Answers · Stretching springs

Loading a spring — answer key

Check meaning, not exact wording. These are teaching points, not official Cambridge marks.

1. SQ1 · Practice

Options as numbered on the pupil sheet: 1 — stretch: it gets longer; 2 — squash: it gets shorter; 3 — bend: it curves; 4 — twist: it turns round on itself; 5 — no change: it is too hard to change.

Model answer. The pull on the catapult rubber makes it longer: it stretches. The racket pushes on the ball, and the ball gets shorter in that direction: it is squashed. The fish pulls the tip of the rod down, so the rod curves: it bends. Winding the propeller turns one end of the rubber motor while the other end is held still, so the rubber twists. The books push down on the middle of the steel shelf, and it curves down a little: it bends. Steel is hard, but every solid changes shape a little when a force acts on it, so “no change” is never right.

A full answer includes

I matched the catapult rubber to stretch.
I matched the tennis ball to squash.
I matched the fishing rod to bend.
I matched the rubber motor of the toy plane to twist.
I matched the steel shelf to bend.
I did not choose “no change” for the steel shelf.

Also accept

catapult rubber – stretch; tennis ball – squash; fishing rod – bend; rubber motor of the toy plane – twist; steel shelf – bend.

Watch for

“The steel shelf does not change shape, because steel is too hard.” — The books push down on the shelf, and it sags a little in the middle: it bends. Every solid changes shape a little when a force acts on it.
“The fishing rod stretches.” — The rod hardly gets longer. Its tip is pulled down and the whole rod curves: it bends.
“The tennis ball bends.” — The racket pushes on one side of the ball and the ball gets flatter, shorter in that direction: it is squashed.
“The rubber motor of the toy plane stretches.” — Winding the propeller turns one end of the rubber round while the other end stays still. That is a twist.

2. SQ2 · Practice

Model answer. The hanger was on the spring when the zero reading was taken, so the stretch it causes is already in the zero reading, and its weight is not part of the load. The load is the weight of the masses added: 6 × 100 g = 600 g = 0.600 kg. Load = 0.600 kg × 9.8 N/kg = 5.88 N, or 5.9 N to 2 significant figures.

Checked automatically

Load: 5.88 N (within ±0.05 N)
- Common wrong answer 6.37 : You counted the hanger too, 650 g. The hanger was already on for the zero reading, so only the 600 g added afterwards is the load.
- Common wrong answer 600 : 600 g is the mass added. The load is a force: change the mass to kilograms and multiply by g = 9.8 N/kg.
- Common wrong answer 5880 : You multiplied 600 g by 9.8. Change grams to kilograms first: 600 g = 0.600 kg.
- Common wrong answer 0.6 : 0.600 kg is the mass added. The load is its weight: 0.600 kg × 9.8 N/kg.
- Common wrong answer 6.0 : Use g = 9.8 N/kg, not 10 N/kg: 0.600 kg × 9.8 N/kg.

A full answer includes

I counted only the masses added after the zero reading: 600 g = 0.600 kg.
Method: load = 0.600 kg × 9.8 N/kg.
Answer: 5.88 N (5.9 N).
I did not add the 50 g hanger to the load.

Also accept

5.88 N (or 5.9 N), the weight of the 600 g added; the hanger is not counted.

Watch for

“The load is 650 g, the hanger and the masses together.” — The hanger was already on for the zero reading, so the extension measured from it is caused only by the 600 g added.
“The load is 600.” — A load is a force, in newtons. 600 g is a mass: change it to kilograms and multiply by g.

3. SQ3 · Practice

Model answer. The zero reading, 9.5 cm, is where the pointer was before the load went on. The extension is how much longer the spring got, so subtract: 14.3 cm − 9.5 cm = 4.8 cm. The reading, 14.3 cm, is only where the pointer is now.

Checked automatically

Extension: 4.8 cm (within ±0.05 cm)
- Common wrong answer 14.3 : That is the reading, where the pointer is now. The extension is how far it moved: 14.3 cm − 9.5 cm.
- Common wrong answer 23.8 : You added the two readings. The extension is the difference: take away the zero reading.
- Common wrong answer 9.5 : That is the zero reading, before the load went on. Subtract it from the new reading.
- Common wrong answer -4.8 : Take the zero reading away from the new reading: 14.3 cm − 9.5 cm. An extension is positive.

