Why a straight line on a spinning plate comes out curved
Tonight's idea
Set a paper plate spinning on a turntable. Run a pen across it through the centre, straight and steady as seen from the room. Stop the plate and look: the line on the plate is curved. Nothing pushed the pen sideways. The paper turned underneath it, so each new bit of ink landed on a different part of the plate.
On a spinning surface, motion that is straight from outside looks curved from on board. This is the Coriolis effect. A pen crossing the centre of a plate that turns anticlockwise draws a path that bends to the right. On a clockwise plate it bends to the left.
Earth spins too. Seen from above the North Pole it turns anticlockwise, so moving air there drifts to the right. Seen from above the South Pole it turns clockwise, so air drifts to the left. Air heading into a low-pressure centre is turned aside and ends up circling it, anticlockwise in the north and clockwise in the south: the swirl of a big storm. At the equator the effect on winds fades to nothing.
Watch one: half a turn while the pen crosses
A plate of radius 10 cm turns anticlockwise once every 4 s, which is \(360^\circ \div 4 = 90^\circ\) each second. A pen crosses it through the centre in 2 s, so it moves \(20 \div 2 = 10\) cm each second. (An idealisation: perfectly steady pen, perfectly steady plate.) During the crossing the plate turns \(2 \times 90^\circ = 180^\circ\), half a turn.
| time (s) | 0 | 0.5 | 1 | 1.5 | 2 |
| pen to centre (cm) | 10 | 5 | 0 | 5 | 10 |
| plate has turned | \(0^\circ\) | \(45^\circ\) | \(90^\circ\) | \(135^\circ\) | \(180^\circ\) |
Call the pen's starting spot on the rim \(S\). Seen from the plate, the pen's line swings round clockwise. Before the centre, the ink lands on the spoke that is the turned angle clockwise from the spoke to \(S\). After the centre, measure from the spoke opposite \(S\). At 2 s the ink is 10 cm out, \(180^\circ\) clockwise from opposite \(S\). That is \(S\) itself: the rim has carried \(S\) half way round to meet the pen. The loop bends to the right all the way.
Your turn
- The same plate (radius 10 cm, anticlockwise, once every 4 s), but now the pen crosses through the centre in 1 s. (a) How far does the plate turn during the crossing? (b) Make the table for 0, 0.25, 0.5, 0.75 and 1 s. (c) Sketch the curve on a circle. Where on the rim does the pen leave, compared with \(S\)?
- Back to the 2 s crossing, but the plate now turns clockwise at the same rate. (a) How does the curve compare with the figure, and which way does it bend? (b) Where does the pen leave the plate? (c) Looking down on which pole does Earth turn this way?
- (a) Earth turns once in about 24 hours. How many degrees is that each hour? (b) A pendulum at the North Pole keeps swinging along the same line while Earth turns underneath. Seen from the ground, how fast does that line seem to turn, and which way (looking down)? (c) At latitude \(L\) the rate is \(15^\circ \times \sin L\) each hour. Find the rate at \(30^\circ\) north and the time for one full turn. And at the equator?
- True or false: "Sinks drain anticlockwise in the north and clockwise in Australia, because of the Coriolis effect." Before you decide: a sink empties in about 30 s. How many degrees does Earth turn in that time? And in the 3 days that a big storm can last?
Stretch: the tea leaves
Stir a cup of tea with loose leaves in it. The leaves gather in the middle of the bottom. (a) The tea goes round about once a second, Earth once a day. How many times slower is Earth? (b) Stir the other way and the leaves still gather in the middle. Why does that rule out the Coriolis effect? (c) So what gathers them? Hint: the tea at the very bottom rubs on the cup and goes round more slowly.
Answers
- (a) \(1 \times 90^\circ = 90^\circ\), a quarter turn. (b) The pen now moves 20 cm each second. Distances from the centre: 10, 5, 0, 5, 10 cm. The plate has turned \(0^\circ\), \(22.5^\circ\), \(45^\circ\), \(67.5^\circ\), \(90^\circ\). (c) The curve leaves \(S\), bends gently to the right through the centre, and ends 10 cm out, \(90^\circ\) clockwise from the spot opposite \(S\). That is a quarter of the way round the rim from \(S\), going anticlockwise. On a plate that did not turn, the pen would leave opposite \(S\). The faster crossing gives the plate less time to turn, so the line is less curved.
- (a) It is the mirror image of the figure's loop, flipped top to bottom, and it bends to the left all the way. (b) At \(S\) again. Half a turn brings \(S\) round to meet the pen whichever way the plate turns. (c) The South Pole. That is why moving air drifts to the left in the Southern Hemisphere.
- (a) \(360^\circ \div 24 = 15^\circ\) each hour. (More exactly, Earth turns once against the stars in about 23 h 56 min, which makes \(15.04^\circ\) each hour.) (b) About \(15^\circ\) each hour, clockwise seen from above. The pendulum is the pen and Earth is the anticlockwise plate. (c) \(15^\circ \times \sin 30^\circ = 7.5^\circ\) each hour, so a full turn takes \(360 \div 7.5 = 48\) hours. At the equator \(\sin 0^\circ = 0\): the line of the swing does not turn at all.
- False. Thirty seconds is \(\frac{1}{120}\) of an hour, so Earth turns \(15^\circ \div 120 = 0.125^\circ\), an eighth of a degree. In 3 days it turns 3 full turns, \(1080^\circ\). The storm is huge and lasts long enough for Earth's turning to shape it. In a sink the effect is tiny next to the swirl left over from filling, the shape of the basin and the way the plug is pulled. Those decide the direction, and sinks drain both ways in both hemispheres. Only very careful experiments, with a wide round tank left to settle for many hours, have picked up Earth's turning.
Stretch. (a) A day is \(24 \times 3600 = 86{,}400\) s, so Earth turns 86,400 times more slowly than the tea. Its effect over a few seconds in a cup is far too small to see. (b) The Coriolis effect pushes moving things to one side (to the right in the north). So stirring the other way would reverse its push, outward one way and inward the other, but the leaves go to the middle both ways. (c) Spinning tea presses outward on the wall of the cup, so the pressure is higher near the wall. Most of the tea goes round fast enough to hold its place against that pressure. The slow layer at the bottom does not, so it is pushed inward along the bottom, rises in the middle and flows back out near the top. The inward flow along the bottom sweeps the leaves to the middle, and they are too heavy to rise with it. Albert Einstein published this explanation in 1926.