Mechanical Reasoning curriculum

A force has a size and a direction, so you cannot just add the numbers. Two 100-newton pulls in the same direction give 200 newtons, in opposite directions they give zero, and at an angle they give something in between. Always ask which way each force points.

Several forces on one point can be replaced by a single resultant with the same effect. Going the other way, a slanted force can be split into perpendicular components, such as along and across a ramp, which usually makes a problem much easier.

A part can carry load by being pulled (tension), pushed (compression) or slid across itself (shear). A rope works only in tension, a column mostly in compression, and a bolt across a joint in shear. Each fails differently: snapping, crushing or buckling, and slicing.

Mass is how much matter an object has, and it is the same anywhere. Weight is the pull of gravity on that mass, so it changes with location: an object weighs less on the Moon but keeps the same mass. A scale measures weight, so its reading would change on the Moon.

The normal force is a surface's push straight out against whatever presses on it. It equals an object's weight only on a level surface with nothing else pushing up or down. On a slope it is smaller, and pressing down on the object makes it larger.

Force is measured in newtons or pounds-force, and torque or work in newton-meters, foot-pounds or inch-pounds. One foot-pound is 12 inch-pounds, so a value off by that factor is a unit error. Check that the units of an answer make sense before trusting the number.

Scalars, such as distance, speed, mass and energy, have size only. Vectors, such as displacement, velocity and force, also have direction. Only vectors need you to track which way they point, and only vectors can cancel each other out.

An object keeps doing what it is doing, resting or moving in a straight line at constant speed, until a net force changes that. When a truck brakes, an unsecured load keeps moving forward; nothing pushes it, the truck simply stops under it.

Acceleration equals net force divided by mass. Double the net force and the acceleration doubles; double the mass and it halves. What matters is the net force, the total after all opposing forces are subtracted.

Whenever one object pushes on another, the second pushes back equally in the opposite direction. The two forces act on different objects, so they never cancel each other. A hammer hits a nail and the nail hits the hammer just as hard.

An object moving at a steady speed in a straight line has zero net force on it. A car cruising at constant speed still needs the engine, but only to balance friction and air resistance, not to keep it moving.

Without air resistance, all objects fall with the same acceleration, whatever their weight. A feather falls slower than a hammer on Earth only because air resists it more; in a vacuum they land together.

In an elevator speeding up on the way up, the floor must push harder than your weight, so you feel heavier and a scale reads more. As it slows on the way up, or speeds up going down, you feel lighter. At constant speed you feel normal.

Draw the object on its own and add every outside force acting on it: weight, supports, ropes, friction and any pushes or pulls. Leave out forces the object exerts on other things. This one habit turns most confusing setups into simple sums.

A body at rest needs two things: the forces must add to zero, so it does not slide, and the moments must add to zero, so it does not turn. Some trick setups balance the forces but not the moments, and the object still rotates.

A member loaded at just two points, such as a straight strut pinned at both ends, carries force only along the line joining those two points, whatever its shape. Spotting one tells you the direction of a force without calculating it.

When three forces that are not parallel hold an object in balance, their lines of action must meet at a single point. If they did not, they would make the object turn.

An ideal rope carries the same tension along its whole length. When two angled ropes hold a load, each carries more than you might expect, and the flatter the angle, the higher the tension; a nearly flat sling can see forces many times the load.

A roller support pushes only straight out from its surface, a pin can push in any direction but lets the part rotate, and a fixed support resists pushing in any direction and rotation too. The type of support decides which reaction forces you solve for.

A load spread along a beam, such as snow on a roof, can be replaced by one equal force acting at the center of the spread load. For a uniform load, that point is the middle of the loaded length.

Torque is the turning effect of a force: the force times its perpendicular distance from the pivot. The same force applied twice as far from the pivot produces twice the torque.

Only the perpendicular distance from the pivot to the line of the force counts. A force aimed straight through the pivot has no moment arm, so it produces no torque, however hard you push.

Pulling a wrench at an angle wastes part of the force, because only the part perpendicular to the handle turns it. You get the most torque by pulling at a right angle to the handle.

For a balanced beam or seesaw, clockwise moments equal counterclockwise moments about the pivot. Multiply each force by its distance from the pivot, set the two sides equal, and solve for the unknown.