A full answer includes

Method: extension = reading − zero reading = 14.3 cm − 9.5 cm.
Answer: 4.8 cm.
I did not give 14.3 cm, the reading, as the extension.

Also accept

4.8 cm (14.3 cm − 9.5 cm).

Watch for

“The extension is 14.3 cm, the reading on the rule.” — The spring was already at 9.5 cm before the load went on. The extension is the change: 14.3 cm − 9.5 cm.

4. SQ4 · Practice

Options as numbered on the pupil sheet: 1 — Add a 100 g mass; when the spring is still, read the pointer at eye level and subtract the zero reading; 2 — Plot load against extension, including (0, 0), and draw a ruled best-fit line; 3 — Clamp the stand to the bench, hang the spring from the clamp and fix a metre rule upright beside it; 4 — Repeat with one more 100 g mass each time, recording the load and the extension in a table; 5 — Hang the empty hanger on the spring and read the pointer at eye level: the zero reading.

Model answer. First set up safely: clamp the stand to the bench, hang the spring from the clamp and fix a metre rule upright beside it. Hang the empty hanger on and read the pointer at eye level: this is the zero reading. Add a 100 g mass, wait until the spring is still, read the pointer at eye level again, and subtract the zero reading to get the extension. Repeat, one more 100 g mass at a time, recording the load and the extension in a table. Last, plot the load against the extension, including (0, 0), and draw a ruled best-fit line.

A full answer includes

Step 1: I set up the clamped stand, the spring and the rule (a).
Step 2: I took the zero reading with the empty hanger on (b).
Step 3: I added a mass, read the pointer when the spring was still and subtracted the zero reading (c).
Step 4: I repeated with one more mass each time and recorded the results (d).
Step 5: I plotted load against extension and drew a best-fit line (e).
I did not put the zero reading after the first mass.

Also accept

a, b, c, d, e: set up; zero reading; add a mass, read and subtract the zero reading; repeat and record; plot and draw a best-fit line.

Watch for

“Read the pointer straight after adding each mass.” — The spring bounces when a mass goes on. Wait until it is still, or the pointer is caught at the wrong place.
“Plot the readings, not the extensions.” — Load against reading is a straight line that does not go through the origin. Plot the extension, the reading minus the zero reading.
“Take the zero reading after the first mass goes on.” — The zero reading is where the pointer is before any mass is added. Every extension is measured from it, so it comes first.

5. SQ5 · Practice

Model answer. (1) stretches: a pull makes the strip longer, a change of its size and its shape. (2) elastic: an elastic object goes back to its original shape when the force is taken away. (3) load and (4) newtons: the load is a force, the weight of the marbles, so it is measured in newtons. (5) zero reading: the reading with the empty bag already on, before any load is added. (6) extension: the increase in length. (7) proportional: twice the load giving twice the extension, every time, means the extension is proportional to the load.

A full answer includes

(1) I wrote stretches.
(2) I wrote elastic.
(3) I wrote load.
(4) I wrote newtons.
(5) I wrote zero reading.
(6) I wrote extension.
(7) I wrote proportional.
I did not write grams for (4).

Also accept

stretches; elastic; load; newtons; zero reading; extension; proportional.

Watch for

“(1) squashes.” — A squash makes an object shorter. The marbles pull down on the strip and it gets longer: it stretches.
“(4) grams.” — Grams measure mass. The load is a force, the weight of the marbles, so it is measured in newtons.
“(5) original length.” — The original length is measured with nothing hanging on the strip. Here the empty bag is already on, so this is the zero reading.
“(7) equal.” — The extension in cm does not equal the load in N. Proportional means that doubling the load doubles the extension.

6. SQ6 · Practice

Model answer. b. A straight line shows proportion only if it goes through the origin, so that zero load gives zero on the other axis. Her line meets the reading axis at 5.0 cm, the zero reading. Doubling the load from 1.0 N to 2.0 N takes the reading from 7.2 cm to 9.4 cm, not to 14.4 cm. Taking 5.0 cm off every reading gives the extensions, 2.2, 4.4, 6.6 and 8.8 cm. These are proportional to the load, so a graph of load against extension is a straight line through the origin.

