Question 2 — Deformation of paper cylinders
Roll paper into a cylinder, squash it under a spring-loaded mass, and test whether the paper's width relates to the springs' change in length as w = k(y − p).
RUN IT ON THE BENCH
Your readings feed straight into the answer table below as you record them.
ON YOUR BENCH — FROM THE CONFIDENTIAL INSTRUCTIONS
Question 2 — apparatus
- stand(1)
- boss(1)
- wooden rod(1)
- expendable springs(2)
- sheet of paper(1)
- strip of paper(1)
- paper clip(4)
- small container(1)
- 100 g mass hanger(1)
- 100 g slotted mass(1)
- stopwatch(1)
- metre rule(1)
- 30 cm ruler(1)
THE BENCH — ROLLED PAPER CYLINDER UNDER A SPRING-LOADED MASS · SMALLER SHEET
Why is p smaller than y?
With the mass hanging clear, the springs alone support its whole weight: T = mg, so they stretch to their full length y. Once the cylinder is squashed underneath, it pushes back up on the mass with some normal force N — force balance now reads T + N = mg, so T = mg − N is smaller. Less spring tension means less extension, so the springs are shorter: p < y. The amount they "gave back", (y − p), is a direct read on how much weight the paper itself is carrying.
Why should k come out the same for both sheets?
w = k(y − p) says the width of paper needed to support a given load scales linearly with (y − p), with k as the constant of proportionality — a property of the paper's stiffness against buckling, not of how wide your particular strip happens to be. The wider sheet should show a proportionally larger (y − p) for the same load. If k1 and k2 come out close to each other, that constant relationship is supported; if they're wildly different, w = k(y − p) doesn't hold as a simple proportionality.
Why 2.5 cm above the bench, and not squashed flat?
Squashing the cylinder flat would destroy it — you'd have nothing left to compare against the second sheet, and no way to repeat the reading. Stopping at a fixed height (2.5 cm) gives a controlled, repeatable amount of deformation, so both sheets are tested under the same condition.
YOUR TASKS · 0/3
- ✓Roll and clip the smaller sheet at d ≈ 7.0 cm, then measure y and p
- ✓Repeat with the larger sheet
- ✓Compare k1 and k2 — is w = k(y − p) supported?
RESULTS — BOTH SHEETS
| SMALLER | LARGER | |
|---|---|---|
| w / cm | 6.8 | 21.0 |
| d / cm | — | — |
| y / cm | — | — |
| p / cm | — | — |
| (y − p) / cm | — | — |
YOUR ANSWERS
Select the smaller of the two pieces of paper. Measure and record its width w. Roll it into a cylinder held with two paper clips (Fig. 2.2) and adjust until its diameter d is as close as possible to 7.0 cm. Measure and record d.
| smaller sheet | |
|---|---|
| width w / cm | |
| diameter d / cm |
Set up the apparatus of Fig. 2.3: slide the upper spring's loop onto the wooden rod, hang a 200 g mass from the lower spring, and adjust the boss so the bottom of the mass is approximately 10 cm above the bench, with the paper cylinder centred underneath. Measure and record the length of the springs, y.
| smaller sheet | |
|---|---|
| length of springs y / cm |
Estimate the percentage uncertainty in your value of y. Show your working.
By adjusting the height of the boss, lower the mass to squash the middle of the paper cylinder until the bottom of the mass is 2.5 cm above the bench (Fig. 2.4). Measure and record the new length of the springs, p.
| smaller sheet | |
|---|---|
| length of springs p / cm |
Calculate (y − p).
Using the larger sheet of paper, repeat 2(a), 2(b)(i), 2(c)(i) and 2(c)(ii).
| larger sheet | |
|---|---|
| width w / cm | |
| diameter d / cm | |
| length of springs y / cm | |
| length of springs p / cm | |
| (y − p) / cm |
Using your data and w = k(y − p), calculate two values of k — one for each sheet of paper.
| value | |
|---|---|
| first value of k (smaller sheet) | |
| second value of k (larger sheet) |
Justify the number of significant figures that you have given for your values of k.
It is suggested that the percentage uncertainty in the values of k is 10%. Using this uncertainty, explain whether your results support the relationship in 2(e).
Describe four sources of uncertainty or limitations of the procedure for this experiment. For any measurement uncertainty you describe, state the quantity being measured and a reason for the uncertainty.
Describe four improvements that could be made to this experiment. You may suggest other apparatus or different procedures.