A couple is two equal forces pointing in opposite directions on either side of a center. They cancel as forces, so nothing slides, but they add as turning effects. Turning a steering wheel with both hands uses a couple.

A longer handle multiplies torque, because the same hand force acts farther from the bolt. That is why a breaker bar or cheater pipe loosens a stuck nut, and also why it can snap a bolt or strip a thread if you overdo it.

Before calculating, decide whether a question is about turning (torque, which depends on distance from a pivot) or about pulling in a line (force). Mixing them up, by ignoring the lever arm or adding one where none exists, is the most common misread.

The center of gravity is the point where an object's whole weight can be treated as acting. For a ring or a horseshoe it lies in the empty space, outside the material itself.

An object stays upright while a vertical line through its center of gravity falls inside its base of support. Once that line passes outside the base, the object tips over.

The lower the center of gravity, the farther an object can tilt before tipping. Loading the top shelf of a cabinet or the roof of a vehicle raises the center of gravity and makes it tip over sooner.

In stable equilibrium, a small push brings the object back, like a ball in a bowl. In unstable equilibrium, a small push makes things worse, like a ball on a dome. In neutral equilibrium, it just stays where it is moved, like a ball on a flat floor.

A counterweight places mass on the opposite side of a pivot from the load, so the two moments roughly balance. Cranes, elevators and drawbridges use counterweights to cut the force and power needed to lift.

Moving a load closer to the support or pivot shortens its moment arm, so it puts less turning load on the structure. Carrying a heavy box close to your body is easier for the same reason.

A lever balances when the effort times its distance from the fulcrum equals the load times its distance from the fulcrum. Every lever question reduces to that one equation.

In a first-class lever, the fulcrum sits between the effort and the load, as in a seesaw, a crowbar or scissors. Depending on where the fulcrum sits, it can gain force or gain distance.

In a second-class lever, the load sits between the fulcrum and the effort, as in a wheelbarrow or a nutcracker. The effort is always farther from the fulcrum than the load, so it always gains force.

In a third-class lever, the effort sits between the fulcrum and the load, as in tweezers, a broom or your forearm. The effort is closer to the fulcrum than the load, so it trades force for extra speed and range of movement.

Divide the length of the effort arm by the length of the load arm and you get a lever's mechanical advantage. Whatever force you gain, you give up in distance by the same ratio: lifting a load with half the force means moving your end twice as far.

Linking levers so one drives the next multiplies their mechanical advantages. Bolt cutters use two linked levers to cut through steel with hand pressure.

To read an unfamiliar tool, first find the fulcrum, where the effort goes in and where the load is. Their order tells you the lever class, and their distances tell you the advantage.

A fixed pulley only changes the direction of the pull. Its mechanical advantage is one: lifting 50 pounds still takes 50 pounds of pull, but you can pull down instead of up.

A movable pulley rides on the load, so two rope segments share the weight. That halves the force needed, but you must pull twice as much rope for the same lift.

The mechanical advantage of a pulley system equals the number of rope segments holding up the moving block. Count only the segments attached to or supporting the moving part, not the one you pull on if it runs to a fixed pulley.

A block and tackle combines several fixed and movable pulleys in two blocks to give a large mechanical advantage in a compact package, which is why it is used for heavy lifting on ships and in workshops.

A pulley system that multiplies force by four makes you pull four times as much rope, so the load rises at a quarter of your pulling speed. The work you put in still equals the work done on the load, before losses.

Real pulleys lose some effort to friction in the bearings and to bending stiff rope, so the actual mechanical advantage is less than the ideal count. The more pulleys in a system, the more of these losses add up.

A wheel and axle is a lever that turns in a full circle. Turning the large wheel moves the small axle with more force; the ratio of the wheel radius to the axle radius is the mechanical advantage.

A windlass or winch uses a crank turning a drum. The crank's radius compared with the drum's radius sets the force gain, which is how a person can raise a heavy bucket or anchor.

A screwdriver handle and a doorknob are both wheel-and-axle devices. A fatter handle gives a larger wheel radius, so the same hand force produces more turning force on the screw.