A full answer includes

I chose b.
I found that her line meets the reading axis at 5.0 cm, not at the origin.
I checked that doubling the load from 1.0 N to 2.0 N takes the reading from 7.2 cm to 9.4 cm, not to 14.4 cm.
I wrote that she should plot the extension: the reading minus 5.0 cm.
I did not choose c: a constant gradient alone does not show proportion.

Also accept

b: wrong, because the line meets the reading axis at 5.0 cm, not at the origin; plot the extension instead.

Watch for

“a: any straight line shows proportion.” — Only a straight line through the origin does. Load against reading is straight, but it starts at 5.0 cm.
“c: the line has a constant gradient, so it shows proportion.” — A constant gradient makes the line straight, but proportion also needs zero load to give zero. With no load, the reading is 5.0 cm.
“d: she should join the points dot to dot.” — A ruled best-fit line is right, not dot to dot. The problem is that her line starts at 5.0 cm, not at the origin.

7. SQ7 · Practice

Model answer. (a) Start at 6.5 cm on the extension axis, go straight up to the best-fit line, then straight across to the load axis: about 2.6 N. (b) Start at 3.0 N on the load axis, go across to the line, then down to the extension axis: about 7.5 cm. Use the line, not the nearest point: the line averages out the small errors in each reading. The nearest point to 6.5 cm, at 7.4 cm, would give 2.94 N, which is too big.

Checked automatically

(a) Load for an extension of 6.5 cm: 2.6 N (within ±0.1 N)
- Common wrong answer 6.5 : 6.5 cm is the extension you start from. Go up from 6.5 cm to the line, then across to the load axis.
- Common wrong answer 2.94 : That is the load of the nearest point, at 7.4 cm. Read the line at 6.5 cm instead.
(b) Extension for a load of 3.0 N: 7.5 cm (within ±0.2 cm)
- Common wrong answer 1.2 : You read the graph the wrong way round: 3.0 cm on the extension axis gives 1.2 N. Start at 3.0 N on the load axis.
- Common wrong answer 3.0 : 3.0 N is the load you start from. Go across from 3.0 N to the line, then down to the extension axis.

A full answer includes

(a) I went up from 6.5 cm to the line, then across to the load axis.
(a) Answer: about 2.6 N.
(b) I went across from 3.0 N to the line, then down to the extension axis.
(b) Answer: about 7.5 cm.
I did not use the nearest plotted point, which gives 2.94 N.

Also accept

(a) about 2.6 N; (b) about 7.5 cm, both read from the line.

Watch for

“Use the nearest plotted point.” — The points scatter a little. The best-fit line averages them, so read from the line at exactly 6.5 cm.

8. SQ8 · Practice

Model answer. Ravi is wrong. A force can change the size and the shape of any solid object, even one made of steel. The weight of the people pulls down on the lift, and the cable stretches: it gets a little longer, by an amount too small to see. When they get out, the cable goes back to its original length, because it is elastic. Changing shape does not mean breaking.

A full answer includes

I wrote that a force can change the size and the shape of any solid object, even a steel one.
I wrote that the people’s weight pulls on the cable.
I wrote that the cable stretches: it gets a little longer.
I wrote that the cable is elastic: it goes back to its original length when they get out.
I did not write that a change of shape means the cable breaks.

Also accept

Wrong: every solid changes shape under a force; the people’s weight stretches the cable a little, and it goes back when they get out.

Watch for

“Only soft things such as rubber change shape.” — Every solid changes shape a little when a force acts on it. A steel cable stretches under a load, by an amount too small to see.
“If the cable changes shape, it must break.” — The cable stretches under the load and goes back when the people get out: it is elastic. Changing shape does not mean breaking.

