed point. Every lever problem has three players. The fulcrum is the pivot point the bar rotates around. The load (also called resistance) is the weight or object you are trying to move. The effort is the force you apply. The distance from the fulcrum to where you push is the effort arm, and the distance from the fulcrum to the load is the load arm. Everything about how a lever behaves comes from where these three things sit relative to each other - and that is exactly what defines the three classes of levers.
The Three Classes of Levers
Class 1: the fulcrum sits between the effort and the load. Think of a crowbar prying up a crate, a seesaw, or a pair of scissors. Class 1 levers can multiply force (long effort arm, short load arm) and they reverse the direction of motion - push down on one end, the other end goes up. Class 2: the load sits between the fulcrum and the effort. The classic example is a wheelbarrow - the wheel is the fulcrum, the cargo sits in the middle, and you lift the handles at the far end. A bottle opener and a nutcracker work the same way. Class 2 levers always multiply force, because the effort arm is always longer than the load arm. Class 3: the effort sits between the fulcrum and the load. Tweezers, a fishing rod, a broom, and your own forearm are class 3 levers. These never multiply force - you actually apply more force than the load - but in exchange they multiply speed and range of motion. Memory trick: think F-L-E for what is in the middle - Fulcrum (class 1), Load (class 2), Effort (class 3).
Mechanical Advantage from Arm Lengths
For a lever, the ideal mechanical advantage is simply the effort arm divided by the load arm: MA equals effort arm divided by load arm. If you push on a crowbar 6 feet from the fulcrum and the load sits 1 foot from the fulcrum on the other side, MA equals 6 divided by 1, which is 6. That means a 50 pound push can lift a 300 pound load, because 50 x 6 equals 300. To find the effort needed, divide the load by the MA. Notice the trade-off in action: your end of the bar swings through a big arc while the load barely rises. Class 2 levers always have MA greater than 1, class 3 levers always have MA less than 1, and class 1 levers can go either way depending on where the fulcrum sits.
Balance and Torque: Seesaw Problems
Torque is the turning effect of a force, and for ASVAB purposes it equals force times distance from the fulcrum. A lever balances when the torque on one side equals the torque on the other side. This gives you the balance equation: weight 1 x distance 1 equals weight 2 x distance 2. Every seesaw problem is this one equation. Example: a 100 pound child sits 6 feet from the center of a seesaw. Where must a 150 pound adult sit to balance? Set 100 x 6 equal to 150 x d. The left side is 600, so d equals 600 divided by 150, which is 4 feet. Heavier riders sit closer to the fulcrum; lighter riders sit farther out. If the torques are unequal, the side with the larger torque rotates downward - some questions ask which way the bar tips, and you answer by comparing the two force-times-distance products.
Inclined Plane, Wedge, Screw, and Wheel and Axle
Inclined plane: a ramp. MA equals the length of the slope divided by the vertical height. A 12 foot ramp rising 3 feet has MA equal to 4, so a 400 pound crate can be pushed up with about 100 pounds of force (ignoring friction). Longer, shallower ramps mean easier pushing over a longer distance. Wedge: two inclined planes back to back that move through the material - an axe, a knife, a chisel, a nail. A longer, thinner wedge gives more mechanical advantage. Screw: an inclined plane wrapped in a spiral around a cylinder. The distance between neighboring threads is called the pitch. Finer threads (smaller pitch) give more mechanical advantage but require more turns to advance - the same distance-for-force trade again. Wheel and axle: a large wheel fixed to a smaller shaft so they turn together, like a steering wheel, a doorknob, or a screwdriver handle. MA equals the wheel radius divided by the axle radius. A steering wheel with a 10 inch radius on a 2 inch radius shaft gives MA equal to 5.
Worked example
A 120 pound student sits 4 feet from the fulcrum of a seesaw. How far from the fulcrum must an 80 pound student sit on the other side to balance it?
- Write the balance equation: weight 1 x distance 1 equals weight 2 x distance 2.
- Plug in the known side: 120 x 4 equals 480 foot-pounds of torque.
- Set the other side equal to it: 80 x d equals 480.
- Solve for d: d equals 480 divided by 80, which is 6 feet.
- Check that it makes sense: the lighter student sits farther from the fulcrum. 6 feet is farther than 4 feet, so the answer fits.
Tip: On seesaw problems, always compute the known side's torque first (force x distance), then divide by the other force. And sanity-check: lighter always sits farther out.
Worked example
A worker uses a 9 foot pry bar to lift a heavy rock. The fulcrum is placed 1 foot from the rock, so the worker pushes down 8 feet from the fulcrum. The rock weighs 400 pounds. How much force must the worker apply?
- Identify the arms: the effort arm is 8 feet and the load arm is 1 foot.
