Mechanical reasoning, oriented

A force is a push or a pull with a direction, so two forces add like arrows rather than like plain numbers. Two equal pulls in the same direction double the effect; in opposite directions they cancel; at an angle they give something in between. When a question shows a force at a slant, break it into two pieces at right angles: one pointing the way the load moves and one crosswise. Only the first piece moves the load.

Newton's first law says an object keeps doing what it is already doing until a net force acts. That is why an unsecured load slides forward when a truck brakes: nothing pushed it forward, the truck simply stopped pushing it along. A steady speed therefore needs no net force at all, only enough push to balance friction and drag. The second law links the net force to acceleration: the same push speeds up a light cart more than a heavy one. The third law says forces come in pairs, but the two act on different objects, so they never cancel each other out. Mass is the amount of matter and stays the same everywhere, while weight is the pull of gravity on that mass, which is why your weight would be smaller on the Moon.

To solve a balance problem, isolate one object and draw every force acting on it: its weight, the push of each surface, the pull of each rope. Ignore forces the object exerts on other things. If the object is still, those forces must add to zero in every direction.

Friction opposes sliding. It takes more force to start a slide than to keep one going, which is why a stuck crate breaks loose suddenly. For ordinary dry surfaces, friction depends on how hard the surfaces are pressed together and on the pair of materials, summed up in a coefficient of friction. Rolling replaces sliding with much smaller losses, lubrication separates surfaces with a film, and brakes, belts and your own footing all depend on friction being there.

Rule: before you predict any motion, draw the forces on one object and ask whether they balance; if they do, its motion does not change.

Torque is the turning effect of a force. It equals the force times the perpendicular distance from the pivot to the line of the force. The same push at twice the distance gives twice the turning effect, which is why a long wrench loosens a stuck nut that a short one cannot. A force aimed straight at the pivot produces no torque at all, and a force at a slant does less than one at a right angle to the handle.

An object is balanced when the forces add to zero and the turning effects on each side are equal. On a seesaw, a light child far from the pivot can balance a heavy adult close to it. Multiply each weight by its distance; the side with the larger product goes down.

Every object behaves as if its whole weight acted at one point, its center of gravity. An object tips once a vertical line dropped from that point lands outside the area it stands on. Keeping that point low and the footprint wide resists tipping, which is why a cabinet with heavy items on the top shelf tips sooner, and why cranes carry counterweights on the side opposite the load.

A lever is a rigid bar turning about a fulcrum. Effort times its arm equals load times its arm, so a long effort arm lets a small push move a heavy load. In a first-class lever the fulcrum sits between effort and load, as in a crowbar. In a second-class lever the load sits between, as in a wheelbarrow, and you always gain force. In a third-class lever the effort sits between, as in tweezers or your forearm, and you gain speed and reach while giving up force. Whatever force a lever gains, it costs distance in the same ratio.

Rule: for any balance or lever question, multiply each force by its distance from the pivot and compare the products; never compare the forces alone.

Work is force times the distance moved in the direction of the force. A machine cannot change the amount of work needed, only how it is split between force and distance. That single idea explains pulleys, ramps, screws and wheels.

A fixed pulley only changes the direction of the pull, so its advantage is one. A movable pulley rides on the load, and two rope segments share the weight, so you pull half as hard but pull twice as much rope. In any pulley system, count the rope segments that hold up the moving block; that count is the ideal mechanical advantage. Real systems lose some of it to friction in the sheaves.

A wheel and axle is a lever turned in a circle. Turning the large wheel moves the small axle with more force, which is how a winch, a screwdriver handle or a doorknob works. A ramp lets you raise a load with less force than lifting it straight up because you push it along a longer path. A wedge is a ramp you drive into the work, and a screw thread is a ramp wrapped around a cylinder; a finer thread takes more turns but less force.

Energy comes in two everyday forms here: kinetic energy of motion and potential energy of position. A raised weight stores energy that it gives back as it falls. Energy is conserved, so any answer describing a machine that puts out more energy than it takes in is wrong. Efficiency is useful work out divided by energy in, and the missing part mostly becomes heat. Power is how fast work is done; one horsepower is 550 foot-pounds per second, or about 746 watts.

Rule: whenever a machine reduces the force, look for the extra distance it charges you; if you cannot find it, recheck the setup.

When two gears mesh, the smaller one turns faster and the larger one turns slower but with more turning force. The ratio is the ratio of their tooth counts. Gearing down multiplies torque and divides speed; power stays the same, minus friction. Two meshed external gears turn in opposite directions, and every extra gear in a single line reverses the direction again. An idler gear placed between two others changes direction without changing the overall ratio. Two gears fixed to one shaft turn together, which lets a compound train multiply ratios in a small space. A worm gear can drive its wheel but usually cannot be driven backward by it, which is why hoists use one.

Belts and chains follow the same speed-versus-force rule, using pulley or sprocket diameters instead of teeth. An open belt keeps both pulleys turning the same way, while a crossed belt reverses the driven one. A belt grips by friction, so it can slip under overload; a chain cannot slip, but it needs lubrication and stretches with wear.

