Gears, Fluids and Structures

⚙️ Gears, Fluids and Structures

Part 1 of 7 — Gears, Fluids and Structures on the ASVAB

The rest of Mechanical Comprehension

Mechanical Comprehension (MC) is $15$ questions in $22$ minutes on the CAT-ASVAB, about $1\frac{1}{2}$ minutes each. Besides forces and simple machines, the official topic list includes "mechanical motion," "fluid dynamics" and "structural support." In practice that means pictures like these:

• a gear train: which way does the last gear turn, and how fast?
• belt-driven pulleys of different sizes: which one spins fastest?
• a cam pushing a rod up and down, or springs stretched by weights
• a hydraulic lift or a pump with valves
• objects floating in water
• a beam on two posts (which post holds more?), or shelves and brackets (which holds the most weight?)

Why these are good points

Each picture type has one rule behind it. Learn the rule once and every version of the picture becomes a 30-to-60-second question:

The plan for this lesson

Where MC counts: Marine Corps GT ($\text{AR} + \text{MC} + \text{VE}$) and MM, Air Force M, Navy MEC, MEC2 and NUC, and the Army's mechanical maintenance area, which leans heavily on AS, MC and EI. Strong MC scores open a wide range of mechanical, aviation, engineering and technical jobs.

Warm-up 1. Use the figure. 🎯

Warm-up 2. Use the figure. 🎯

Key Takeaways — Part 1

• This lesson covers gears, belts, cams, springs, fluids, buoyancy and structures, all picture-driven
• Meshed gears turn opposite ways; the post nearer a load carries more of it
• Every picture type has one rule. Learn the rule, then each question takes well under the $1\frac{1}{2}$ minutes you have

Up Next: Part 2 — gears and gear trains.

⚙️ Gears, Fluids and Structures

Part 2 of 7 — Gears and Gear Trains

Rule 1: direction alternates

Two meshed gears always turn in opposite directions. In a chain of gears in a row, the direction alternates:

Shortcut: odd-numbered gears turn the same way as the first; even-numbered gears turn the opposite way.

An idler gear is a gear placed between two others just so they turn the same direction. It does not change the speed ratio between the first and last gear.

Gears fixed on the same shaft turn together: same direction, same rpm.

Rule 2: teeth × rpm is the same for meshed gears

Every tooth that passes on one gear pushes one tooth on the other, so

$\text{teeth}_1 \times \text{rpm}_1 = \text{teeth}_2 \times \text{rpm}_2$

• The smaller gear turns faster; the larger gear turns slower.
• Speed and turning force trade off: a small gear driving a big gear gives less speed and more torque (a low gear for climbing); a big gear driving a small one gives more speed and less torque.

Worked example. A $10$-tooth gear at $300$ rpm drives a $30$-tooth gear. $10 \times 300 = 30 \times \text{rpm}$, so the big gear turns at $100$ rpm, one-third as fast but with three times the torque.

Worked example. In a row of three gears, $A$ ($20$ teeth) drives idler $B$ ($8$ teeth), which drives $C$ ($40$ teeth). Skip the idler for speed: $20 \times \text{rpm}_A = 40 \times \text{rpm}_C$. If $A$ turns at $60$ rpm, $C$ turns at $30$ rpm, in the same direction as $A$.

Gear speed. ✍️ Use the figure.

Gear A has $12$ teeth and turns clockwise at $90$ rpm. It drives gear B, which has $36$ teeth.

1) How fast does gear B turn, in rpm? (number only)

2) How many full turns does gear A make for each one turn of gear B? (number only)

Three in a row. Use the figure. 🎯

Two gears on one shaft. Use the figure. 🎯

Kinds of gears

Worm gear fact: with a single-thread worm, one full turn of the worm moves the worm gear by just one tooth. A $40$-tooth worm gear needs $40$ turns of the worm to go around once.

Name the gear. Use the figure. 🎯

Gear rules. 🔽

Key Takeaways — Part 2

• Meshed gears turn opposite ways; odd-numbered gears in a row match the first
• An idler changes direction only; gears on the same shaft turn together
• teeth × rpm is equal for meshed gears: small gear = fast, large gear = slow but more torque
• Bevel = change of angle; worm = big reduction, no back-driving; rack and pinion = rotation to straight line

Up Next: Part 3 — belts, chains, cams, cranks and springs.