- Compute the mechanical advantage: MA equals effort arm divided by load arm, so MA equals 8 divided by 1, which is 8.
- Find the required effort: effort equals load divided by MA, so effort equals 400 divided by 8, which is 50 pounds.
- Confirm with torque balance: 50 x 8 equals 400, and 400 x 1 equals 400. The torques match, so 50 pounds is right.
Tip: Two ways to solve any lever force problem: divide the load by the MA, or set the two torques equal. They always give the same answer - use whichever comes to mind first, and use the other to check.
Worked example
A mover must load a 300 pound crate onto a truck bed that is 2.5 feet off the ground. He uses a ramp that is 10 feet long. Ignoring friction, how much force does he need to push the crate up the ramp?
- Find the ramp's mechanical advantage: MA equals slope length divided by height, so MA equals 10 divided by 2.5, which is 4.
- Divide the load by the MA: effort equals 300 divided by 4, which is 75 pounds.
- Notice the trade-off: he pushes with only 75 pounds instead of lifting 300, but he must push the crate along 10 feet of ramp instead of raising it 2.5 feet straight up.
Tip: For inclined planes, the only two numbers that matter for MA are slope length and vertical height. The weight of the object comes in only at the last step, when you divide by the MA.
Key takeaway: Simple machines trade distance for force - that single idea explains everything in this chapter. The six simple machines are the lever, inclined plane, wedge, screw, wheel and axle, and pulley. Levers come in three classes based on what sits in the middle: fulcrum (class 1, like a crowbar or seesaw), load (class 2, like a wheelbarrow - always multiplies force), or effort (class 3, like tweezers - trades force for speed and range). A lever's mechanical advantage is effort arm divided by load arm, and the effort needed is load divided by MA. Balance problems use one equation: weight 1 x distance 1 equals weight 2 x distance 2 - the heavier object always sits closer to the fulcrum. For ramps, MA is slope length divided by height; for wheel and axle, it is wheel radius divided by axle radius; a wedge is two inclined planes back to back; a screw is an inclined plane wrapped around a cylinder. Master the torque equation and the MA formulas and you can solve these problems in seconds.
Lesson 3 Pulleys and Mechanical Advantage
Pulleys are everywhere in the military: hoisting equipment onto trucks, raising flags, rigging loads on ships, and lifting engines out of vehicles. The ASVAB loves pulley questions because they test one big idea: mechanical advantage, which is how a simple machine multiplies your force. The best part is that pulley problems follow simple, reliable rules. Once you learn to count rope strands, you can solve almost any pulley question on the test in seconds. Let's break it down step by step.
The Fixed Pulley: Changing Direction, Not Force
A fixed pulley is attached to something that does not move, like a ceiling beam or a flagpole. The wheel spins, but the pulley itself stays put. A single fixed pulley has a mechanical advantage of 1, which means it does NOT reduce the force you need. If the load weighs 100 pounds, you still pull with 100 pounds of force. So why use one? Direction. A fixed pulley lets you pull DOWN to lift a load UP. Pulling down is easier and safer because you can use your body weight and stand on the ground instead of hauling a rope upward. Think of a flagpole: the pulley at the top does not make the flag lighter, it just lets you raise the flag while standing at the bottom.
The Movable Pulley: Cutting the Force in Half
A movable pulley is attached to the load itself, and the pulley moves up with the load as you lift. Here is the key difference: the load now hangs from TWO strands of rope, one on each side of the pulley wheel. Each strand carries half the weight, so you only supply half the force. A single movable pulley has a mechanical advantage of 2. To lift a 100 pound load, you pull with only 50 pounds of force. The trade-off is that you must pull the rope twice as far as the load rises. Nothing is free in mechanics, but spreading the work over a longer pull makes heavy loads manageable.
Counting Strands: The Golden Rule of Pulley Problems
Here is the rule that solves most ASVAB pulley questions: the mechanical advantage equals the number of rope strands directly supporting the load or the movable pulley block. Look at the diagram and count every strand of rope that is holding the load up. Do not count a strand that just runs over a fixed pulley and down to your hand as a free end, unless that strand also pulls upward on the movable block. Two supporting strands means a mechanical advantage of 2. Four supporting strands means a mechanical advantage of 4, and you pull with one fourth of the load's weight. When you see a pulley diagram on the test, ignore the clutter, find the load, and count the strands holding it up. That number is your answer to almost everything.