Rolling resistance is far smaller than sliding friction, which is why wheels make heavy loads easy to move. A wheel does not slide against the ground at the point of contact; it rolls over it.

A larger wheel travels farther in one turn, because its circumference is larger. It covers ground faster for the same turning speed, but needs more torque to turn it.

Wrapping a rope around a drum or post multiplies the holding force, because friction builds up around every part of the wrap. Each extra turn multiplies the effect, so a few turns let one person hold a heavy pull.

The gear ratio is the number of teeth on the driven gear divided by the number on the driving gear. When a small gear drives a larger one, the larger one turns slower but with more turning force.

Two meshed external gears turn in opposite directions. Each additional gear in a simple chain reverses the direction again, so count the gears to find which way the last one turns.

An idler gear sits between two others to change the direction of the last gear. It does not change the overall speed ratio, which depends only on the first and last gears in a simple train.

In a compound gear train, two gears share one shaft, so their ratios multiply. That is how a small, fast motor can drive a heavy, slow load through a few stages of gearing.

Gearing down, from a small gear to a large one, multiplies torque and divides speed by the same ratio. The power passed through stays the same, apart from friction losses.

Spur gears connect parallel shafts, bevel gears turn the motion through an angle, helical gears run more quietly, a rack and pinion turns rotation into straight-line motion, and a worm gear gives a very large reduction in a small space.

A worm drive can turn its gear, but the gear usually cannot turn the worm back. That self-locking holds a load in place without a brake, which is why worm drives are used on hoists and guitar tuning pegs.

A belt drive works like gears, but uses pulley diameters in place of tooth counts. A small driving pulley turning a large driven pulley slows the output and increases its torque.

An open belt keeps both pulleys turning in the same direction. Crossing the belt into a figure eight makes the driven pulley turn the opposite way.

A belt transmits force through friction against the pulley, so it can slip if overloaded or loose. More wrap around the pulley and proper tension increase the grip. A chain meshes with teeth, so it cannot slip.

A chain on sprockets gives positive engagement, so it can carry high torque without slipping. The price is noise, the need for lubrication, and stretch as the chain wears.

An idler or tensioner pulley presses on the belt to take up slack and increase how far it wraps around the drive pulleys. It improves grip without changing the drive ratio.

Flat belts suit light, fast drives; V-belts wedge into grooved pulleys for better grip and more power; and toothed belts mesh like gears, keeping exact timing, as in an engine's camshaft drive.

A ramp lets you raise a load with less force than lifting it straight up, because you push it a longer distance. A longer, shallower ramp needs less force but more distance for the same rise.

On a slope, an object's weight splits into a part pulling it down the slope and a part pressing it into the surface. The gentler the slope, the smaller the part pulling it downhill, which is what makes a ramp work.

A wedge is an inclined plane that moves. Driving it forward pushes its sides outward with much larger force, which is how an axe splits wood and a doorstop holds a door.

A screw thread is an inclined plane wrapped around a cylinder. The pitch, the distance the screw advances per turn, sets its mechanical advantage: a finer pitch means more turns and more force.

A screw jack or clamp gives a huge force gain because each turn advances it only a little. Friction in fine threads also keeps it from turning back under load, so it holds its position.

A coarse thread advances quickly with each turn but needs more force; a fine thread advances slowly but with more force and holds better against vibration.

A four-bar linkage is four rigid links joined by pins. Changing their lengths changes the motion it produces, so it can turn one rotation into a rocking or a specific path, as in windshield wipers.

A crank and slider converts rotation into back-and-forth motion, or the reverse. In a piston engine, the pistons push a connecting rod that turns the crankshaft.

A cam is a shaped wheel; as it turns, a follower riding on its edge rises and falls. The cam's shape sets exactly how far and when the follower moves, as when a camshaft opens an engine's valves.

An eccentric is a disc mounted off-center on a shaft. The offset sets the stroke: the total back-and-forth movement is twice the offset, so doubling the offset doubles the travel.

A ratchet and pawl lets a wheel turn one way while the pawl stops it from turning back. That one-way action is a safety feature on hoists and winches, holding a load if you let go.

In a toggle mechanism, the output force grows very large as two links approach a straight line. Toggle clamps and latches use this to hold firmly, and pushing just past straight locks them in place.

