Deformation of paper cylinders: testing w = k(y − p)
Roll a strip of paper into a cylinder and stand it under a spring-loaded 100 g mass. Squash it by a fixed amount and read how much the springs 'give back' — then test whether a wider strip needs proportionally more give, via w = k(y − p).
WHAT YOU'LL LEARN
- ✓Reading a change in load from a spring's change in length, not from a scale
- ✓Why the springs get SHORTER once the cylinder is bearing part of the weight — force balance with three forces instead of two
- ✓Testing a proposed linear relationship (w = k(y − p)) against two independent measurements rather than assuming it
- ✓Justifying significant figures in a derived quantity by the sig figs of the raw readings that produced it
ON YOUR BENCH
- □Stand, boss and wooden rod (~35 cm)
- □Two expendable springs (k ≈ 25 N/m each), connected in series
- □Sheet of paper, width 21.0 cm, approx. length 30 cm (e.g. A4)
- □Narrower strip, width 6.8 cm, approx. length 30 cm, cut from an A4 sheet
- □4 paper clips + small container to hold them
- □100 g mass hanger + 100 g slotted mass
- □Stopwatch
- □Metre rule (mm scale)
- □30 cm ruler (mm scale)
The protocol, step by step
The same guide is printed and waiting at your bench.
- 01
Roll and clip the smaller strip
Roll the 6.8 cm strip into a cylinder held with two paper clips, and adjust it until its diameter d is as close as possible to 7.0 cm. Measure and record w and d.
- 02
Hang the mass clear and measure y
Slide the springs onto the rod above the cylinder, hang the 100 g mass from the lower spring, and lower the boss until the mass hangs clear — about 10 cm above the bench. Measure the springs' length, y.
- 03
Squash the cylinder and measure p
Lower the boss further so the mass descends onto the middle of the cylinder, squashing it, until the base of the mass is 2.5 cm above the bench. Wait for the mass to settle, then measure the new (shorter) spring length, p. Calculate (y − p).
- 04
Repeat everything with the larger sheet
Swap in the 21.0 cm sheet and repeat the full procedure: roll and clip to d ≈ 7.0 cm, measure y, squash to 2.5 cm above the bench, measure p, and calculate (y − p) again.
- 05
Calculate k for each sheet and compare
Using w = k(y − p), calculate k = w / (y − p) for each sheet. If the relationship holds with one genuine constant, the two values of k should come out close to each other despite the very different widths.
What you should see
- →y depends only on the springs and the 100 g mass, not on which sheet is underneath — expect it to land in roughly the same place (here, around 22 cm) for both sheets.
- →The wider sheet resists being squashed more, so it needs to give up proportionally more spring length: (y − p) for the 21.0 cm sheet comes out several times larger than for the 6.8 cm strip.
- →Two readings alone can't confirm a trend — that's exactly the limitation the mark scheme expects you to name in 2(g)(i).
- →k = w / (y − p) should land close to the same value for both sheets if w = k(y − p) genuinely holds with one constant, independent of paper width.
| Smaller sheet (w = 6.8 cm) | Larger sheet (w = 21.0 cm) | |
|---|---|---|
| Diameter d / cm | 7.0 | 7.0 |
| Length of springs y / cm | 22.0 | 22.0 |
| Length of springs p / cm | 19.7 | 15.0 |
| (y − p) / cm | 2.3 | 7.0 |
| k = w / (y − p) | 3.0 | 3.0 |
⚠ BEFORE YOU START
- •Paper clips have sharp ends — handle them carefully when fitting them to the rolled cylinder.
- •Keep fingers clear while lowering the boss: it's a pinch point between the descending mass and the cylinder.
- •Make sure the stand's base is clear of the bench edge so a knock doesn't topple it while loaded.
Try it, then run it for real.
Practise the whole thing on the virtual bench, then book real lab time and do it with your own hands.