Block and Tackle: Stacking Pulleys for Big Lifts
A block and tackle combines fixed and movable pulleys into one system. A block is a frame holding one or more pulley wheels, and the tackle is the rope threaded through them. One block is fixed to a support overhead, the other block is attached to the load and moves with it. Threading the rope back and forth between the blocks creates multiple supporting strands. A block and tackle with 4 supporting strands gives a mechanical advantage of 4: a 400 pound crate needs only 100 pounds of pull. Sailors, riggers, and mechanics have used block and tackle systems for centuries to move loads far heavier than any person could lift alone. The same counting rule applies: strands supporting the load equal the mechanical advantage.
The Force-Distance Trade-Off and Efficiency
Mechanical advantage never gives you free work. If a pulley system has a mechanical advantage of 4, you use one fourth of the force, but you must pull 4 times as much rope. To lift a load 2 feet with that system, you pull 8 feet of rope through your hands. Force goes down, distance goes up, and the total work stays the same. That is the trade-off in every simple machine. In the real world, there is one more factor: friction. Rope rubbing over pulley wheels and axles wastes some of your effort as heat, so the actual mechanical advantage is always a little less than the ideal number you get from counting strands. Efficiency measures this: a system that is 80 percent efficient delivers only 80 percent of your effort to the load, so you must pull somewhat harder than the ideal calculation says. More pulleys mean more lifting power, but also more friction. On the ASVAB, assume the ideal numbers unless the question mentions friction or efficiency.
Worked example
A block and tackle diagram shows a crate hanging from a movable block. Four strands of rope support the movable block, and the free end of the rope comes off a fixed pulley to the worker's hands. The crate weighs 200 pounds. How much force must the worker pull with, and how much rope must be pulled to raise the crate 3 feet?
- Count the strands supporting the load: the diagram shows 4 strands holding up the movable block, so the mechanical advantage is 4.
- Find the force: divide the load by the mechanical advantage. 200 pounds divided by 4 equals 50 pounds of pull.
- Find the rope distance: multiply the lift height by the mechanical advantage. 3 feet x 4 equals 12 feet of rope pulled.
- Check the trade-off: 50 pounds x 12 feet equals 600 foot-pounds of work in, and 200 pounds x 3 feet equals 600 foot-pounds of work out. They match, so the answer makes sense.
Tip: Always check your answer with the work equation: force in x distance in should equal load x lift height in an ideal system. If the two sides do not match, recount your strands.
Worked example
A mechanic uses a single movable pulley to lift a 300 pound engine out of a vehicle. How much force is needed, and how far must the rope be pulled to raise the engine 2 feet?
- Identify the pulley type: a single movable pulley attached to the load has 2 supporting strands, so the mechanical advantage is 2.
- Find the force: 300 pounds divided by 2 equals 150 pounds of pull.
- Find the rope distance: 2 feet of lift x 2 equals 4 feet of rope pulled.
- Sanity check: 150 pounds is still a hard pull for one person, which is why real engine hoists use more pulleys or a chain fall for a bigger mechanical advantage.
Tip: A single movable pulley always cuts the force in half and doubles the rope distance. Memorize that pair: half the force, twice the pull.
Worked example
A hoisting rig has an ideal mechanical advantage of 4, but friction in the pulleys makes it only 80 percent efficient. How much force is actually needed to lift a 320 pound load?
- Start with the ideal force: 320 pounds divided by 4 equals 80 pounds if the system were perfect.
- Account for friction: at 80 percent efficiency, only 80 percent of your pull reaches the load, so you must pull harder than the ideal 80 pounds.
- Divide the ideal force by the efficiency: 80 pounds divided by 0.80 equals 100 pounds of actual pull.
- Interpret the result: friction costs the worker an extra 20 pounds of effort. The actual mechanical advantage is 320 divided by 100, which is 3.2 instead of the ideal 4.
Tip: Friction always makes real force HIGHER than the ideal calculation. If an efficiency answer comes out easier than the ideal number, you divided in the wrong direction.
Key takeaway: A fixed pulley changes the direction of your pull but not the force, with a mechanical advantage of 1. A movable pulley rides on the load and gives a mechanical advantage of 2. For any pulley system, count the rope strands supporting the load: that count is the mechanical advantage. Divide the load weight by the mechanical advantage to get the required force, and multiply the lift height by the mechanical advantage to get the rope distance you must pull. That force-distance trade-off means total work stays the same. Friction reduces real-world performance, so actual mechanical advantage is always a bit less than the ideal strand count. Master strand counting and the trade-off rule, and pulley questions become some of the easiest points on the test.
Lesson 4 Gears, Wheels, and Torque
Almost every machine that spins uses gears, belts, or chains to move power from one shaft to another. The ASVAB loves these questions because they all come down to a few simple rules: count the teeth, compare the sizes, and remember that speed and torque always trade places. Once you know those rules, you can solve gear problems in seconds. Let's break it down.