Linkages convert motion. A crank and slider turns rotation into back-and-forth motion and back again: in an engine, the offset of the crankshaft's rod journal turns the piston's up-and-down travel into rotation. A cam and follower turns steady rotation into a timed lift, as the cam lobes do when they open an engine's valves. A ratchet allows motion in one direction only.

A drivetrain chains these stages. A transmission picks a ratio so the engine runs at a useful speed, with a low gear for starting. A differential lets the outer wheel turn faster than the inner one on a curve. Clutches and brakes work by friction and must shed heat, and shear pins and slip clutches are placed to fail first and protect costlier parts.

Rule: to find direction, step through each mesh and crossed belt in turn; to find speed, multiply the ratios; never trust how a gear train looks.

Momentum is mass times velocity, so a slow heavy truck can carry as much as a fast light car. In a collision with no outside force, the total momentum before equals the total after. When a gun fires, the gun recoils backward with momentum equal and opposite to the bullet's. Kinetic energy is conserved only in an elastic collision; when objects stick together, some becomes heat, sound and damage. Stopping something takes force over time, so stretching the stopping time lowers the force, which is how crumple zones and padding protect you.

Every point on a spinning wheel turns at the same rate, but points farther out move faster because they travel a bigger circle. A heavy rim makes a wheel hard to start or stop, which is how a flywheel works: through its inertia it smooths out an engine's separate power strokes and stores energy between them.

To move in a circle, an object needs a force pulling it toward the center. There is no outward force on the object itself; if a whirling stone's string is cut, it flies off along a straight line, not straight outward.

Rotating machines fail in predictable ways. Mass that is off center shakes the machine with a force that grows quickly with speed. Bearings carry loads either across the shaft or along it, and a bearing used against the wrong kind of load wears out fast. Misaligned shafts force the coupling to bend on every turn. If a machine runs at a speed matching one of its structure's natural frequencies, small forces build into large vibrations.

Rule: in a collision, add up momentum before and after; in a spinning machine, ask where the mass sits and what holds it on its circle.

Pressure is force per unit of area. In a liquid it increases with depth and with the liquid's density, and at a given depth it is the same whether the container is a narrow pipe or a wide tank. Gauge pressure is measured relative to the surrounding air; absolute pressure adds the atmosphere back in.

Pressure applied to a confined liquid is passed on undiminished throughout it. Because force equals pressure times area, a small piston pushing on a large one multiplies the force by their area ratio, while the large piston moves a shorter distance. Liquids barely compress, which is what makes hydraulic systems firm; gases compress easily, so air trapped in a line makes it spongy.

An object in a fluid is pushed up by a force equal to the weight of the fluid it displaces. If the object weighs more than the fluid it displaces, it sinks, and it seems lighter under water by that displaced weight. If it weighs less, it rises and floats at the depth where it displaces exactly its own weight, so adding cargo makes a boat sit lower.

In a full pipe, the same volume passes every point each second, so the fluid speeds up where the pipe narrows. Where it moves faster, its pressure is lower; a Venturi narrowing uses this to measure flow. Smooth layered flow becomes turbulent at higher speeds, and friction in long or narrow pipes costs pressure, as does lifting the fluid higher. Drag on a moving body grows with the square of its speed and depends on its shape.

Pumps come in two families. A positive displacement pump moves a fixed volume with each stroke and must be protected against overpressure. A centrifugal pump flings fluid outward with a spinning impeller, must be primed with liquid before it can pump, and can be damaged by cavitation when vapor bubbles form and collapse at a low-pressure inlet. Check valves let flow go only one way, and turbines run the idea backward, using a moving fluid to spin a shaft.

Rule: in a fluid question, find the depth for pressure, the displaced weight for floating, and the narrowest point for speed.

Stress is force divided by the area carrying it, and strain is how much a part stretches compared with its length. In the elastic range, strain is proportional to stress, so a spring stretches twice as far under twice the load. Beyond that range, the part keeps a permanent bend. Loads can pull a part (tension), squeeze it (compression) or slide one layer past another (shear). Repeated loading can crack metal even below the load it could survive once, and such fatigue cracks tend to start at notches and scratches.

In a structure, a beam under load is squeezed on one face and stretched on the other, and making it deeper stiffens it far more than making it wider. Triangles hold their shape, which is why trusses are built from them. A long, slender column fails by bowing sideways before it crushes.

Most materials expand when heated, which is why bridges have expansion joints. Temperature measures how hot something is; heat is energy that flows because of a temperature difference. Heat moves by conduction through solids, by convection with moving fluid and by radiation. An engine takes in heat, turns part of it into work through its intake, compression, power and exhaust strokes, and must reject the rest; no engine can turn all of its heat into work.

Good shop practice follows the same mechanics. Secure work with clamps or a vise so both hands are free and the force goes into the cut. A sharp edge concentrates force, so it cuts with less effort.

Machine safety starts with reading the mechanism. Find where effort goes in and work comes out, then follow the force from one to the other. Guards keep hands, hair and clothing out of points of operation and the nip points where rollers or gears meet. Turning off the power is not enough: springs, raised parts and pressure can still hold energy, so it must be locked out and released before anyone services the machine.

Rule: to predict what fails first, look for the part carrying the most stress or the weak link placed there on purpose.