It takes more force to start something sliding than to keep it sliding, because static friction is larger than kinetic friction. That is why a stuck object breaks loose suddenly once you push hard enough.

Friction depends on how hard the surfaces are pressed together and on the materials, not on the area of contact, for ordinary dry surfaces. A box on its side slides as easily as one standing upright.

The coefficient of friction is the friction force divided by the normal force. It has no units and describes a particular pair of surfaces, such as rubber on dry concrete, not either surface alone.

A rolling wheel still meets resistance, because the wheel and the surface deform slightly as it rolls. Rolling resistance is much smaller than sliding friction but never zero, and soft or underinflated tires increase it.

A lubricant keeps a thin film of fluid between moving surfaces so they do not rub directly, cutting friction and wear. Using the wrong lubricant, too thin, too thick or incompatible, can leave parts less protected than the right one would.

Friction is often what makes things work: brakes stop wheels, clutches and belts transmit power, nails hold wood, and your shoes grip the floor. Without friction, you could not walk.

Work done against friction turns into heat and wears material away. That is why brakes can overheat and lose grip on long descents, and why bearings without lubrication fail.

Work is force times the distance moved in the direction of the force. Holding a heavy box still does no mechanical work, however tiring it is, because the box does not move. Carrying it across a level floor does no work against gravity either.

A moving object carries kinetic energy, which rises with the square of its speed. Potential energy is stored energy of position, such as a raised weight or a stretched spring. A swinging pendulum trades one for the other on every swing.

Energy cannot be created or destroyed, only changed from one form to another. A machine can never put out more energy than goes in, so any answer describing a machine that runs itself forever is wrong.

A machine that reduces the force you need makes you apply it over a longer distance. The work you put in equals the work it does plus what is lost to friction. Mechanical advantage is never free energy.

Power is how fast work is done: work divided by time. Lifting the same load to the same height takes the same work whether you go fast or slow, but doing it faster takes more power.

Efficiency is the useful work out divided by the total energy in, usually given as a percentage. No real machine reaches 100 percent; the missing share goes mostly into heat from friction.

Power is measured in watts or horsepower; one horsepower is about 746 watts. A kilowatt is 1,000 watts, so a 1-kilowatt motor delivers about 1.34 horsepower.

Momentum is mass times velocity. A slow, heavy truck can have as much momentum as a fast, light car, and the object with more momentum takes more force or more time to stop.

Changing an object's momentum takes a force applied over time. Spreading the stop over a longer time lowers the force, which is why crumple zones, airbags and bending your knees on landing all protect you.

In a collision, the total momentum of the objects before equals the total after, as long as no outside force acts. Kinetic energy, by contrast, is usually not conserved, because some becomes heat, sound and damage.

In an elastic collision the objects bounce apart and keep their total kinetic energy, as in an ideal pool-ball strike. In an inelastic collision some energy is lost, and when the objects stick together the loss is greatest.

When a gun fires a bullet forward, the gun recoils backward, because the total momentum must stay zero. A rocket works the same way, pushing exhaust out the back to move itself forward, which is why it works in space.

A hammer works because its heavy head carries momentum that is stopped in a very short time, producing a large force for an instant. Steady pressure on a nail, with no impact, produces far less force.

Every point on a spinning wheel turns at the same rate, but points farther from the center travel faster, because they cover a bigger circle each turn. The rim of a fan blade moves much faster than the hub.

Moment of inertia is resistance to changes in spinning, and it depends on where the mass sits, not just how much there is. Mass far from the axis makes something much harder to start or stop spinning.

A flywheel is a heavy wheel that stores energy in its spin. It smooths out an uneven power source, such as an engine's separate power strokes, by absorbing energy between pulses and releasing it as needed.

A spinning object keeps its angular momentum unless something twists it. Pulling mass toward the axis makes it spin faster, which is why a skater pulling in their arms speeds up.

To move in a circle, an object needs a force pulling it toward the center. There is no outward force on the object; the feeling of being thrown outward is your body trying to keep going straight.

A spinning wheel resists changes to the direction of its axis. That gyroscopic effect helps keep a moving bicycle upright and a spinning top balanced.