Meshed Gears Reverse Direction
When two gears mesh directly, their teeth interlock like fingers. As the first gear (the driver) turns clockwise, it pushes the second gear (the driven gear) counterclockwise. Every direct mesh flips the direction of rotation. So in a row of meshed gears, the direction alternates: gear 1 clockwise, gear 2 counterclockwise, gear 3 clockwise, and so on. Quick rule: with an odd number of gears in a straight line, the first and last gears turn the same direction; with an even number, they turn opposite directions. On the test, trace the rotation gear by gear and you will never get lost.
Gear Ratio from Tooth Counts
The gear ratio compares the driven gear to the driver gear. Ratio equals teeth on the driven gear divided by teeth on the driver gear. If a 10-tooth gear drives a 30-tooth gear, the ratio is 30 divided by 10, which is 3 to 1. That means the driver must turn 3 full times to spin the driven gear once. Speed works in reverse of tooth count: the gear with more teeth always turns slower. To find output speed, multiply the driver speed by the driver teeth, then divide by the driven teeth. Teeth in equals teeth out - the number of teeth passing the mesh point is the same for both gears every second.
The Speed-Torque Trade-Off
Gears never give you something for free. When a small gear drives a big gear, the big gear turns slower but with more torque (turning force). When a big gear drives a small gear, the small gear spins faster but with less torque. Whatever factor you lose in speed, you gain in torque, and vice versa. A 3 to 1 reduction gives one third the speed and three times the torque. This is exactly like the gears on a bicycle or a truck: low gear equals slow and strong for climbing hills, high gear equals fast and weak for cruising on flat ground.
Idler Gears and Gear Trains
An idler gear sits between the driver and the driven gear. Here is the key fact the ASVAB tests: an idler changes only the direction of the final gear - it has no effect on the overall gear ratio, no matter how many teeth it has. The ratio still depends only on the first and last gears. A gear train is a series of gears working together. In a compound gear train, two gears share the same shaft, so they turn at the same speed. To find the total ratio of a compound train, multiply the ratios of each stage together. Compound trains let machines multiply torque enormously in a small space.
Belts, Pulleys, Sprockets, and Chains
Belts and pulleys work like gears, but you compare diameters instead of teeth. A small pulley driving a large pulley slows the output and boosts torque, just like gears. Output speed equals driver speed times driver diameter divided by driven diameter. One big difference: two pulleys connected by an open belt turn in the SAME direction, because there is no meshing to flip the rotation. To reverse the direction, you cross the belt into a figure-eight shape - a crossed belt makes the pulleys turn in opposite directions. Sprockets and chains (like on a bicycle) behave like a belt drive that cannot slip: both sprockets turn the same direction, and the speed ratio comes from the tooth counts, just like gears.
Worm Gears, Cams, and Cranks
A worm gear is a screw-shaped shaft (the worm) meshed with a toothed wheel. Each full turn of the worm advances the wheel by just one tooth, so a worm meshed with a 40-tooth wheel gives a 40 to 1 reduction - huge torque, very slow output. Bonus: most worm drives self-lock, meaning the wheel cannot drive the worm backward, which is why they are used in hoists and guitar tuners. A cam is an egg-shaped or lobed wheel that converts rotary motion into up-and-down (reciprocating) motion - think of the camshaft opening valves in an engine. A crank does the opposite job in an engine: the crankshaft turns the up-and-down motion of the pistons into rotation.
Torque Equals Force Times Lever Arm
Torque is twisting force, and the formula is simple: torque equals force times the distance from the pivot (the lever arm). Push with 50 pounds on a wrench 1 foot long and you get 50 pound-feet of torque. Use a 2-foot wrench with the same 50 pounds and you get 100 pound-feet - double the twist for the same effort. That is why mechanics slip a pipe (a cheater bar) over a wrench handle to break loose a stuck bolt: the longer arm multiplies the torque. It is also why door handles sit far from the hinges and why steering wheels are wide. More distance from the pivot means more torque from the same push.
Worked example
A 12-tooth gear turning at 90 rpm drives a 36-tooth gear. How fast does the 36-tooth gear turn, and what happens to the torque?
- Find the gear ratio: driven teeth divided by driver teeth equals 36 divided by 12, which is 3. The ratio is 3 to 1.
- The bigger gear turns slower by that factor. Output speed equals 90 rpm divided by 3, which is 30 rpm.
- Check with teeth per minute: the driver passes 12 x 90 = 1080 teeth per minute, and 1080 divided by 36 teeth equals 30 rpm. It matches.
- Torque trades the other way: the 36-tooth gear delivers about 3 times the torque of the driver.
- Direction: the two gears mesh directly, so the driven gear turns the opposite way from the driver.
