Mechanical Reasoning Guide

Stability & Centre of Gravity

Every object has one point where all its weight seems to act. Whether it stands, tips or balances depends on where that point lies. Mechanical aptitude tests ask which. Can you tell?

The board's gradient is 8 in 10. Which box stays standing?

Ten rules are all it takes. Read on, then try the test.

Where it appears

Questions about balance, tipping and the centre of gravity belong to mechanical reasoning (or mechanical comprehension) tests, beside the gears, levers and pulleys. Each shows a picture and asks what happens next. You will meet them in:

Bennett Mechanical Comprehension TestThe classic test of mechanical understanding, used in hiring for technical and engineering roles. Stability and centre of gravity are among its everyday-physics topics.

Differential Aptitude Tests (DAT)Its Mechanical Reasoning subtest, used in school and career guidance.

ASVABThe Mechanical Comprehension section of the US military's entrance test.

Technician and apprenticeship testsMechanical aptitude tests for maintenance, manufacturing and trade roles, and the mechanical sections of employers' aptitude batteries.

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: find the centre of gravity, then ask whether it is still over the base.

Tipping: seven rules

A body tips over when the line straight down from its centre of gravity leaves its base. A uniform box has its centre of gravity at its middle: half the width across, half the height up. The question is which box tips last, how steep a board can be, where the centre of gravity of a pile is, or how far a load can reach out over an edge.

1Wide and low is hard to tip

For a uniform box it is the shape that counts: the bigger the width ÷ height, the steeper a board must be before the box tips. Work out the ratio for every box and pick the biggest.

hardest to tip = biggest width ÷ height

Three traps: the widest box, the shortest box and the heaviest box. In the test none of them is the answer. A wide box that is tall is not stable, a short box with a narrow base tips, and weight does not decide it.

1.00, 0.60, 0.64 and 0.57: A is hardest to tip, though C is the widest and heaviest and B the shortest

2The board's gradient where a box tips

Tilt the board until the centre of gravity is straight over the lower corner. Half the width across and half the height up give the same ratio as the whole box, so the box tips when the gradient reaches width ÷ height. Written as a rise per 10 across, multiply by 10.

tips at: width ÷ height × 10 in 10

The ratio upside down (height over width) is the slip, and so is giving the width on its own, or half the width over the whole height.

6 wide and 10 high: it tips when the board rises 6 in 10

3Which box stays standing?

A box stays up while its width ÷ height is more than the board's gradient. Turn the gradient into a number (8 in 10 is 0.8), work out each box's ratio, and look for the one that is above it.

The widest and the shortest box are traps: in the test neither is the answer. Check each ratio against the gradient: only one is above it.

On an 8 in 10 board only D (1.00) stays; A, B and C are under 0.8

4Blocks: weigh the centres

For two blocks one on the other, find each block's centre of gravity (the middle of its height), then take the average of the heights, counted by weight. The heavier block pulls the centre of gravity towards its own.

height = (weight × height + weight × height) ÷ total weight

Half the total height is the slip: that is the middle of the pile, not its centre of gravity. Averaging the two centres without their weights, or taking the heavier block's centre only, are the others.

The 30 kg block's centre is 6 cm up, the 60 kg block's is 15 cm: (30 × 6 + 60 × 15) ÷ 90 = 12 cm

5A loaded pile tips on a gentler board

Put a heavy block on a short one and the centre of gravity rises, so the pile tips on a gentler board. Find the pile's centre of gravity (rule 4), then tip it the way rule 2 does: half the base width across, the centre's height up.

gradient = half the width ÷ height of the centre of gravity × 10

Using the middle of the pile's height instead of the weighted centre is the slip, and so is only counting the lower block, or averaging the two blocks' centres without their weights.

Centre of gravity 2.67 cm up, half the width 4 cm: 4 ÷ 2.67 × 10 = 15 in 10

6A plank over a table's edge

Push a plank out over the edge with a weight on its far end. It is on the point of tipping when the turning effects about the edge are equal: the plank's own weight, acting at its middle, against the weight on the end. Write each one as weight times distance from the edge.

plank × (x − half length) = load × (length − x)

Half the plank's length is what you get if you forget the load. And do not give the length still on the table: ask how far out it is.

