Centre of gravity — answer key

1. MQ16 · Practice

Model answer. Every part of an object has weight, but the object behaves as if all its weight acted at one point, its centre of gravity. For a uniform rule that point is at the middle; for a hammer it is near the heavy head, so it is not always the middle and not always the lowest point.

A full answer includes

• I chose b.
• The centre of gravity is the point through which the whole weight of the object seems to act.

Also accept

• b: the point through which the whole weight of the object seems to act.

Watch for

• “a: it is always the middle of the object.” — Only for a uniform object such as the metre rule. A hammer’s centre of gravity is near its heavy head.
• “d: it is the lowest point of the object.” — The centre of gravity can be high up, like the top-heavy box in the stability picture. It is where the weight seems to act.

2. MQ17 · Practice

Model answer. The rule is uniform, so its centre of gravity is at its 50 cm mark, 0.10 m to the right of the pivot. The 0.60 N weight is 0.30 m to the left, so its anticlockwise moment is 0.60 N × 0.30 m = 0.18 N m. The rule balances, so the clockwise moment of its own weight is the same: W × 0.10 m = 0.18 N m, so W = 1.8 N.

Checked automatically

• Weight of the rule: 1.8 N (within ±0.05 N)
  • Common wrong answer 0.3 : You used 0.60 m, the distance to the end of the rule. The rule’s weight acts at the 50 cm mark, 0.10 m from the pivot.
  • Common wrong answer 0.6 : The rule’s weight acts closer to the pivot, so it must be bigger than 0.60 N to balance.
  • Common wrong answer 0.2 : The distances are the wrong way round: 0.60 N × 0.30 m = W × 0.10 m.

A full answer includes

• The 0.60 N weight is 40 − 10 = 30 cm = 0.30 m from the pivot: its moment is 0.60 N × 0.30 m = 0.18 N m.
• The rule’s weight acts at the 50 cm mark, 0.10 m from the pivot: W × 0.10 m = 0.18 N m.
• W = 1.8 N.

Also accept

• 1.8 N

Watch for

• “0.30 N, using the distance from the pivot to the end of the rule.” — The rule’s weight acts at its centre of gravity, the 50 cm mark, not at its end.
• “0.60 N, the same as the hanging weight.” — The rule’s weight acts closer to the pivot than the 0.60 N weight, so it must be bigger to give the same moment.

3. MQ18 · Practice

Options as numbered on the pupil sheet: 1 — Draw a pencil line on the card along the thread; 2 — Make three small holes near the edge of the card, well apart; 3 — Mark where the two lines cross, and check that a line from the third hole passes through it; 4 — Hang a plumb line from the same pin and wait until the card and the thread are still; 5 — Hang the card from the second hole and draw a second line in the same way; 6 — Hang the card from a pin through one hole, so that it can swing freely.

Model answer. Make three small holes near the edge of the card, well apart. Hang the card from a pin through one hole so that it swings freely, and hang a plumb line from the same pin. When the card and the thread are still, the centre of gravity is somewhere on the thread’s line, so draw that line on the card. Hang the card from the second hole and draw a second line in the same way. The centre of gravity is where the two lines cross; a line from the third hole should pass through the same point.

A full answer includes

• Make three holes near the edge; hang the card on a pin so it swings freely; hang a plumb line from the pin and wait until all is still;
• draw a line along the thread; hang the card from the second hole and draw a second line; the centre of gravity is where the lines cross, checked with the third hole.

Also accept

• holes; hang on a pin; plumb line, wait until still; draw the line; second hole, second line; crossing point, check with the third hole.

Watch for

• “Draw the line as soon as the card is hung up.” — Wait until the card and the plumb line are completely still, or the line will be in the wrong place.
• “One line is enough: the centre of gravity is halfway along it.” — One line only shows that the centre of gravity is somewhere on it. You need a second line to find the point.

4. MQ19 · Practice

Model answer. Each time the card hangs still, its centre of gravity is straight below the pin, on the plumb line. So it is on the first line and also on the second line. The only point on both lines is where they cross, Q. P is on the first line only, R is on neither line, and S is the hole, where the pin is.

A full answer includes

• I chose Q.
• The centre of gravity lies on both lines, so it is where they cross.

Also accept

• Q, where the two lines cross.

Watch for

• “P, because it is on the first line.” — The centre of gravity is on the first line, but also on the second. Only the crossing point is on both.
• “S, because the card hangs from there.” — The card hangs from the pin at S, and its centre of gravity hangs below the pin, on the plumb line.

5. MQ20 · Practice

Model answer. The centre of gravity is where the whole weight of an object seems to act. A tipped object turns about the edge of its base; it topples once the vertical line from its centre of gravity passes outside the base. A low centre of gravity and a wide base let it tip further before that happens, so it is more stable.