Tip: Speed and tooth count are always opposites: more teeth means slower rotation but more torque.
Worked example
In a compound gear train, gear A (12 teeth) drives gear B (24 teeth). Gear C (10 teeth) is on the same shaft as gear B and drives gear D (30 teeth). If gear A turns at 600 rpm, how fast does gear D turn?
- Stage 1: A drives B. Ratio equals 24 divided by 12, which is 2. Gear B turns at 600 divided by 2, which is 300 rpm.
- Gears B and C share one shaft, so gear C also turns at 300 rpm.
- Stage 2: C drives D. Ratio equals 30 divided by 10, which is 3. Gear D turns at 300 divided by 3, which is 100 rpm.
- Total reduction: multiply the stage ratios, 2 x 3 = 6. Check: 600 rpm divided by 6 equals 100 rpm. It matches.
- The output shaft is 6 times slower but delivers roughly 6 times the torque of the input.
Tip: In compound trains, multiply the stage ratios together. Gears on the same shaft always spin at the same speed.
Worked example
A motor pulley 4 inches in diameter spins at 500 rpm and drives a 10-inch pulley with an open belt. How fast does the big pulley turn, and which direction?
- For pulleys, use diameters the way you use teeth for gears. Output speed equals driver speed times driver diameter divided by driven diameter.
- Multiply: 500 rpm x 4 inches = 2000. Divide by the driven diameter: 2000 divided by 10 equals 200 rpm.
- Sanity check: the driven pulley is 2.5 times bigger (10 divided by 4), so it should turn 2.5 times slower. 500 divided by 2.5 equals 200 rpm. It matches.
- Direction: an open belt does not flip rotation, so both pulleys turn the same direction. Only a crossed belt would reverse it.
Tip: Belt-driven pulleys turn the same way unless the belt is crossed. Meshed gears always turn opposite ways.
Key takeaway: Meshed gears reverse direction at every mesh; belts and chains keep the same direction unless the belt is crossed. Gear ratio equals driven teeth divided by driver teeth, and the gear with more teeth turns slower but with more torque - speed and torque always trade places. Idlers change direction only, never the ratio. Compound gear trains multiply their stage ratios, worm gears give big reductions and self-lock, cams turn rotation into up-and-down motion, and cranks do the reverse. Torque equals force times lever arm, which is why a longer wrench loosens a stubborn bolt with the same push. Learn these rules and rotation questions become free points.
Lesson 5 Fluids and Hydraulics
Ever wonder how a small hand pump can lift a 3-ton truck, or why pressing a brake pedal with one foot can stop a vehicle moving at highway speed? The answer is fluid power. Liquids and gases can carry force from one place to another and even multiply it along the way. The military runs on fluid power - hydraulic lifts, brake systems, aircraft landing gear, and fuel pumps all use the ideas in this chapter. The good news: it all comes down to one simple formula, pressure equals force divided by area, plus a few common-sense rules. Master those and the ASVAB fluid questions become easy points.
Pressure = Force / Area
Pressure is how concentrated a force is. The formula is P = F / A, where force is in pounds and area is in square inches, giving pressure in pounds per square inch (psi). The same force feels very different depending on the area it acts on. Push a thumbtack with 10 pounds of force and the tiny point creates enormous pressure - it sinks into the wall. Push a book against the wall with the same 10 pounds and nothing happens, because the force spreads over a large area. Two rearranged versions of the formula are just as useful: F = P x A (force equals pressure times area) and A = F / P. On the ASVAB, always check whether the question asks for pressure, force, or area, then pick the right version. Bigger area with the same force means lower pressure; smaller area with the same force means higher pressure.
Pascal's Principle: Pressure Travels Everywhere Equally
Pascal's principle says that pressure applied to an enclosed fluid is transmitted equally and undiminished to every part of the fluid and to the walls of the container. Squeeze a sealed water bottle in the middle and the pressure rises at the top, the bottom, and everywhere in between - all at the same time and by the same amount. This works because liquids are practically incompressible: you cannot squeeze water into a smaller volume, so any push you give it has to go somewhere. This one rule is the foundation of every hydraulic system. It means the pressure at the small input piston equals the pressure at the large output piston, and that fact is what lets hydraulics multiply force.
Hydraulic Systems Multiply Force
A hydraulic system connects a small piston to a large piston through a sealed fluid line. Because pressure is the same everywhere in the fluid (Pascal's principle), the force at each piston equals that shared pressure times the piston's area. A larger piston therefore produces a larger force. Here is the three-step method: first, find the pressure from the input side (P = input force / input area). Second, that same pressure acts on the output piston. Third, multiply pressure by the output area to get the output force (F = P x A). The force multiplies by the ratio of the areas. A hydraulic jack uses this to lift vehicles, and car brakes use it to turn light pedal pressure into strong clamping force at the wheels. But there is no free lunch - just like levers, hydraulics trade distance for force. If the large piston has 10 times the area, it pushes with 10 times the force but moves only one tenth as far. That is why you pump a jack handle many times to raise a car a few inches.