4 × 5 = 20 on the overhang, 10 × 2 = 20 on the table: pushed out 12 cm

7Bricks: half, a quarter, a sixth ...

Stack bricks so each sticks out past the one below. From the top, the first can be pushed out half its length, the next a quarter, the next a sixth, and so on: L ÷ 2, L ÷ 4, L ÷ 6 ... The total reach is their sum.

Pushing every brick out half its length would topple the pile. The centre of gravity of everything above must stay over the edge of the brick below.

Four bricks of 24 cm: 12 + 6 + 4 + 3 = 25 cm

Flat plates: three rules

A plate of even thickness balances at the centre of gravity of its area. Cut it into rectangles; each rectangle's centre is its middle. The question is which lettered point is the centre of gravity, or how far it is from the left or bottom edge.

8Weigh the parts by their area

Split the plate, find each part's area and the middle of each part, then average the distances counted by area. Do it once for the distance across (x) and once for the distance up (y).

x = (area × x + area × x) ÷ total area

The middle of the plate's outline, the centre of the biggest part alone, and the parts' centres averaged without their areas, are the slips. And the centre of gravity can lie outside the metal: here it is in the corner of the L.

Parts of 9 and 12 cm², total 21: x = 6.5 cm and y = 2.5 cm, point D

9One distance at a time

A question may ask only how far across (or up) the centre of gravity is. Then you only need that one coordinate. Measure from the edge it names.

Look at which edge the question names: from the left is x, from the bottom is y. The same slips apply as in rule 8.

Parts of 16 and 8 cm², total 24: 2 cm from the left edge

10A hole counts as negative area

For a plate with a piece cut out, take the whole plate as one part and the cut-out as a part of negative area. The centre of gravity is pushed away from the hole, and it can even lie in the gap.

Choosing the point on the hole's side of the middle is the slip: a cut-out removes weight there, so the balance point moves the other way. When a distance is asked for, measuring it from the far edge (the mirror image) is another.

Whole plate 72, cut-out −36, total 36: the centre of gravity is (3, 5), point A, in the gap

Two questions, solved

1A tipping question

How far out can the plank be pushed before it tips?

The question

Try it first, then open the answer.

Show the answer: 11 cm

Push the plank x cm out. Its own weight acts at its middle, x − 7 cm beyond the edge; the 8 kg load is 14 − x cm behind it.
It tips when the turning effects about the edge are equal: 6 × (x − 7) = 8 × (14 − x).
That gives x = 11. Check: 6 × 4 = 24 and 8 × 3 = 24.

The other options, and the slip each one is:

3 cm The length still on the table, not how far out7 cm Half the plank's length: the load forgotten22 cm Not what the geometry gives

The working, on the picture

2A flat-plate question

Which point is the centre of gravity of the plate?

The question

Try it first, then open the answer.

Show the answer: point A

Parts (areas, cm²): 80, cut-out −40; total 40
x = (80×4 − 40×3.5) ÷ 40 = 4.5 cm
y = (80×5 − 40×4) ÷ 40 = 6 cm
The centre of gravity is at (4.5, 6): point A.

The other options, and the slip each one is:

B The middle of the plate's outline, not its centre of gravityC The wrong side of the middle: a cut-out pushes the centre of gravity away from itD Not what the geometry gives

The working, on the picture

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 tipping or about a flat plate, drawn half and half. Tipping climbs from which of four boxes is hardest to tip, to the gradient where a box tips, the centre of gravity of two blocks, which box stays standing on a board, how far a plank can be pushed over an edge, how far a pile of bricks can reach, and the gradient where a loaded pile tips. Plates climb from which point is the centre of a symmetrical plate, to an L, T or stair-shaped plate, how far the centre is from the left edge and from the bottom edge, which point is the centre of a plate with a hole, a plate of three parts, and how far across the centre of a plate with a hole is.
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 Stability test

Free and adaptive, no account needed. Start with which box is hardest to tip and climb to loaded piles, bricks and plates with holes.

Start the Stability test