Levers, Pulleys, Inclined Planes and Screws

🏗️ 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. $\text{MA} = 3{,}000 \div 60 = 50$.

The catch: you trade force for distance

A machine cannot give you more work out than you put in. For an ideal machine (no friction):

$\text{effort} \times \text{effort distance} = \text{load} \times \text{load distance}$

So if a machine lets you push with one-fourth the force, you must push four times as far. That is why:

• a long ramp needs less force than a short steep one, but you walk farther;
• a pulley system with MA $4$ makes you pull $4$ feet of rope for every $1$ foot the load rises;
• a jack with a huge MA needs many strokes to lift the truck a few inches.

Worked example. A block and tackle with MA $4$ lifts a $200$-lb engine. Ideal effort $= 200 \div 4 = 50$ lb. To raise the engine $3$ feet, you pull $4 \times 3 = 12$ feet of rope. Check the work: $50 \times 12 = 600$ ft·lb in, $200 \times 3 = 600$ ft·lb out. ✓

Real machines lose some work to friction, so the real effort is a little more than the ideal answer. Unless a question mentions friction or efficiency, use the ideal numbers. Some machines (like tweezers or a broom) have an MA less than 1: they cost force but gain speed and distance at the business end.

Warm-up. 🎯

Warm-up 3. Use the figure. 🎯

Key Takeaways — Part 1

• The six simple machines: lever, pulley, wheel and axle, inclined plane, wedge, screw
• MA = load ÷ effort
• Ideal machines: effort × effort distance = load × load distance. Less force always means more distance
• Machines never reduce the work; friction makes real effort a bit higher than ideal

Up Next: Part 2 — levers: classes, fulcrum math and balance problems.

🏗️ 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.

In each lever, look for what sits between the other two.

The lever rule

A lever balances when the turning effect on each side of the fulcrum is equal:

$\text{effort} \times \text{effort arm} = \text{load} \times \text{load arm}$

Each arm is the distance from the fulcrum to where that force acts. Always measure from the fulcrum, not from the end of the bar.

$\text{MA} = \frac{\text{effort arm}}{\text{load arm}}$

Worked example (1st class). A $5$-foot pry bar rests on a fulcrum $1$ foot from a $200$-lb rock. The effort arm is $5 - 1 = 4$ ft.

$\text{effort} \times 4 = 200 \times 1 \quad \Rightarrow \quad \text{effort} = 50 \text{ lb}, \qquad \text{MA} = \frac{4}{1} = 4$

Worked example (seesaw). A $50$-lb child sits $6$ ft from the center. Where must a $75$-lb child sit to balance? $75 \times d = 50 \times 6 = 300$, so $d = 4$ ft. The heavier person sits closer to the fulcrum.

Several weights on one side: add up each weight × its distance on that side. The side with the bigger total goes down.

Moving the fulcrum toward the load makes the load arm shorter and the effort arm longer, so the effort needed drops.

Pry bar. ✍️ Use the figure.

An $8$-foot bar lifts a $120$-lb load at one end. The fulcrum is $2$ feet from the load, so the effort is applied $6$ feet from the fulcrum.

1) The effort needed to balance the load, in pounds (number only):

2) The mechanical advantage of this lever (number only):

Balance the beam. ✍️ Use the figure.

A $60$-lb box sits $4$ feet to the left of the fulcrum. An $80$-lb box goes on the right side.

1) How far to the right of the fulcrum must the $80$-lb box sit for the beam to balance, in feet? (number only)

2) If a $120$-lb box were used on the right instead, how far from the fulcrum would it have to sit, in feet? (number only)

The wheelbarrow. Use the figure. 🎯

Which way does it turn? Use the figure. 🎯

Name the class. 🔽

Key Takeaways — Part 2

• F-L-E, 1-2-3: fulcrum, load or effort in the middle makes a 1st-, 2nd- or 3rd-class lever
• effort × effort arm = load × load arm, with arms measured from the fulcrum
• MA = effort arm ÷ load arm; 3rd-class levers always have MA below 1
• Heavier weights balance closer to the fulcrum
• With several weights, add weight × distance on each side; the bigger total goes down

Up Next: Part 3 — pulleys and how to count the ropes that hold the load.

🏗️ 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.

Three pulley setups. ✍️ Use the figure.

Each setup lifts a $120$-lb load. A is a single fixed pulley. B is a single movable pulley with one end of the rope tied to the ceiling. C is a block and tackle with $3$ rope segments running between the top block and the bottom block; the free end comes down from the top block.

Ignoring friction, what effort is needed in each setup, in pounds? (numbers only)

1) Setup A:

2) Setup B:

3) Setup C:

Rope pulled. 🎯

Count carefully. Use the figure. 🎯

Pulley facts. 🔽

Key Takeaways — Part 3

• Fixed pulley: MA $1$, changes direction. Movable pulley: MA $2$
• MA = number of rope segments holding up the moving block
• The free end counts only if it leaves the moving block and pulls up
• Effort = load ÷ MA; rope pulled = MA × load rise

Up Next: Part 4 — inclined planes and wedges.

🏗️ 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.

Roll the barrel. ✍️ Use the figure.

A $400$-lb barrel is rolled up a ramp $12$ feet long to a platform $3$ feet high.