Water Pressure, Depth, and Buoyancy
In any liquid, pressure increases with depth because the deeper you go, the more liquid is stacked above you, pressing down. In fresh water, pressure rises about 0.433 psi for every foot of depth - roughly 1 psi for every 2.3 feet. Only depth matters, not the width or shape of the container: a narrow 10-foot pipe full of water has the same pressure at the bottom as a 10-foot-deep lake. This depth pressure also explains buoyancy. Water pushes harder on the bottom of a submerged object than on its top (the bottom is deeper), so there is a net upward push called the buoyant force. Archimedes' principle says the buoyant force equals the weight of the fluid the object displaces. If an object weighs less than the water it pushes aside - meaning it is less dense than water - it floats. If it weighs more, it sinks. That is why a steel ship floats: its hollow hull can push aside a volume of water that would outweigh the ship, so its average density is less than water and it settles only until the displaced water weighs exactly as much as the ship.
Flow, Pneumatics, Pumps, and Valves
When fluid flows through a pipe, the same amount must pass every point each second. So when a pipe narrows, the fluid must speed up to squeeze the same flow through the smaller opening - narrower means faster. That is why putting your thumb over a garden hose makes the water jet out. Pneumatic systems work like hydraulics but use compressed air or gas instead of liquid. Air is compressible, so pneumatics are springy and less precise, but they are lighter, cleaner, and safe if a line leaks - think air tools, air brakes on trucks, and tire inflators. Hydraulic fluid does not compress, so hydraulics deliver stronger, stiffer, more exact force - think jacks, excavators, and aircraft controls. Pumps put fluid in motion and build pressure; valves control the fluid. A check valve allows flow in one direction only, a relief valve opens automatically to release dangerous excess pressure, and a shutoff valve simply starts or stops flow.
Worked example
A hydraulic jack has a small input piston with an area of 2 square inches and a large output piston with an area of 20 square inches. You push down on the small piston with 50 pounds of force. How much force does the large piston produce?
- Step 1: Find the pressure created at the input piston. P = F / A = 50 pounds / 2 in2 = 25 psi.
- Step 2: Apply Pascal's principle - that same 25 psi acts on every part of the fluid, including the large piston.
- Step 3: Find the output force. F = P x A = 25 psi x 20 in2 = 500 pounds.
- Check with the area ratio: the output piston is 20 / 2 = 10 times larger, so the force is multiplied by 10. 50 x 10 = 500 pounds. It matches.
Tip: Shortcut: output force = input force x (output area / input area). But remember the trade-off - to lift the load 1 inch, the small piston must travel 10 inches.
Worked example
In a car's brake system, your foot pushes the master cylinder piston (area 1 square inch) with 100 pounds of force. Each wheel cylinder piston has an area of 4 square inches. How much force does each wheel cylinder apply to the brakes?
- Step 1: Find the system pressure from the input side. P = F / A = 100 pounds / 1 in2 = 100 psi.
- Step 2: Pascal's principle sends that 100 psi through the brake fluid to every wheel cylinder equally - that is why all four brakes engage at the same time.
- Step 3: Find the force at each wheel. F = P x A = 100 psi x 4 in2 = 400 pounds.
- So a 100-pound push from your foot becomes 400 pounds of braking force at each wheel.
Tip: This also shows why air in brake lines is dangerous: air compresses, so your pedal push squeezes the bubble instead of transmitting pressure, and the pedal feels spongy.
Worked example
You need a hydraulic press to deliver 300 pounds of output force, but you can only supply 40 pounds of input force on a piston with an area of 2 square inches. How large must the output piston be?
- Step 1: Find the pressure you can create. P = F / A = 40 pounds / 2 in2 = 20 psi.
- Step 2: The output piston sees that same 20 psi and must produce 300 pounds.
- Step 3: Rearrange F = P x A to solve for area: A = F / P = 300 pounds / 20 psi = 15 in2.
- Check: 20 psi x 15 in2 = 300 pounds. A 15 square inch output piston does the job.
Tip: When a fluid problem asks for a missing piece, write P = F / A first, then rearrange for whatever the question wants. The pressure is always the bridge between the two pistons.
