🏗️ Levers, Pulleys, Inclined Planes and Screws

Part 1 of 7 — Simple Machines and Mechanical Advantage

Why simple machines are all over the MC subtest

Mechanical Comprehension (MC) is $15$ questions in $22$ minutes on the CAT-ASVAB, about $1\frac{1}{2}$ minutes each. A large share of them show a simple machine: a lever with weights on it, a set of pulleys, a ramp, a jack, a crank. Each one is solved with the same small idea, so this lesson pays off fast.

The six simple machines

Mechanical advantage (MA)

A machine's mechanical advantage tells you how much it multiplies your force:

$\text{MA} = \frac{\text{output force (load)}}{\text{input force (effort)}}$

Worked example. A jack lets you lift a $3{,}000$-lb corner of a truck with a $60$-lb push on the handle. $\text{MA} = 3{,}000 \div 60 = 50$.

🏗️ Levers, Pulleys, Inclined Planes and Screws

Part 2 of 7 — Levers

Three parts, three classes

Every lever has a fulcrum (the pivot), a load (what you move) and an effort (where you push or pull). The class depends on which one is in the middle.

Memory hook "F-L-E, 1-2-3": the thing in the middle is Fulcrum for 1st class, Load for 2nd, Effort for 3rd.

🏗️ Levers, Pulleys, Inclined Planes and Screws

Part 3 of 7 — Pulleys

Fixed and movable pulleys

The one rule you need: count the supporting ropes

$\text{MA} = \text{number of rope segments that hold up the moving block}$

Count every segment of rope attached to or wrapped around the moving (lower) block. Do not count the free end if it comes down from a fixed pulley; that segment just changes direction. Do count the free end if it leaves the moving block and you pull up on it.

Ideal effort and rope pulled:

$\text{effort} = \frac{\text{load}}{\text{MA}}, \qquad \text{rope pulled} = \text{MA} \times \text{distance the load rises}$

Worked example. A block and tackle has $4$ rope segments holding the lower block, which carries a $600$-lb engine. Effort $= 600 \div 4 = 150$ lb. To lift the engine $2$ feet, pull $4 \times 2 = 8$ feet of rope.

Why use a single fixed pulley at all if its MA is $1$? Because pulling down is easier than lifting up: you can use your body weight, as when raising a flag or hoisting a bucket onto a roof.

🏗️ Levers, Pulleys, Inclined Planes and Screws

Part 4 of 7 — Inclined Planes and Wedges

The inclined plane (ramp)

Rolling or sliding a heavy load up a ramp takes less force than lifting it straight up, because you spread the same work over a longer distance.

$\text{MA} = \frac{\text{length of the ramp (slope)}}{\text{height of the ramp}}$

$\text{effort} \times \text{length} = \text{load} \times \text{height} \quad \Rightarrow \quad \text{effort} = \frac{\text{load} \times \text{height}}{\text{length}}$

Use the length along the slope, not the horizontal distance along the ground.

Worked example. A $300$-lb generator is pushed up a ramp $9$ feet long to a truck bed $3$ feet high. $\text{MA} = 9 \div 3 = 3$, and the ideal effort is $300 \div 3 = 100$ lb.

Key ideas:

• A longer ramp to the same height needs less force but more distance.
• A steeper ramp (shorter, same height) needs more force.
• The work (load × height) is the same either way; friction makes the real effort a bit more than ideal.
• Roads that switch back and forth up a mountain are long inclined planes: gentler grade, longer drive.

🏗️ Levers, Pulleys, Inclined Planes and Screws

Part 5 of 7 — Screws and Wheel and Axle

Wheel and axle

A wheel and axle is a large wheel (or crank, or handle) fixed to a smaller shaft so they turn together. It works like a lever that goes all the way around: the center of the axle is the fulcrum.

$\text{effort} \times \text{wheel radius} = \text{load} \times \text{axle radius}, \qquad \text{MA} = \frac{\text{wheel radius}}{\text{axle radius}}$

(Diameters work just as well as radii, as long as you use the same kind for both.)

Worked example. A well winch has a crank handle $12$ inches from the center and a drum (axle) of radius $2$ inches. $\text{MA} = 12 \div 2 = 6$. Lifting a $90$-lb bucket takes $90 \div 6 = 15$ lb on the handle. The handle travels $6$ times as far as the rope.

Where you see it: steering wheels, doorknobs, screwdrivers (a fatter handle is easier to turn), faucet handles, winches and capstans, the crank on a hand drill.

Turning it the other way: if the force is applied to the axle to turn the wheel (like a car's drive axle turning its tires), you lose force but gain speed and distance at the rim.

🏗️ Levers, Pulleys, Inclined Planes and Screws

Part 6 of 7 — Simple Machines Test Strategy and Traps

Timing

Mechanical Comprehension: $15$ questions in $22$ minutes, about $88$ seconds each. Simple-machine questions are some of the most reliable points on the subtest, because one formula solves each type. Aim for 45 to 75 seconds each, with one line of scratch work.

• You cannot go back on the CAT, so finish each question before moving on.
• Treat every question as scored; tryout questions look exactly the same.
• If time runs short, keep working real answers. The CAT penalizes a run of wrong answers at the end of a subtest heavily.

One formula per machine

All six are the same idea: force × distance in = force × distance out.

Sanity checks that catch most mistakes

• With MA above 1, the effort must be smaller than the load. If your effort comes out bigger, you multiplied when you should have divided.
• The heavier side of a balanced lever is always closer to the fulcrum.
• A machine never puts out more work than you put in.

🏗️ Levers, Pulleys, Inclined Planes and Screws

Part 7 of 7 — Simple Machines Mastery Set

Every simple machine appears in this set: levers of all three classes, pulleys, ramps, wedges, wheels and axles, and screws. Work each one with its formula, then sanity-check: with MA above $1$, the effort must come out smaller than the load. Allow about $1\frac{1}{2}$ minutes per question.