Linkages
Levers, rods and cranks pass a movement along and change it on the way. Mechanical aptitude tests ask which way it goes and how far. Can you tell?
The handle is pushed down in each. In which does X move down?
Twelve rules are all it takes. Read on, then try the test.
Where it appears
Questions about levers joined by rods, and about cranks and cams, belong to mechanical reasoning (or mechanical comprehension) tests, beside the gears, pulleys and springs. Each shows a picture and asks what happens next. You will meet them in:
Those tests usually give a picture with no numbers to count and no time to spare. Here every picture carries its numbers, so each answer is exact, and the questions grow harder as you get better. It trains the habit those questions reward: follow the movement from the handle, part by part, and ask what each part does to its direction and to its distance.
Levers and rods: seven rules
A lever is a bar that turns about a fixed pivot (the triangle). A rod joins the end of one lever to the next and passes the movement on unchanged. A bell crank is two arms at a right angle on one pivot. The question is which way the end X moves, how far it moves, or how hard it pushes.
1Pivot between the ends: they move opposite ways
When the pivot is between a lever's two ends, one end goes down as the other goes up. When the pivot is at one side of both ends, they move the same way.
2The end moves in proportion to its arm
An end twice as far from the pivot moves twice as far. Multiply the handle's movement by the out arm ÷ in arm.
distance = handle's movement × (out arm ÷ in arm)3Levers in a row: multiply
A rod hands the movement to the next lever, which multiplies it again. Work along the chain from the handle, one lever at a time, and keep each answer whole before the next.
4Count the levers that turn the movement round
Each lever with its pivot between the ends turns the movement round; each lever with the pivot at a side does not. An odd number of turns-round sends X the opposite way to the handle, an even number sends it the same way: two turns-round cancel.
5What a lever gains in distance it loses in force
The end that moves further pushes with less force. For force, multiply by the arms the other way up: in arm ÷ out arm. Work along a chain lever by lever.
force = handle's force × (in arm ÷ out arm)6A bell crank turns a movement through a right angle
A rod on one arm moves up or down; the rod on the other arm then moves left or right (and the other way round). Turn the crank in your head the way the first rod pushes its arm, and watch where the other arm's end goes.
7A bell crank's distance is in proportion to its arms
The same rule as a lever: the distance the second rod moves is the first rod's movement times (its arm ÷ the first arm). Only the direction needs the turning.
Cranks and cams: five rules
A wheel carries a crank pin, some distance from its centre. A rod joins the pin to a piston on a straight guide, so as the wheel turns the piston runs to and fro. A stroke is one way along the guide. The question is how long the stroke is, how many strokes, which of four pistons has the longest, or how far a cam lifts its follower.
8The stroke is twice the pin's distance from the centre
The pin swings from one side of the centre to the other, so the piston travels twice the pin's distance between its two furthest places. Turn it round to find the pin: it is half the stroke from the centre.
stroke = 2 × the pin's distance9A turn of the wheel is two strokes
In one turn the piston goes out and comes back: two strokes. For so many turns a second over so many seconds, multiply all three.
10The longest stroke is the largest pin distance
For four pistons, work out each stroke from its pin. Do not judge by the picture: the wheels and the rods differ in size, but only the pin counts.
11A cam lifts its follower by the longest radius less the shortest
The follower rests on the cam's edge, so it is lowest on the shortest radius and highest on the longest. How far it rises is the difference.
rise = longest radius − shortest radius12A crank that drives a lever: stroke, then ratio
A rod joins the pin to the end of a lever. That end moves the crank's stroke, 2 × the pin's distance, and the other end moves in proportion to its arm (rule 2).
Two questions, solved
1A question about levers and rods
The handle moves up 8 cm. How far does the end X move?
Try it first, then open the answer.
Show the answer: 9 cm
- Each lever multiplies the movement by (its out arm ÷ its in arm); a rod passes it on.
- Lever 1: 8 × (6 ÷ 4) = 12 cm.
- Lever 2: 12 × (3 ÷ 2) = 18 cm.
- Lever 3: 18 × (2 ÷ 4) = 9 cm.
- So X moves 9 cm.
The other options, and the slip each one is:
2A question about a crank
How far does the end X of the lever move, from its lowest to its highest?
Try it first, then open the answer.
Show the answer: 12 cm
- The crank moves the lever's end 2 × 4 = 8 cm, from lowest to highest.
- The other end moves 9 ÷ 6 times as far: 8 cm × (9 ÷ 6) = 12 cm.
The other options, and the slip each one is:
How the practice test runs
- 3:00 on the clock, and it only runs while a question is on screen.
- Levels: each question is about levers and rods or about a crank, drawn half and half. Levers and rods climb from which way a bell crank moves a rod, to how far the end of one lever moves, which of four linkages moves X up, how far the end of two levers moves, how far a bell crank moves a rod, how far three levers move it, and the force at the end of two levers. Cranks climb from the length of a stroke, to the strokes in so many turns, which of four pistons has the longest stroke, how far a cam lifts its follower, how far the pin is from the centre, the strokes in so many seconds, and a crank that drives a lever.
- The ladder: until your first miss, every right answer jumps a whole level; after it, three right answers in a row jump a level and a miss drops one. Your level carries over to your next session.
- Scoring: a right answer earns the question's points (harder questions pay more), a wrong one costs half as many, so guessing does not pay.
- Every question is new, and the review after a session works every answer out on its picture and names the slip behind each wrong option.
Try the Linkages test
Free and adaptive, no account needed. Start with a bell crank and a piston and climb to chains of levers, cams, and cranks that drive levers.
Start the Linkages test