Key takeaway: Pressure is force divided by area (P = F / A), measured in psi. Pascal's principle says pressure in an enclosed fluid transmits equally in all directions, which lets hydraulic systems multiply force: find the pressure at the small piston, then multiply by the large piston's area - the force gain equals the area ratio, and the big piston moves proportionally less. Water pressure grows with depth (about 0.433 psi per foot in fresh water), and that depth difference creates buoyancy: objects float when they weigh less than the water they displace. In pipes, narrower sections mean faster flow. Hydraulics use incompressible liquid for strong, precise force; pneumatics use compressible air for lighter, springier power. Pumps create flow and pressure; check valves allow one-way flow and relief valves vent excess pressure. You now have every tool the ASVAB uses for fluid questions - work the arithmetic step by step and these are reliable points.
Lesson 6 Springs, Structures, and Materials
So far you have studied levers, gears, pulleys, and fluids. This chapter zooms out to the stuff machines and structures are made of. Why does a spring stretch a predictable amount? Why do bridges use triangles everywhere? Why does a wide, low stance keep a crane from tipping over? These are the kinds of questions the ASVAB loves, and the good news is that almost every one of them can be answered with a few simple, common-sense rules. No heavy math here, just clear thinking about forces and shapes. Let's get into it.
Hooke's Law: Springs Stretch in Proportion to Force
A spring is honest: pull on it twice as hard and it stretches twice as far. That simple idea is Hooke's law. In plain words, the stretch (or squeeze) of a spring is directly proportional to the force applied to it, as long as you do not overload it. If 10 pounds stretches a spring 1 inch, then 20 pounds stretches it 2 inches and 5 pounds stretches it half an inch. Every spring has a stiffness, often called the spring constant. A stiff spring (like a truck suspension spring) needs a lot of force for a little stretch. A soft spring (like the one in a ballpoint pen) stretches easily. One warning: Hooke's law only holds up to the spring's elastic limit. Stretch a spring too far and it stays bent out of shape permanently. On the test, assume springs behave proportionally unless the question says otherwise.
Springs in Series vs. Springs in Parallel
How you connect springs changes how the combination behaves. In series means end to end, like links in a chain. The same force passes through each spring, so each one stretches its full amount and the stretches add up. Two identical springs in series stretch twice as far as one spring alone under the same load, so the combination acts softer. In parallel means side by side, both attached between the same two points, sharing the load. Each spring carries only part of the force, so each stretches less. Two identical springs in parallel each carry half the load, so the combination stretches half as much and acts stiffer. Memory hook: series = softer, parallel = stiffer. Think of a truck with multiple leaf springs stacked side by side: parallel springs make a stiff suspension that can carry heavy cargo.
Three Ways to Load a Material: Tension, Compression, and Shear
Every force on a solid part is basically one of three types. Tension is a pulling or stretching force. A tow rope, a crane cable, and the string of a bow are all in tension. Ropes and cables can ONLY carry tension; you cannot push a rope. Compression is a squeezing or pushing force. The legs of a table, a building column, and the jack holding up a car are in compression. Shear is a sliding force, where two parts of a material are pushed in opposite directions past each other, like the blades of scissors cutting paper or a bolt being loaded sideways where two plates meet. Real structures often mix these: when a beam bends, its top surface is squeezed (compression) while its bottom surface is stretched (tension). If a test question asks what kind of force a cable, rope, or chain carries, the answer is tension.
Center of Gravity and Stability: Why Wide Bases Win
The center of gravity (CG) is the balance point of an object, the single spot where all its weight seems to act. Stability follows one rule: an object stays upright as long as its center of gravity is over its base of support. Tip the object until the CG passes beyond the edge of the base, and gravity takes over and knocks it down. Two things make an object more stable. First, a wider base, because the CG has to travel farther sideways before it passes the edge. Second, a lower center of gravity, because a low CG swings outward less for the same amount of tilt. This is why a race car is low and wide while a tall bookcase tips easily, why you spread your feet on a moving bus, and why a forklift must keep its load low while driving. To make anything harder to tip: widen the base, lower the weight.
Triangles, Trusses, and Beams: How Structures Carry Loads
Look at any bridge or radio tower and you will see triangles everywhere. Here is why: a triangle is the only shape that cannot change its angles without changing the length of a side. Push on a square frame and it collapses sideways into a diamond; push on a triangle and nothing moves unless a side stretches or crushes. Engineers connect triangles into a truss, a framework that turns a heavy load into simple tension and compression in each straight member, which is exactly what materials handle best. Beams work differently: a beam resting on two supports carries a load by bending. The load is shared between the supports, but not always equally. The support CLOSER to the load carries MORE of the weight. Stand near the left end of a plank and the left support does most of the work. Put the load exactly in the middle and each support carries half. Spreading a load out along a beam (instead of concentrating it at one point) reduces the worst bending and is easier on the beam.
