Why spaghetti snaps sideways
Tonight's idea
Stand a strand of dry spaghetti on a table and press gently on the top. It stays straight and pushes back. Press harder, a little at a time. Still straight. Then, at one particular push, it bows out sideways. This is buckling, and that push is the critical load.
Below the critical load the straight shape is stable: nudge the strand and it springs back. Above it, the straight shape cannot last, and a tiny extra push makes a big bow. Tiny flaws decide which way it bows.
For strands of the same material and thickness, held the same way at the ends, the critical load goes like \(1 \div (\text{length} \times \text{length})\). Half the length holds 4 times the load.
This rule is a model for long, thin, straight strands. Real strands are never perfectly straight, so they start to bow a little early.
A gradual cause with a sudden effect is the subject of catastrophe theory. It sorts the simplest ways that a smooth change can produce a sudden jump into a few standard types.
Watch one: shorter is much stronger
These measurements are made up. Suppose a 20 cm strand buckles when the load on top reaches 40 g. Predict the critical load for 10 cm and for 5 cm of the same spaghetti, held the same way.
The rule says: divide the length by some number, and the load is multiplied by that number twice.
10 cm is 20 cm divided by 2. So the load is multiplied by \(2 \times 2 = 4\): \(40 \times 4 = 160\) g.
5 cm is 20 cm divided by 4. So the load is multiplied by \(4 \times 4 = 16\): \(40 \times 16 = 640\) g.
Check with one number that never changes: load \(\times\) length \(\times\) length. \(40 \times 20 \times 20 = 16{,}000\), \(160 \times 10 \times 10 = 16{,}000\) and \(640 \times 5 \times 5 = 16{,}000\). So for this spaghetti, load \(= 16{,}000 \div (\text{length} \times \text{length})\), in grams and centimetres.
Your turn
- Same made-up spaghetti: 20 cm buckles at 40 g. Fill in the predicted critical loads.
Length (cm) 4 8 16 20 25 Critical load (g) 40 - (a) A strand must hold 9 times the load of the 20 cm strand. How long should it be? (b) A friend says "half the length, so double the load". Use two columns of your table to show what is wrong. (c) Work out the rule's load for a 2 cm piece. Why should you not trust it much?
- To test the rule, you stand a strand upright, balance a light paper cup on top, and add coins one at a time until it bows. (a) What one thing do you change between tests? Name three things to keep the same. (b) Why test each length several times? (c) The cup weighs as much as 2 coins. A 20 cm strand bows with 6 coins in the cup. How many coins in the cup does the rule predict for a 10 cm strand?
- Stand a thin plastic ruler on its long edge and wedge it between two heavy books, so that it bows away from you and its ends cannot slide apart. Push the middle of the bow slowly towards you, harder and harder. (a) What do you expect to see? (b) Take your finger away slowly. What happens? (c) For the spaghetti, the smooth change is the load. What is it here, and what changes suddenly?
Stretch: thicker rods
Thickness matters even more than length. For solid round rods of the same material and length, held the same way, the model says the critical load goes like thickness \(\times\) thickness \(\times\) thickness \(\times\) thickness. (a) A rod is twice as thick. How many times the load does it hold? (b) A rod is twice as thick and twice as long. How many times the load? (c) A rod is 3 times as thick. How many times as long can it be and still hold the same load as the thin one?
Answers
- Use load \(= 16{,}000 \div (\text{length} \times \text{length})\). 4 cm: \(16{,}000 \div 16 = 1000\) g. 8 cm: \(16{,}000 \div 64 = 250\) g. 16 cm: \(16{,}000 \div 256 = 62.5\) g. 25 cm: \(16{,}000 \div 625 = 25.6\) g. Notice that 4, 8 and 16 cm each double the length, and each load is a quarter of the one before. The 25 cm strand is longer than 20 cm, so it holds less than 40 g.
- (a) 9 is \(3 \times 3\), so divide the length by 3: \(20 \div 3 \approx 6.7\) cm. The load is \(9 \times 40 = 360\) g. (b) 16 cm holds 62.5 g and 8 cm holds 250 g. Half the length holds 4 times the load, not double. The same happens from 8 cm to 4 cm: 250 g to 1000 g. (c) \(16{,}000 \div (2 \times 2) = 4000\) g, which is 4 kg. The rule is a model for long, thin strands. A 2 cm piece is short and stubby, and it may crush or crack before it ever bows. The 40 g was also made up.
- (a) Change only the length. Keep the same: the kind and thickness of spaghetti, the way the two ends are held, and the way you add the load (gently, in the middle of the cup, with the strand upright). Straight, undamaged strands matter too. (b) Strands differ a little and so do your hands, so one test can mislead. Several tests show how much the results scatter. (c) Count the cup. The load at 20 cm is \(6 + 2 = 8\) coins' worth. Half the length holds 4 times the load: \(4 \times 8 = 32\) coins' worth. Take off the cup: \(32 - 2 = 30\) coins. Forgetting the cup gives \(4 \times 6 = 24\), which is too few.
- (a) At first the bow flattens a little and pushes back. Then, at one particular push, it snaps through all at once and bows towards you. (b) It stays there. It does not go back the way it came. To get the first shape back you have to push from the other side, and it snaps again. The ruler has two stable shapes, and which one you see depends on what happened before. (c) The smooth change is how hard you push. The sudden change is the shape, which jumps from a bow away from you to a bow towards you. This is a true jump. The spaghetti is a gentler cousin: in the ideal model it bows without a jump, but very fast. With ends free to tilt, a load 1% above critical bows a 20 cm strand sideways by about 1.8 cm.
Stretch. (a) \(2 \times 2 \times 2 \times 2 = 16\) times the load. (b) Twice as thick multiplies the load by 16. Twice as long divides it by \(2 \times 2 = 4\). Together: \(16 \div 4 = 4\) times the load. (c) 3 times as thick multiplies the load by \(3 \times 3 \times 3 \times 3 = 81\). To cancel that, the length must be multiplied by 9, since \(9 \times 9 = 81\). So it can be 9 times as long. Why four thicknesses? A rod twice as thick has 4 times as much material in its cross-section, and that material sits further from the centre line, where it resists bending better. That counts for another factor of 4, and \(4 \times 4 = 16\).