Why does a spinning ball follow a curved pattern?
August 1, 2005 4:19 AM   Subscribe

Beckham-physics filter. The curling ball.

I like to kick a ball in the yard sometimes and particularly enjoy trying to swerve or curl the ball around stationary objects. So I'll kick it with the instep on my right foot and from above it goes in a counterclockwise and will (on those occasions I get it right !) move from right to left in the air in a sort of curving arc* . And of course the opposite occurs if I use the outstep, spinning it in a clockwise direction (ie. it curls from right to left)

If we accept for the exercise that there's no prevailing wind, why is it that a spinning ball will curl - what forces are at play? And sometimes the ball seems to spin at a similar speed but doesn't actually follow a curling path (less often). Why? Bonus points for explaining why a ball will sometimes dip +/- weave --- not in the backyard, but you can see it for instance when David Beckham does a freekick around a wall of players over 25-40 yards distance towards the goal.
(Is the direction of spin the single variable here +\- weight or...?)
*[It's not a true flat (horizontal) arc of course because of gravity ie. the ball curls but also falls]
posted by peacay to Science & Nature (13 answers total)
Best answer: "The first explanation of the lateral deflection of a spinning object was credited by Lord Rayleigh to work done by the German physicist Gustav Magnus in 1852. Magnus had actually been trying to determine why spinning shells and bullets deflect to one side, but his explanation applies equally well to balls. Indeed, the fundamental mechanism of a curving ball in football is almost the same as in other sports such as baseball, golf, cricket and tennis.

Consider a ball that is spinning about an axis perpendicular to the flow of air across it. The air travels faster relative to the centre of the ball where the periphery of the ball is moving in the same direction as the airflow. This reduces the pressure, according to Bernouilli's principle. The opposite effect happens on the other side of the ball, where the air travels slower relative to the centre of the ball. There is therefore an imbalance in the forces and the ball deflects. This lateral deflection of a ball in flight is generally known as the "Magnus effect".

The forces on a spinning ball that is flying through the air are generally divided into two types: a lift force and a drag force. The lift force is the upwards or sidewards force that is responsible for the Magnus effect. The drag force acts in the opposite direction to the path of the ball.

The drag force, FD, on a ball increases with the square of the velocity, v, assuming that the density, r, of the ball and its cross-sectional area, A, remain unchanged: FD = CDrAv2/2. It appears, however, that the "drag coefficient", CD, also depends on the velocity of the ball. For example, if we plot the drag coefficient against Reynold's number - a non-dimensional parameter equal to rv D /µ, where D is the diameter of the ball and µ is the kinematic viscosity of the air - we find that the drag coefficient drops suddenly when the airflow at the surface of the ball changes from being smooth and laminar to being turbulent.

When the airflow is laminar and the drag coefficient is high, the boundary layer of air on the surface of the ball "separates" relatively early as it flows over the ball, producing vortices in its wake. However, when the airflow is turbulent, the boundary layer sticks to the ball for longer. This produces late separation and a small drag.

The Reynold's number at which the drag coefficient drops therefore depends on the surface roughness of the ball. For example, golf balls, which are heavily dimpled, have quite a high surface roughness and the drag coefficient drops at a relatively low Reynold's number (~ 2 x 104). A football, however, is smoother than a golf ball and the critical transition is reached at a much higher Reynold's number (~ 4 x 105) .

The upshot of all of this is that a slow-moving football experiences a relatively high retarding force. But if you can hit the ball fast enough so that the airflow over it is turbulent, the ball experiences a small retarding force (see right). A fast-moving football is therefore double trouble for a goalkeeper hoping to make a save - not only is the ball moving at high speed, it also does not slow down as much as might be expected. Perhaps the best goalkeepers intuitively understand more physics than they realize.

In 1976 Peter Bearman and colleagues from Imperial College, London, carried out a classic series of experiments on golf balls. They found that increasing the spin on a ball produced a higher lift coefficient and hence a bigger Magnus force. However, increasing the velocity at a given spin reduced the lift coefficient. What this means for a football is that a slow-moving ball with a lot of spin will have a larger sideways force than a fast-moving ball with the same spin. So as a ball slows down at the end of its trajectory, the curve becomes more pronounced."

posted by cyphill at 5:16 AM on August 1, 2005

That pretty much explains it. The shorter version is this: When you see the diagram of an airfoil (wing cross section), they always show that lift is generated by the fact that airflow over the wing is a longer path than under the wing and therefore the it needs to move more quickly over the top to get around the wing. When the air moves more quickly, the pressure drops and a force toward that side of the wing (lift) is generated.

Same principle applies here. The airflow around the ball is affected by the spin of the ball. On the side where the spin is in the same direction of travel, the relative flow is slower, where it is in the opposite direction of travel, it is faster, so an aerodynamic force acts on the ball to make it curve.

This is the same principle that makes a tennis ball with topspin drop more quickly than the classic ballistic trajectory while a ball with back spin floats longer than the ballistic trajectory would predict.

I'm too lazy to google it, but I bet you would find similar discussions with diagrams if you looked up what happens during a pitched curveball in the game of American baseball.
posted by Doohickie at 6:03 AM on August 1, 2005

But for extra credit, why does a curling stone curl the way it does?

(You are the one who brought up 'curling'...)
posted by Chuckles at 8:07 AM on August 1, 2005

Similar AskMe question (believe it or not).
posted by Otis at 9:49 AM on August 1, 2005

For a truly awesome example of the curveball kick, check out Roberto Carlos.
posted by essexjan at 10:06 AM on August 1, 2005

How the hell do you download anything from that 3dnuta.com site? I even created an account but it still says "You are not authorized to view this" when I try to load the Roberto Carlos clip (or any other clip on the site for that matter.) Does anyone have a link that actually works?
posted by Rhomboid at 12:18 PM on August 1, 2005

Sorry Rhomboid, I had no problem with it, I just clicked on it and off it went, no login needed or anything. But you'll find another link to the goal here.
posted by essexjan at 1:19 PM on August 1, 2005

Can you imagine being in the wall when he kicked that... I think it pretty much changed direction by 90 deg before it hit the post.

posted by Chuckles at 2:22 PM on August 1, 2005

This shot, from the page that essexjan linked to, is pretty amazing, too. Check out the goalie's reaction in the shot from in back of the goal.
posted by kirkaracha at 2:54 PM on August 1, 2005

Look up info about baseball curveballs - same idea.
posted by thedevildancedlightly at 4:32 PM on August 1, 2005

Response by poster: Thanks everybody. I love Roberto! The 'canon on treestumps'. He's in a couple of the ads linked in this thread the other day. I have a great compilation of shots kirkaracha - I just don't remember where I got it from and don't have a server to post it. It's like a best of the best in the English Premier League over a few seasons, set to music.
Now that I know how these great shots are made, I will of course be able to perform them much better and more consistently *cough*.

[By the way, in that Carlos shot, who were Brazil playing? Peru?? Dark blue tops, white shorts and red sox] There's possibly a good askme in there: 'please find me the best goals for download' or similar.
posted by peacay at 6:53 PM on August 1, 2005

posted by mr.marx at 9:32 PM on August 1, 2005

Response by poster: aaaah. Thanks mr.marx. The red through me. I saw the Spanish (or Portuguese) signs behind the goals and kept thinking South America - which is true enough of course....for one team. I left my lateral thinking hat in my other computer.
posted by peacay at 9:43 PM on August 1, 2005

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