1) The mechanical advantage of the ramp (number only):

2) The ideal effort needed, in pounds (number only):

3) The same barrel goes up a $6$-foot ramp to the same $3$-foot height. The ideal effort, in pounds (number only):

Two ramps. Use the figure. 🎯

Wedges

A wedge is an inclined plane that moves into the material. As it is driven forward, its sloped sides push the material apart with a much larger sideways force.

$\text{MA of a wedge} \approx \frac{\text{length of the wedge}}{\text{thickness at its back}}$

• A long, thin wedge has a large MA: it splits with less force but must be driven farther.
• A short, thick wedge has a small MA: it needs more force.
• Examples: an axe or splitting maul, a chisel, a knife blade, a doorstop, a nail point, the teeth of a saw.

Worked example. A splitting wedge is $10$ inches long and $2$ inches thick at the back. Its ideal MA is $10 \div 2 = 5$. Sharpening an axe makes its edge thinner, which raises its MA, so it bites in with less force.

Two wedges. Use the figure. 🎯

Ramps and wedges in practice. 🎯

Key Takeaways — Part 4

• Ramp MA = slope length ÷ height; effort = load × height ÷ length
• Longer ramp, same height: less force, more distance, same work
• A wedge is a moving inclined plane; MA ≈ length ÷ thickness; long and thin = more MA
• Axes, chisels, knives, nail points and doorstops are wedges

Up Next: Part 5 — screws and the wheel and axle.

🏗️ 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.

The winch. ✍️ Use the figure.

A crank handle turns $15$ inches from the center of a drum whose radius is $3$ inches. The rope on the drum lifts a $100$-lb bucket.

1) The mechanical advantage of the winch (number only):

2) The ideal force on the handle to lift the bucket, in pounds (number only):

3) If the handle were moved in to $10$ inches from the center, the ideal force would be how many pounds? (number only)

Wheel and axle checks. 🎯

Screws

A screw is an inclined plane wrapped around a cylinder. Each full turn moves the screw (or a nut on it) forward by one pitch: the distance from one thread to the next.

$\text{distance moved} = \text{number of turns} \times \text{pitch}, \qquad \text{pitch} = \frac{1}{\text{threads per inch}}$

Worked example. A vise screw has $8$ threads per inch, so its pitch is $\frac{1}{8}$ inch. To close the jaws $1\frac{1}{2}$ inches takes $1\frac{1}{2} \div \frac{1}{8} = 12$ turns.

Mechanical advantage of a screw jack

In one turn, your hand travels once around a circle (the circumference of the handle's path), while the load rises just one pitch:

$\text{MA} = \frac{\text{distance the handle travels in one turn}}{\text{pitch}}$

Worked example. A jack handle's end travels $48$ inches around in one turn, and the screw's pitch is $\frac{1}{4}$ inch. $\text{MA} = 48 \div \frac{1}{4} = 192$. A $40$-lb push on the handle could ideally lift $40 \times 192 = 7{,}680$ lb. Friction in real screws is large, which is why a jack also holds the load without slipping back.

Finer threads (smaller pitch) give a larger MA but need more turns.

The screw jack. ✍️ Use the figure.

A screw jack has a pitch of $\frac{1}{4}$ inch. The end of its handle travels $60$ inches around in one full turn.

1) How many full turns raise the load $2$ inches? (number only)

2) What is the jack's ideal mechanical advantage? (number only)

3) With that MA, what ideal push on the handle lifts a $3{,}000$-lb load, in pounds? (number only)

Screws and wheels. 🔽

Key Takeaways — Part 5

• Wheel and axle: effort × wheel radius = load × axle radius; MA = wheel radius ÷ axle radius
• Bigger handle, wheel or crank = less force, more distance
• Screw: distance = turns × pitch; pitch $= \frac{1}{\text{TPI}}$
• Screw jack MA = handle travel per turn ÷ pitch; finer thread = more MA, more turns
• Real screws have lots of friction, which also keeps them from slipping back

Up Next: Part 6 — test strategy and the classic simple-machine traps.

🏗️ 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.

The traps test writers like

Trap 1. Use the figure. 🎯

Traps 2 and 3. 🎯

Speed drill. Under 30 seconds each. ⏱️

Key Takeaways — Part 6

• Every simple machine: force × distance in = force × distance out
• Lever arms from the fulcrum; ramp length along the slope; pulleys count supporting segments on the moving block
• With MA above 1, effort is smaller than the load; machines never save work
• 3rd-class levers trade force for speed and reach (MA below 1)

Up Next: Part 7 — a mixed mastery set across all six simple machines.

🏗️ 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.

Question 1. Use the figure. 🎯

Question 2. Use the figure. 🎯

Questions 3 and 4. ✍️

3) A $320$-lb crate is pushed up a frictionless ramp $16$ feet long to a dock $4$ feet high. What is the ideal effort, in pounds? (number only)

4) A real ramp of the same size has friction, and its efficiency is $80\%$. What effort is actually needed, in pounds? (number only)

Question 5. Use the figure. 🎯

Questions 6 to 10. 🎯

Lesson Wrap-Up

Next steps: take the exit quiz for this topic, then finish Mechanical Comprehension with Gears, Fluids and Structures.