9. SQ9 · Practice

Model answer. (a) With 800 g hanging out from the rod, the stand is top-heavy and could tip over. He clamps its base to the bench with a G-clamp, or puts a heavy weight on it. (b) A catch tray under the hanger, so that a mass that slips off lands in the tray, not on his feet. (c) Looking down from above, the pointer seems to sit against a different mark on the rule, so every reading is wrong: this is parallax. He should put his eye level with the pointer and look straight across at the rule. (d) He should wait until the spring stops bouncing and is still. A bouncing pointer is caught above or below the place where it settles.

A full answer includes

(a) I wrote that the stand could tip over as the masses go on.
(a) I wrote: clamp the stand to the bench, or put a heavy weight on its base.
(b) I wrote a catch tray.
(b) I gave the reason: a mass that slips off lands in the tray, not on his feet.
(c) I wrote that from above, the pointer seems to sit against a different mark: parallax.
(c) I wrote that his eye should be level with the pointer, looking straight across at the rule.
(d) I wrote: wait until the spring is still.
(d) I gave the reason: a bouncing pointer is caught above or below where it settles.
I did not write that looking from above only makes the scale harder to see.

Also accept

(a) it could tip over: clamp it to the bench; (b) a catch tray, for falling masses; (c) parallax: eye level with the pointer; (d) wait until it is still.

Watch for

“(c) Looking from above is fine if you look carefully.” — From above, the pointer seems to sit against a different mark however carefully you look. Only an eye level with the pointer gives the true reading.
“(a) The stand is heavy enough on its own.” — With 800 g hanging out from the rod, the stand can tip over. Clamp it to the bench or weigh its base down.
“(d) Read it quickly, before it moves.” — A bouncing spring is still moving when you read it. Wait until it is still, then read.

10. SQ10 · Practice

Model answer. A. For the extension to be proportional to the load, the graph must be a straight line, so that each extra newton gives the same extra extension, and it must go through the origin, so that zero load gives zero extension. Only A has both. B is straight but starts part of the way along the extension axis. C curves, so equal steps of load do not give equal steps of extension. D shows the same load whatever the extension.

A full answer includes

I chose A.
I checked that A is a straight line.
I checked that A goes through the origin.
I did not choose B, which is straight but does not start at the origin.

Also accept

a: A, a straight line through the origin.

Watch for

“B, because it is a straight line.” — B is straight, but it does not go through the origin. A straight line shows proportion only if it starts at (0, 0).
“C, because the load keeps going up.” — In C the line curves, so equal steps of extension do not give equal steps of load. Proportion needs a straight line.
“D, because the load stays the same.” — In D the extension grows while the load does not change, so the extension does not depend on the load at all. Proportion needs a straight line that slopes up through the origin.

11. SQ11 · Practice

Model answer. With 100 g the extension is 8.7 cm − 5.5 cm = 3.2 cm. The extension is proportional to the load, and 250 g is two and a half times 100 g, so the extension is 2.5 × 3.2 cm = 8.0 cm. The pointer starts at the zero reading, so it reads 5.5 cm + 8.0 cm = 13.5 cm. Multiplying the reading, 2.5 × 8.7 cm, would also multiply the 5.5 cm the pointer was already down the rule.

Checked automatically

(a) Extension with 250 g: 8.0 cm (within ±0.1 cm)
- Common wrong answer 3.2 : That is the extension for 100 g. 250 g is two and a half times as much, so the extension is two and a half times as big.
- Common wrong answer 13.5 : That is the pointer reading. The extension is the reading minus the zero reading, 5.5 cm.
- Common wrong answer 2.45 : 2.45 N is the load of 250 g. The question asks for the extension.
- Common wrong answer 21.75 : You multiplied the reading, 8.7 cm, by 2.5. Multiply the extension instead: the reading includes the zero reading.
(b) Pointer reading with 250 g: 13.5 cm (within ±0.5 cm)
- Common wrong answer 8.0 : That is the extension. The pointer starts at the zero reading, so add 5.5 cm to it.
- Common wrong answer 21.75 : The reading is not proportional to the load. Find the extension first, then add the zero reading.
- Common wrong answer 27.25 : You multiplied the reading by 2.5 and then added the zero reading again. Multiply the extension, 3.2 cm, instead.