The force needed to hold a spinning part together grows with the square of its speed. A grinding wheel or flywheel spun far beyond its rating can break apart violently, which is why rated speeds must never be exceeded.

Within its elastic range, a spring stretches in proportion to the force pulling it: twice the load, twice the stretch. That proportional behavior is what lets a spring scale measure weight.

A spring's rate, or stiffness, is the force needed per unit of stretch or compression. A stiff spring needs more force to stretch the same amount. The rate is different from the largest load the spring can safely carry.

Springs side by side, in parallel, share the load and make a stiffer combination. Springs end to end, in series, each stretch under the full load, so the combination is softer. This is the opposite of what many people first guess.

Stress is the force on a material divided by the area carrying it, and strain is how much it stretches compared with its length. A thin rod sees much higher stress than a thick one under the same load, so it fails sooner.

Below its elastic limit, a material springs back to its original shape when the load is removed. Beyond that limit, it stays permanently bent. Designers keep working loads well below that limit.

Metal can break under repeated loading even when every load is below what it could survive once. Cracks tend to start where stress concentrates, at holes, sharp corners and scratches, then grow with each cycle.

Choosing a material means balancing strength, stiffness, toughness, weight and cost. No material is best at all of them; steel is strong and stiff but heavy, while aluminum is lighter but less stiff.

When a beam bends under load, one face is squeezed and the opposite face is stretched. Between them, a neutral axis carries neither. In a simple beam supported at both ends, the top is in compression and the bottom in tension.

A beam's depth matters far more than its width for resisting bending. That is why floor joists stand on edge and why I-beams put most of their material in flanges far from the center.

A beam sags far more as its span grows; sag rises steeply with length, so doubling the span makes the sag many times larger. Adding a support in the middle is often the cheapest way to stiffen it.

A cantilever is supported at only one end, like a diving board or a balcony. The bending is greatest at the support, so that is where it must be strongest.

A truss is built from triangles, which cannot change shape without changing a member's length. When loads are applied at the joints, each straight member carries pure tension or pure compression.

A long, slender column fails by bowing sideways, buckling, long before the material would crush. Bracing a column partway along its length shortens the free length that can buckle and raises the load it can carry.

Every load follows a path through a structure to the ground. Tracing that path shows which parts are critical. A redundant structure has more than one path, so losing one member does not bring it down.

Pressure is force divided by the area it acts on. The same force on a smaller area gives a higher pressure, which is why a sharp knife cuts and why snowshoes keep you on top of snow.

Pressure in a liquid increases with depth and with the liquid's density. At a given depth, it does not depend on how wide the container is or how much liquid it holds.

Containers of very different shapes, filled to the same depth with the same liquid, have the same pressure at the bottom. A narrow tube and a wide tank give equal readings, because pressure depends only on depth.

Gauge pressure is measured relative to the atmosphere, so an open gauge reads zero. Absolute pressure includes the atmosphere. A tire gauge reads gauge pressure; the absolute pressure inside the tire is higher by about one atmosphere.

A suction pump works by letting atmospheric pressure push liquid up a pipe, so it cannot lift water higher than the atmosphere can support. A siphon flows only if the outlet sits below the surface it draws from.

A manometer measures pressure by how high it pushes a column of liquid. A Bourdon gauge uses a curved tube that straightens as pressure rises, moving a needle; most read gauge pressure, relative to the surrounding air.

Pressure applied to a liquid in a closed system is passed on equally to every part of it. That is the principle behind every hydraulic jack, lift and brake.

In a hydraulic system, force is multiplied by the ratio of the piston areas. Because area grows with the square of the diameter, a piston three times as wide has nine times the area and gives nine times the force.

The large piston in a hydraulic system moves a much shorter distance than the small one, in proportion to the area ratio. The work in still equals the work out, just as with a lever.

Car brakes and hydraulic jacks multiply a small force into a large one. An air bubble in a brake line ruins this, because the air compresses, so the pedal goes soft instead of applying the brakes.

Hydraulic systems work because liquids barely compress, so a push at one end arrives fully at the other. Air or other gas in the line compresses and absorbs the stroke, making the system spongy and weak.

Pneumatic systems use compressed air, which gives speed and a cushioned action but less force and precise positioning. Hydraulic systems use oil, which gives great force and rigid control.