⚙️ Gears, Fluids and Structures

Part 3 of 7 — Belts, Chains, Cams, Cranks and Springs

Belt-driven pulleys

A belt connects two pulleys so they turn together. Every inch of belt that runs off one pulley runs onto the other, so

$\text{diameter}_1 \times \text{rpm}_1 = \text{diameter}_2 \times \text{rpm}_2$

• The smallest pulley in a belt system turns fastest.
• An open belt (a simple loop) turns both pulleys the same direction.
• A crossed belt (a figure 8) turns them in opposite directions.
• Pulleys fixed on the same shaft turn together, at the same rpm.

Worked example. A motor turns a $3$-inch pulley at $1{,}200$ rpm. A belt drives a $9$-inch pulley. $3 \times 1{,}200 = 9 \times \text{rpm}$, so the big pulley turns at $400$ rpm.

Chains and sprockets

A chain works exactly like a belt, but with teeth: teeth × rpm is the same for both sprockets, and both turn the same direction. On a bicycle with a $48$-tooth front sprocket and a $16$-tooth rear sprocket, the rear wheel turns $48 \div 16 = 3$ times for each turn of the pedals.

Belt drive. ✍️ Use the figure.

A $4$-inch pulley turns clockwise at $300$ rpm and drives a $12$-inch pulley with an open belt.

1) How fast does the $12$-inch pulley turn, in rpm? (number only)

2) If the driver speeds up to $600$ rpm, how fast does the $12$-inch pulley turn? (number only)

Crossed belt. Use the figure. 🎯

Which pulley is fastest? Use the figure. 🎯

Cams, cranks and ratchets

Counting cam lifts: the follower rises once for each lobe (bump) on the cam, every revolution. A cam with $3$ lobes turning at $100$ rpm lifts its follower $300$ times a minute.

Two cams. Use the figure. 🎯

Springs

A spring stretches (or compresses) in proportion to the force on it (Hooke's law):

$F = kx$

where $x$ is the stretch and $k$ is the spring's stiffness. Double the weight, double the stretch.

Worked example. A spring stretches $2$ inches under a $10$-lb weight. Under $25$ lb it stretches $\frac{25}{10} \times 2 = 5$ inches.

So two springs in series act like a softer spring, and two side by side act like a stiffer one.

Springs. ✍️ Use the figure.

All springs are identical and every weight is the same. In setup A, the single spring stretches $3$ inches.

1) In setup B (two springs in series), how far does the weight move down in total, in inches? (number only)

2) In setup C (two springs side by side sharing a bar), how far does each spring stretch, in inches? (decimal or fraction)

Key Takeaways — Part 3

• Belts: diameter × rpm is equal; smallest pulley is fastest; open = same direction, crossed = opposite
• Chains and sprockets: teeth × rpm is equal, same direction
• Cam: rotary to up-and-down; lifts per minute = lobes × rpm. Crank: back-and-forth to rotary. Ratchet: one way only
• Springs: stretch ∝ force; series = more stretch, parallel = less stretch

Up Next: Part 4 — fluids, Pascal's principle and hydraulics.

⚙️ Gears, Fluids and Structures

Part 4 of 7 — Fluids, Pascal's Principle and Hydraulics

Pressure in a liquid grows with depth

The deeper you go in a liquid, the more liquid is stacked above you, so the pressure increases with depth.

• At the same depth, the pressure is the same in every direction and does not depend on the shape or width of the container. A narrow pipe and a wide tank filled to the same height have the same pressure at the bottom.
• A water tower is tall so the water at the bottom has high pressure to push through the town's pipes.
• Water squirts out of a hole near the bottom of a tank faster and farther than out of a hole near the top.
• Dams are built thicker at the bottom, where the water pressure is greatest.

Pascal's principle

Pressure applied to a confined fluid is passed on equally to every part of the fluid and the walls of its container.