A full answer includes

• weight; outside; low; wide.

Also accept

• weight; outside; low; wide.

Watch for

• “(1) mass.” — The centre of gravity is where the weight, a force, seems to act. Weight is what has a moment about a pivot.
• “(3) high.” — A high centre of gravity makes the line from it pass outside the base after only a small tip, so the object is less stable.

6. MQ21 · Practice

Model answer. A tipped object turns about the edge of its base. For X, the tall narrow box, the vertical line from G passes outside that edge, so the weight has a moment that turns it further over, and it topples. For Y, short and wide, and for Z, with heavy sand at the bottom, the line from G still passes inside the base, so the weight turns them back onto their bases.

A full answer includes

• I chose X.
• The vertical line from X’s centre of gravity falls outside its base, so its weight turns it further over.
• For Y and Z, the line falls inside the base, so they rock back.

Also accept

• X: the vertical line from its centre of gravity falls outside its base.

Watch for

• “Z, because it is the heaviest.” — Being heavier does not decide it. Z’s sand keeps its centre of gravity low, so the line from G stays inside the base.
• “Y, because it is short.” — A short, wide box has a low centre of gravity and a wide base: the line from G is well inside the base.

7. MQ22 · Practice

Model answer. The card’s weight acts at its centre of gravity. If that point is to one side of the pin, the weight’s line of action misses the pin, so the weight has a moment about the pin and turns the card. The card stops turning only when the weight has no moment, when its line of action passes through the pin. That happens when the centre of gravity is straight below the pin.

A full answer includes

• If the centre of gravity is to one side of the pin, the weight’s line of action does not pass through the pin, so the weight has a moment that turns the card.
• The card stops only when the weight has no moment: its line of action passes through the pin, so the centre of gravity is straight below it.

Also accept

• If the centre of gravity is to one side, the weight has a moment about the pin and turns the card; it stops when the weight’s line passes through the pin, so G is straight below it.

Watch for

• “The pin holds the centre of gravity in place.” — The pin holds the card at the hole. It is the weight’s moment that turns the card until the centre of gravity is below the pin.
• “It is luck where it stops.” — It always stops in the same place, because only there does the weight have no moment about the pin.

8. MQ23 · Practice

Model answer. (a) Until everything is still, the thread swings either side of the vertical line through the pin, so a line drawn then would miss the centre of gravity. (b) Lines from holes close together are nearly parallel, so a small error in one line moves their crossing point a long way; holes far apart give lines crossing at a wide angle. (c) Hang the card from a third hole and draw a third line: it should pass through the same point. The card should also balance on a pencil point placed there.

A full answer includes

• (a) While they swing, the thread is not where the vertical line through the pin will be, so the line would be in the wrong place.
• (b) Lines from holes far apart cross at a wide angle, so a small error in one line moves the crossing point only a little.
• (c) Hang the card from a third hole: its plumb line should pass through the same point (or balance the card on a point there).

Also accept

• (a) so the line is the true vertical through the pin; (b) so the lines cross at a wide angle and small errors matter less; (c) a third hole’s line passes through the same point.

Watch for

• “(b) So the card does not tear.” — The holes are near the edge for that. They are far apart so that the lines cross at a wide angle.
• “(c) Measure the middle of the card with a ruler.” — An irregular card’s centre of gravity is not at a measured middle. Check with a third hole or by balancing the card on a point.

9. MQ24 · Practice

Model answer. Weight high up raises the centre of gravity of the bus. As the platform tilts, the bus turns about the wheels on the lower side. With a high centre of gravity, the vertical line from it passes outside those wheels after only a small tilt. Then the weight’s moment turns the bus further over instead of back, so it topples at a smaller angle.

A full answer includes

• The passengers on the top deck raise the centre of gravity of the bus.
• When the bus tilts, a higher centre of gravity makes the vertical line from it pass outside the wheels on the lower side after a smaller tilt.
• Then the weight’s moment turns the bus further over, so it topples.

Also accept

• The full top deck raises the centre of gravity, so the vertical line from it passes outside the base (the lower wheels) at a smaller tilt.

Watch for

• “The bus is heavier, so it falls over.” — A heavier bus with the extra weight low down would be just as stable. What matters is that the weight is high.
• “The passengers push the bus over.” — They only add weight. The weight is high up, which raises the centre of gravity.

10. MQ25 · Practice

Model answer. An object is more stable when its centre of gravity is low and its base is wide. A heavier base and sand in the bottom of the base both lower the centre of gravity, and a wider base makes it tip further before the line from the centre of gravity passes outside the base. A taller stem and a heavy shade at the top raise the centre of gravity, and a narrower base makes it topple sooner.