An object in a fluid is pushed up by a force equal to the weight of the fluid it displaces. That holds whatever the object is made of.

An object floats if its average density is less than the fluid's and sinks if it is greater. A steel ship floats because its hull encloses so much air that its average density is less than water's.

A floating object sinks until it has displaced its own weight of water. Add cargo and it settles deeper, displacing extra water equal in weight to the cargo, so the waterline rises.

An object under water seems lighter, because the buoyant force pushes up on it. Its reading on a scale drops by exactly the weight of water it displaces.

The same object floats higher in denser fluid. Salt water is denser than fresh water, so a ship rides higher at sea than in a river.

Whether a floating hull rights itself depends on where the upward buoyant force acts compared with its center of gravity. A low center of gravity, from heavy ballast low in the hull, helps it return upright when tipped.

In a pipe full of liquid, the same volume passes every point each second. Where the pipe narrows, the fluid must move faster to keep that volume flowing.

Along a streamline, where a fluid moves faster its pressure is lower. That trade between speed and pressure explains how a carburetor draws in fuel and contributes to explanations of lift.

A Venturi is a narrowing in a pipe. The fluid speeds up through it and its pressure drops, which can be used to measure flow or to draw in another fluid by suction.

In laminar flow, fluid moves in smooth layers; in turbulent flow, it mixes in swirling eddies. Higher speeds, wider pipes and less viscous fluids push the flow toward turbulence.

Thick, viscous fluids resist flowing, and long, narrow pipes add friction. Both take more pressure to push the same flow, which is why pumps must work harder on thick oil or through small, long lines.

A pump's pressure is spent in two ways: lifting fluid to a higher level, and overcoming friction in pipes and fittings. Both reduce the pressure left at the outlet.

Air and water resistance rise steeply with speed, so going faster costs much more power. Streamlined shapes, rounded at the front and tapered at the back, reduce that resistance.

A positive displacement pump traps a fixed volume of fluid and pushes it out with each stroke or turn. Its flow stays nearly the same whatever the outlet pressure, so blocking its outlet can build dangerous pressure without a relief valve.

A centrifugal pump spins an impeller that flings fluid outward and builds pressure. Its flow drops as the pressure it must work against rises.

A centrifugal pump must be filled with liquid, primed, before it can pump, because it cannot pull liquid through air. If inlet pressure falls too low, the liquid boils into vapor bubbles that collapse violently, cavitation, damaging the impeller.

A pump does not pull liquid up; atmospheric pressure pushes it into the low-pressure inlet. That caps how high any pump can lift water by suction, no matter how powerful it is.

Compressing a gas raises its pressure and its temperature, so compressors need cooling. A receiver tank stores compressed air, smoothing out changes in demand.

Valves start, stop and direct flow. A check valve lets fluid move in only one direction, which prevents backflow and is what makes a reciprocating pump push fluid forward instead of back and forth.

Fans and blowers move air against resistance from ducts and filters. The more restriction there is, the less air they deliver, so a clogged filter or kinked duct cuts the airflow.

A turbine works like a pump in reverse: moving fluid pushes on its blades and turns a shaft. Water wheels, wind turbines and steam turbines all take energy out of a moving fluid.

A heat engine takes in heat at a high temperature, turns part of it into work, and rejects the rest at a lower temperature. A four-stroke engine runs intake, compression, power and exhaust strokes in each cycle.

An engine must shed the heat it cannot turn into work. Coolant carries heat to a radiator, airflow carries it away, and oil both lubricates moving parts and carries heat from them.

Most materials expand when heated and shrink when cooled. Bridges have expansion joints and rails have gaps so that this movement does not buckle the structure.

Two metals bonded together expand by different amounts when heated, so the strip bends. A bimetallic strip in a thermostat uses that bending to open or close a switch.

Heat moves by conduction through solids, by convection with moving fluid, and by radiation as infrared light. Insulation slows conduction, a fan speeds convection, and a shiny shield reflects radiation.

Temperature measures how hot something is; heat is energy transferred because of a temperature difference. A large tub of warm water can hold more heat than a small cup of boiling water.

No heat engine can turn all of its heat into work; some must always be rejected at the cold end. Waste heat is therefore a built-in part of every engine's design, not a fault.