Liquids barely compress, so a push on one end of a sealed system shows up instantly at the other end. That is how hydraulic brakes, jacks and lifts work. Air bubbles ruin the effect, because gas does compress: air in brake lines makes the pedal feel spongy.

Hydraulic lifts

Because the pressure is the same under both pistons,

$\frac{F_1}{A_1} = \frac{F_2}{A_2} \qquad \Rightarrow \qquad F_2 = F_1 \times \frac{A_2}{A_1}$

A small force on a small piston makes a large force on a large piston.

The trade-off: the volume of liquid pushed out of the small cylinder equals the volume that enters the large one, so

$A_1 \times d_1 = A_2 \times d_2$

The small piston moves much farther than the large one. Force is multiplied; work is not.

Worked example. A car lift has a small piston of $3$ square inches and a large piston of $90$ square inches. A $100$-lb push on the small piston gives $100 \times \frac{90}{3} = 3{,}000$ lb on the large piston. To raise the car $1$ inch, the small piston must move $30$ inches.

Watch areas, not widths. A piston's area grows with the square of its diameter. A piston with twice the diameter has four times the area, so it multiplies force by $4$, not $2$.

Hydraulic lift. ✍️ Use the figure.

A $50$-lb force pushes down on a small piston with an area of $2$ square inches. The large piston has an area of $40$ square inches.

1) The pressure in the fluid, in psi (number only):

2) The upward force on the large piston, in pounds (number only):

3) How far must the small piston move to raise the large piston $1$ inch? (number only, inches)

Diameter trap. 🎯

Holes in a tank. Use the figure. 🎯

Liquids in motion

Flow through a pipe: in a full pipe, the same volume of liquid passes every point each second. Where the pipe gets narrower, the liquid must move faster to keep up. That is why putting your thumb over the end of a hose makes the water shoot out faster.

Pumps and valves: pumps use one-way (check) valves so liquid can move only forward. A simple piston pump draws liquid in through an inlet valve on the up stroke and pushes it out through an outlet valve on the down stroke.

Siphon: a tube can carry liquid up over the rim of a tank and down the other side, as long as the tube's outlet is lower than the liquid's surface in the tank. Air pressure on the tank's surface keeps pushing the liquid through.

The narrowing pipe. Use the figure. 🎯

Key Takeaways — Part 4

• Liquid pressure increases with depth, is the same at the same depth, and does not depend on container shape
• Pascal: pressure on a confined fluid is passed on equally everywhere
• Hydraulic lift: $F_2 = F_1 \times \frac{A_2}{A_1}$; the small piston moves farther: $A_1 d_1 = A_2 d_2$
• Use areas: twice the diameter = 4 times the force
• Liquids barely compress (air in brake lines = spongy brakes); narrower pipe = faster flow

Up Next: Part 5 — buoyancy and structural support.

⚙️ Gears, Fluids and Structures

Part 5 of 7 — Buoyancy and Structural Support

Buoyancy: Archimedes' principle

Any object in a fluid is pushed up by a buoyant force equal to the weight of the fluid it pushes aside (displaces).

• If the object is less dense than the fluid, it floats. It sinks only until it has displaced its own weight of fluid.
• If it is denser, it sinks, but it still weighs less in the fluid than in air: its apparent weight = real weight − buoyant force.
• A steel ship floats because its hollow hull pushes aside a huge volume of water, enough to weigh as much as the ship.
• Salt water is denser than fresh water, so a boat floats higher in the ocean than in a lake.
• Of several same-size blocks floating in the same water, the one that sits deepest is the densest (heaviest).

Worked example. A $50$-lb block is lowered completely into water and pushes aside $30$ lb of water. The buoyant force is $30$ lb, so the block's apparent weight is $50 - 30 = 20$ lb, and it sinks. A $20$-lb log that would displace $30$ lb of water if pushed under instead floats, sinking only until it displaces $20$ lb.

Floating blocks. Use the figure. 🎯

Buoyancy checks. 🎯

Beams on two supports

When a load sits on a beam held up by two posts, the post closer to the load carries more of it. Each post's share is proportional to the load's distance from the other post:

$\text{share on post A} = \text{load} \times \frac{\text{distance from load to post B}}{\text{distance between posts}}$

The two shares always add up to the whole load. A load in the middle is shared equally, and so is the beam's own weight if the beam is uniform.