A full answer includes

Method: the extension for 100 g is 8.7 cm − 5.5 cm = 3.2 cm.
(a) Method: 250 g is 2.5 × 100 g, so the extension is 2.5 × 3.2 cm.
(a) Answer: 8.0 cm.
(b) Method: reading = zero reading + extension = 5.5 cm + 8.0 cm.
(b) Answer: 13.5 cm.
I did not multiply the reading, 8.7 cm, by 2.5.

Also accept

(a) 8.0 cm; (b) 13.5 cm (14 cm).

Watch for

“(a) 13.5 cm.” — That is where the pointer will be. The extension is the reading minus the zero reading, 5.5 cm.
“(b) 2.5 × 8.7 cm = 21.75 cm.” — The reading is not proportional to the load, because it includes the 5.5 cm zero reading. Multiply the extension, then add the zero reading.

12. SQ25 · Practice

Model answer. (a) From the original length, the extension is 11.5 cm − 6.0 cm = 5.5 cm. That extension includes the 0.5 cm made by the hanger, so the load must include the hanger’s weight as well: 50 g + 500 g = 0.550 kg, and 0.550 kg × 9.8 N/kg = 5.39 N. (b) From the length with the empty hanger on, the extension is 11.5 cm − 6.5 cm = 5.0 cm. The hanger was already on, so the load is only the 500 g added: 0.500 kg × 9.8 N/kg = 4.9 N. Both ways give 0.98 N for each centimetre of extension (5.39 N ÷ 5.5 cm and 4.9 N ÷ 5.0 cm), so both pairs lie on the same straight line through the origin. Mixing them, 5.5 cm with 4.9 N, would put the point off the line: the extension and the load must start from the same place.

Checked automatically

(a) Extension from 6.0 cm: 5.5 cm (within ±0.05 cm)
- Common wrong answer 11.5 : 11.5 cm is the length now. The extension is the increase: 11.5 cm − 6.0 cm.
- Common wrong answer 5.0 : That is measured from 6.5 cm, the length with the hanger on. Part (a) measures from the original length, 6.0 cm.
(a) Load for that extension: 5.39 N (within ±0.05 N)
- Common wrong answer 4.9 : From the original length, the extension includes the hanger’s stretch, so the load includes the hanger’s weight too: 550 g.
- Common wrong answer 5.5 : Use g = 9.8 N/kg, not 10 N/kg: 0.550 kg × 9.8 N/kg.
- Common wrong answer 550 : 550 g is a mass. The load is its weight in newtons: 0.550 kg × 9.8 N/kg.
(b) Extension from 6.5 cm: 5.0 cm (within ±0.05 cm)
- Common wrong answer 5.5 : That is measured from 6.0 cm. The partner measures from 6.5 cm, the length with the empty hanger on.
(b) Load for that extension: 4.9 N (within ±0.05 N)
- Common wrong answer 5.39 : The hanger was already on when he started measuring, so only the 500 g added counts.
- Common wrong answer 500 : 500 g is a mass. The load is its weight in newtons: 0.500 kg × 9.8 N/kg.

A full answer includes

(a) Method: extension = 11.5 cm − 6.0 cm.
(a) Answer: 5.5 cm.
(a) I included the hanger in the load: 550 g = 0.550 kg.
(a) Method: load = 0.550 kg × 9.8 N/kg.
(a) Answer: 5.39 N (5.4 N).
(b) Method: extension = 11.5 cm − 6.5 cm.
(b) Answer: 5.0 cm.
(b) Method: load = 0.500 kg × 9.8 N/kg, without the hanger.
(b) Answer: 4.9 N.
I explained that both ways give 0.98 N for each centimetre of extension.
I did not pair 5.5 cm with 4.9 N.

Also accept

(a) 5.5 cm and 5.39 N; (b) 5.0 cm and 4.9 N.

Watch for

“(a) 5.5 cm with a load of 4.9 N.” — Measured from the original length, the extension includes the stretch made by the hanger, so the load must include the hanger’s weight too: 550 g, 5.39 N.
“(b) 11.5 cm.” — 11.5 cm is the length now. The extension is the increase in length: 11.5 cm − 6.5 cm = 5.0 cm.