A full answer includes

• I chose a, b and e.
• A heavier base and sand in the base lower the centre of gravity; a wider base lets it tip further before toppling.
• A taller stem or a heavy shade at the top raise the centre of gravity; a narrower base makes it less stable.

Also accept

• a, b and e: heavier base, wider base, sand in the base.

Watch for

• “d: a heavy shade makes the whole lamp heavier, so more stable.” — Weight high up raises the centre of gravity. The lamp becomes less stable.
• “c: a taller stem holds it up better.” — A taller stem raises the centre of gravity, so the lamp topples after a smaller tip.

11. MQ26 · Practice

Model answer. (a) The rule is uniform, so its weight acts at its centre of gravity, the 50 cm mark, 0.20 m to the right of the pivot. Its clockwise moment is 1.5 N × 0.20 m = 0.30 N m. The 2.0 N weight on the short side must give the same anticlockwise moment: 2.0 N × d = 0.30 N m, so d = 0.15 m, at the 15 cm mark. (b) The rule is in equilibrium, so the pivot pushes up as hard as everything pulls down: 1.5 N + 2.0 N = 3.5 N.

Checked automatically

• (a) Distance of the 2.0 N weight from the pivot: 0.15 m (within ±0.005 m)
  • Common wrong answer 0.2 : Equal distances balance only equal weights. Work out 1.5 N × 0.20 m and divide by 2.0 N.
  • Common wrong answer 0.525 : The rule’s weight acts at its centre of gravity, the 50 cm mark, not at its far end.
  • Common wrong answer 15 m: 15 m is far longer than the rule. 15 is the distance in centimetres: 15 cm = 0.15 m.
• (b) Upward force of the pivot: 3.5 N (within ±0.05 N)
  • Common wrong answer 2 : You left out the rule’s own weight, 1.5 N.
  • Common wrong answer 0.5 : The forces do not cancel each other: both weights pull down and the pivot holds them both up.

A full answer includes

• (a) The rule’s weight acts 0.20 m from the pivot: 1.5 N × 0.20 m = 0.30 N m = 2.0 N × d, so d = 0.15 m (at the 15 cm mark).
• (b) 1.5 N + 2.0 N = 3.5 N.

Also accept

• (a) 0.15 m from the pivot (the 15 cm mark); (b) 3.5 N.

Watch for

• “(a) 0.20 m, the same distance as the rule’s weight.” — Equal distances balance only equal weights. The 2.0 N weight is heavier, so it hangs closer.
• “(b) 2.0 N.” — You left out the rule’s own weight, 1.5 N. The pivot holds that up too.

12. MT2 · Apply

Model answer. (a) The walker and the rucksack together have one centre of gravity. The rucksack moves it backwards, behind her feet. If the vertical line from it passed behind her heels, her weight would have a moment that turns her over backwards. Leaning forward moves the centre of gravity back above her feet, so the line stays inside her base. (b) With her feet wide apart, her base is wider, so when the train jolts she can tip further before the vertical line from her centre of gravity passes outside her base. (c) Pack the heavy things low and close to her back, which keeps the centre of gravity low and over her feet. (d) Hang the plate on the hook by one hole, so it can swing freely, and wait until it is still. Its centre of gravity is then straight below the hook. Hang a plumb line from the hook and draw along it on the plate with chalk. Hang the plate by a second hole and draw a second line the same way. The centre of gravity is on both lines, so it is where they cross. To check, hang the plate from the third hole: its plumb line should pass through the same crossing.

A full answer includes

• (a) The rucksack moves the combined centre of gravity backwards; leaning forward brings it back above her feet, so the vertical line from it stays inside her base.
• (b) Feet apart make a wider base, so she can tip further before the vertical line from her centre of gravity passes outside it.
• (c) Put the heavy things low down and close to her back, to keep the centre of gravity low and over her feet.
• (d) Hang the plate from one hole so it swings freely; when it is still, hang a plumb line from the hook and chalk along it; repeat from a second hole; the centre of gravity is where the two lines cross; check by hanging it from the third hole: the new line passes through the crossing.

Also accept

• (a) the rucksack moves the centre of gravity back; leaning forward keeps it above the feet; (b) a wider base; (c) heavy things low and close to the back; (d) hang it from each hole in turn, chalk the plumb line each time; the centre of gravity is where the lines cross; check with the third hole.

Watch for

• “(a) She leans forward to push the rucksack up.” — Leaning forward keeps the combined centre of gravity above her feet, inside her base.
• “(b) Feet apart make her lower, so heavier.” — Her weight does not change. The wider base lets her tip further before the line from her centre of gravity passes outside it.
• “(d) The centre of gravity is halfway between two of the holes.” — The plate is of uneven shape, so its centre of gravity need not be halfway between two holes. Find it where the two plumb lines cross.