Shafts carry torque from one component to another, couplings join shafts, and keys lock gears or pulleys to shafts. A designer often makes one of these the deliberate weak point, so it fails before costlier parts.

Clutches connect and disconnect power, and brakes slow or stop motion; both usually work by friction. Both turn energy into heat, so they must shed that heat or they wear and lose grip.

A transmission picks a gear ratio to match engine speed to the load. Starting from rest needs a low gear to multiply torque, while cruising uses a high gear for speed at lower engine revolutions.

On a turn, the outer wheel travels farther than the inner one, so it must turn faster. A differential splits the engine's drive between the two wheels while letting them turn at different speeds.

Torque is multiplied at each stage of a drivetrain: by the transmission gear, then by the final drive. Tracing each ratio from the engine to the wheel gives the turning force available to move the vehicle.

A shear pin or slip clutch is a deliberate weak point. If the machine jams, it breaks or slips first, protecting more expensive parts such as gears and engines from damage.

If a rotating part's mass is off center, it creates a force that rotates with it and shakes the machine. That force grows with the square of the speed, so a small imbalance becomes violent vibration at high speed.

Plain bearings support a shaft on a sliding surface, while ball and roller bearings use rolling elements to cut friction. Each type is built for particular loads and speeds.

A radial load pushes across a shaft, while a thrust load pushes along it. Bearings are designed for one or the other or a mix, and a bearing used against the wrong kind of load wears out quickly.

When two connected shafts are not lined up, the coupling forces them into line on every turn. Misalignment is a leading cause of vibration, seal leaks and early bearing failure.

Every structure has natural frequencies at which it vibrates easily. If a machine runs at a speed that matches one, small forces build into large vibrations; that is resonance, and it can damage equipment quickly.

Damping absorbs vibration energy and turns it into heat, while isolation mounts, such as rubber pads or springs, keep a machine's vibration from passing into its base.

A box-end wrench grips all six sides of a nut, so it is less likely to round the corners than an open-end wrench, which grips only two. Overtightening can stretch a bolt or strip its threads, so use the specified torque.

Tightening a bolt stretches it slightly, like a stiff spring, and that tension clamps the joint together. This preload, not the bolt shank filling the hole, is what holds the parts in place.

A flat washer spreads a fastener's load over a wider area and protects the surface. Lock washers, locking nuts and thread-locking compounds help keep fasteners from loosening under vibration.

Chisels, drill bits and saw teeth are wedges. A sharp edge concentrates force on a small area, so it cuts with less force and produces less heat than a dull one.

A rule measures roughly, calipers measure inside, outside and depth dimensions more precisely, and a micrometer measures finer still. A square checks that surfaces meet at right angles. Choose the tool that matches the precision you need.

Clamps, vises and jigs hold the workpiece firmly, so your force goes into the cut rather than into moving the work. Secure work is also safer, because loose work can spin or fly off a machine.

To choose between two tools for a job, compare how each applies force, whether it can reach the work, and how it might fail. The right tool gives the most leverage with the least risk of slipping or breaking.

To understand an unfamiliar machine, first locate its input, the place you push, turn or pull, and its output, the part that does the job. Everything in between is a chain that changes force, speed or direction.

Follow the load from the input through each part, link by link, to the output. At each joint, gear or pivot, ask how the force or motion changes; most questions about a machine you have never seen give way to this.

To find which way a part moves, step through the mechanism one connection at a time. Each pair of meshed external gears reverses direction, and a crossed belt does the same.

When asked which setup takes less effort, compare their mechanical advantages, such as lever arm ratios, pulley rope counts or gear ratios, rather than relying on which looks easier.

Moving machines create pinch points where parts come together, nip points where rollers or gears meet, and dangers from rotating shafts. Guards are designed to keep hands, hair and clothing out of those zones.

Turning off the power does not make a machine safe. Springs can stay compressed, raised parts can fall, and pressure can remain in lines. Before servicing, every energy source must be locked out and any stored energy released or blocked.

To predict what fails first, look for the part carrying the highest stress, or the deliberate weak link the designer added. Shear pins, fuses and slip clutches are placed to fail before expensive parts do.