Worked example. A $900$-lb machine sits on a $9$-ft beam, $3$ ft from post A and $6$ ft from post B. Post A carries $900 \times \frac{6}{9} = 600$ lb; post B carries $900 \times \frac{3}{9} = 300$ lb. Check: $600 + 300 = 900$. ✓

Which post holds more? ✍️ Use the figure.

A $600$-lb load sits on a $12$-ft beam, $4$ ft from post A and $8$ ft from post B.

1) The load carried by post A, in pounds (number only):

2) The load carried by post B, in pounds (number only):

Structural strength

• Triangles are rigid. A square or rectangular frame with pinned corners can fold (rack) into a slanted parallelogram. Add one diagonal brace and it becomes two triangles that cannot change shape. That is why bridges, roof trusses, towers and gates are built from triangles.
• Shelf brackets: a diagonal brace that runs from low on the wall out to the outer end of the shelf makes a large triangle and supports the most weight. A short brace near the wall leaves the outer end hanging.
• Beams are stronger on edge. A $2 \times 8$ joist set on its narrow edge bends far less than the same board laid flat, because its material is spread farther from the center of bending. An I-beam puts most of its steel in the top and bottom flanges for the same reason.
• Load path: weight travels from the roof to the rafters, down the walls and studs, to the foundation and the ground. Every member along the way must carry the load above it.

Which shelf holds the most? Use the figure. 🎯

Rigid frames. Use the figure. 🎯

Flat or on edge? Use the figure. 🎯

Key Takeaways — Part 5

• Buoyant force = weight of fluid displaced; a floating object displaces its own weight
• Same-size floaters: the one riding deepest is the heaviest; boats float higher in salt water
• Apparent weight under water = real weight − buoyant force
• Beam on two posts: the nearer post carries more; share = load × (distance to the other post) ÷ span
• Triangles are rigid; a brace to the shelf's outer end is strongest; boards are stiffer on edge

Up Next: Part 6 — test strategy and traps for gears, fluids and structures.

⚙️ Gears, Fluids and Structures

Part 6 of 7 — Gears, Fluids and Structures Test Strategy and Traps

Timing

Mechanical Comprehension: $15$ questions in $22$ minutes, about $88$ seconds each. Gear-direction, belt and float questions are often answerable in 20 to 40 seconds once you know the rule; save your time for the multi-step ones (compound gears, hydraulic distances, beam shares).

• You must answer each question to move on, and you cannot go back.
• Treat every question as scored; tryout items look identical.
• If time runs short, keep working real answers. A string of wrong answers at the end of a subtest is penalized heavily on the CAT.

Fast routines

The traps test writers like

Spot the trap. 🎯

Speed drill. Under 30 seconds each. ⏱️

Key Takeaways — Part 6

• Gears: count for direction, teeth × rpm for speed, skip idlers, same shaft = same rpm
• Belts: open = same, crossed = opposite, smallest = fastest
• Hydraulics: areas, not diameters; the small piston moves farther
• Pressure depends on depth; floaters displace their own weight
• Beams: the nearer post carries more; series springs stretch more

Up Next: Part 7 — a mixed mastery set for gears, fluids and structures.

⚙️ Gears, Fluids and Structures

Part 7 of 7 — Gears, Fluids and Structures Mastery Set

This set mixes gear trains, belts, chains, cams, springs, hydraulics, buoyancy and structures. Use the fast routines from Part 6, and give yourself about $1\frac{1}{2}$ minutes per question.

Question 1. Use the figure. 🎯

Question 2. Use the figure. 🎯

Question 3. ✍️ Use the figure.

A $1{,}200$-lb load sits on a $10$-ft beam, $6$ ft from post A and $4$ ft from post B.

1) The load carried by post A, in pounds (number only):

2) The load carried by post B, in pounds (number only):

Questions 4 to 7. 🎯

Questions 8 to 10. 🎯

Lesson Wrap-Up

Next steps: take the exit quiz for this topic. You have now covered all of Mechanical Comprehension. Next comes Assembling Objects, the last CAT-ASVAB subtest.