13. SQ12 · Practice

Model answer. a: reading from above, not at eye level, makes the pointer seem to sit against a different mark, a parallax error. b: the spring did not start at the 0 cm mark, so this gives the reading, not the extension. c: a bouncing pointer is caught above or below where it settles. d, e and f are good practice. Measuring from the length of the bare spring, with nothing on it, is right as long as the hanger’s weight is counted in the load, because the hanger’s stretch is then part of the extension. Taking the zero reading with the empty hanger on is the other right way, and reading the millimetre marks makes the extensions more precise.

A full answer includes

I chose a.
I chose b.
I chose c.
I did not choose d: measuring from the bare spring is right when the hanger is counted in the load.
I did not choose e or f: they are good practice.

Also accept

a, b, c

Watch for

“b is fine: the 0 cm mark is the start of the rule.” — The spring did not start at 0 cm. Measuring from 0 cm gives the reading, not the extension.
“d makes the extensions wrong: the hanger is never part of the load.” — Measured from the bare spring, the extension includes the stretch made by the hanger, so the hanger’s weight belongs in the load. The point still lies on the same straight line.
“e makes the extensions wrong: the hanger must be off for the zero reading.” — With the hanger on for the zero reading, its stretch is already included, so the load is just the masses you add. That is the method used here.
“f makes the extensions wrong: a metre rule is read to the nearest centimetre.” — A metre rule has millimetre marks, so it can be read to the nearest 0.1 cm. That makes the extensions more precise, not wrong.

14. ST1 · Apply

Model answer. (a) The weight of each load pulls down on the spring, so it stretches: the force changes its size and its shape, and it gets longer. (b) Each 10 N moves the pointer by the same 1.6 cm (3.6 − 2.0, 5.2 − 3.6, and so on), because the extension is proportional to the load. So each 5 N moves it 1.6 cm ÷ 2 = 0.8 cm, and the marks should be 0.8 cm (8 mm) apart, all equally spaced, starting from 2.0 cm for 0 N. (c) The extension is 6.0 cm − 2.0 cm = 4.0 cm. Each 1.6 cm is 10 N, and 4.0 cm ÷ 1.6 cm = 2.5, so the weight is 2.5 × 10 N = 25 N. (d) The heavier hook only moves the zero reading, to 2.6 cm. The bag still stretches the spring by 4.0 cm, so the pointer reads 2.6 cm + 4.0 cm = 6.6 cm.

A full answer includes

(a) I wrote that the loads stretch the spring: it gets longer.
(a) I wrote that both its size and its shape change.
(b) Method: each 10 N moves the pointer by the same 1.6 cm, so 5 N moves it 1.6 cm ÷ 2.
(b) Answer: 0.8 cm (8 mm).
(b) I did not halve a reading, such as 3.6 cm, instead of the step between readings.
(c) Method: extension = 6.0 cm − 2.0 cm = 4.0 cm.
(c) Method: 4.0 cm ÷ 1.6 cm = 2.5 steps of 10 N.
(c) Answer: 25 N.
(d) I kept the extension at 4.0 cm: the same bag gives the same extension.
(d) Method: reading = 2.6 cm + 4.0 cm.
(d) Answer: 6.6 cm.
I did not use 6.0 cm as the extension in (c), which gives 37.5 N.

Also accept

(a) stretch it: they change its size and shape; (b) 0.8 cm (8 mm) apart; (c) 25 N; (d) 6.6 cm.

Watch for

“(b) 3.6 cm ÷ 2 = 1.8 cm.” — 3.6 cm is where the pointer is with 10 N, including the 2.0 cm it read with no load. Halve the step between readings, 1.6 cm, instead.
“(b) The marks get closer together for bigger loads.” — Each 10 N moves the pointer by the same 1.6 cm, because the extension is proportional to the load. So the marks are equally spaced.
“(c) 6.0 cm gives 6.0 ÷ 1.6 × 10 = 37.5 N.” — 6.0 cm is the reading. Take away the zero reading first: the extension is 4.0 cm, which gives 25 N.
“(d) 6.0 cm, the same as before.” — The heavier hook has moved the zero reading down from 2.0 cm to 2.6 cm. The bag adds the same 4.0 cm to the new zero reading.