Thermal Expansion, Bearings, and Counterweights
Three quick topics that show up on the test. Thermal expansion: nearly all materials expand when heated and shrink when cooled. Bridges have expansion joints and gaps so hot summer days do not buckle the deck. Mechanics use this on purpose: to slide a tight-fitting bearing or gear onto a shaft, heat the part so its hole grows, slip it on, and let it cool into a super-tight shrink fit. To loosen a stuck metal lid or nut, heat it so it expands away from what it grips. Bearings: a bearing is a machine part that lets things rotate or slide with very little friction, usually by putting rolling balls or rollers between the moving surfaces, because rolling friction is far lower than sliding friction. Lubrication (oil or grease) helps the same way, by separating metal surfaces with a slippery film so they never scrape directly. Less friction means less wear, less heat, and less wasted effort. Counterweights: a counterweight is a weight placed on the opposite side of a pivot to balance a load, exactly like a see-saw. Tower cranes carry huge concrete counterweights behind the cab, elevators hang a counterweight on the other end of the cable, and drawbridges use them too. The balance rule is the lever rule from earlier chapters: weight times distance on one side equals weight times distance on the other. Counterweights mean the motor only has to move the difference, not the whole load.
Worked example
A spring stretches 2 inches when a 10-pound weight hangs from it. How far will it stretch with a 25-pound weight? (Assume the spring stays within its elastic limit.)
- Hooke's law says stretch is proportional to force, so first find the stretch per pound.
- 10 pounds causes 2 inches of stretch, so each pound causes 2 divided by 10 = 0.2 inches of stretch.
- Multiply by the new load: 25 pounds x 0.2 inches per pound = 5 inches.
- Sanity check: 25 pounds is 2.5 times the original 10 pounds, so the stretch should be 2.5 times the original 2 inches. 2.5 x 2 = 5 inches. It checks out.
Tip: Set up spring problems as a simple ratio: new stretch = old stretch x (new force divided by old force). If the force goes up 2.5 times, the stretch goes up 2.5 times.
Worked example
Under a 20-pound load, one spring stretches 4 inches. How far does the load drop if you hang it from (a) two of these springs connected end to end (series), and (b) two of these springs side by side sharing the load (parallel)?
- Series (end to end): the full 20 pounds pulls through BOTH springs, so each spring stretches its full 4 inches.
- The stretches add: 4 + 4 = 8 inches total. The series pair acts like one softer spring.
- Parallel (side by side): the two springs share the 20 pounds, so each carries only 10 pounds.
- Half the force means half the stretch: each spring stretches 2 inches, and since they stretch together, the load drops just 2 inches. The parallel pair acts like one stiffer spring.
Tip: Series doubles the stretch (softer); parallel halves it (stiffer). If you mix these up, picture a chain of slinky springs (very stretchy) versus a stack of leaf springs on a truck (very stiff).
Worked example
A tool cabinet and a low workbench weigh the same. The cabinet is tall and narrow with heavy drawers up top; the bench is short and wide with a heavy steel base. A soldier bumps into each one with the same sideways push. Which is more likely to tip over, and why?
- Find each object's center of gravity. The cabinet has heavy drawers up high, so its CG is high. The bench has a heavy base, so its CG is low.
- Compare the bases. The cabinet is narrow, so its CG only has to shift a short distance sideways to pass the edge of its base. The bench is wide, so its CG must travel much farther to reach the edge.
- Apply the tipping rule: an object falls over when its center of gravity moves past the edge of its base of support.
- The tall, narrow, top-heavy cabinet tips first. Its high CG swings outward quickly as it tilts, and its narrow base gives the CG only a short trip to the edge. The wide, low bench barely notices the push.
Tip: For any stability question, ask two things: how HIGH is the weight, and how WIDE is the base? Low and wide = stable. Tall and narrow = tippy. You do not need numbers, just this comparison.
Key takeaway: Springs follow Hooke's law: stretch is proportional to force, so double the pull means double the stretch. Springs in series (end to end) add their stretches and act softer; springs in parallel (side by side) share the load and act stiffer. Materials feel three basic forces: tension (pulling, the only force a rope can carry), compression (squeezing, what columns and legs feel), and shear (sliding, like scissors). An object is stable while its center of gravity stays over its base, so a wider base and a lower center of gravity both resist tipping. Triangles cannot change shape without a side changing length, which is why trusses in bridges and towers are built from them. A beam on two supports shares its load, with the closer support carrying more. Metals expand when heated, which mechanics use for shrink fits and freeing stuck parts. Bearings and lubrication cut friction so machines run cooler and last longer, and counterweights balance loads across a pivot so motors only lift the difference. Keep picturing real equipment as you review, and these questions will feel like common sense on test day.