# Relative Strength of the Magnus effect relative to gravity

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• pnachtwey
In summary: It is possible that the extra drag from the chopper makes the ball rise more than a regular table tennis ball.In summary, the force on a table tennis ball due to Magnus effect is high, much higher than that due to gravity. The calculation for this force is accurate, but the acceleration seems too high. It's possible the extra drag from the chopper in the video makes the ball rise more than a regular table tennis ball.
pnachtwey
TL;DR Summary
From viewing golf and table tennis balls with back spin, you can see the balls float or even rise against the force of gravity. This means the force due the Magnus effect is relatively equal to that of gravity but in my simulation, the force due to the Magnus effect is much more. Why, What did I do wrong or are my calculations correct?
I have made a simulation of a table tennis ball being hit and landing on the table. There are 5 differential equations that are integrated to compute the horizontal position, horizontal velocity, vertical position, vertical speed and spin. by integrating 5 differential equations simultaneously I have checked the units so when I divide the forces due to gravity, viscous friction, damping and the Magnus effect by the mass of the table tennis ball, I get units of meters per second squared as expected.
What I didn't expect is that the acceleration, actually deceleration, due to drag and Magnus effect would be many times that of gravity. The calculations look correct but it bothers me that the Magnus effect should be so high.

Often a golfer will drive a ball and you can see it lift during the flight. In this case the upwards force due to the Magnus effect is a little greater than that caused by gravity. However, in my simulation the force due to the Magnus effect is 10 times that due to gravity. A table tennis ball "chopped" with heavy back spin will appear to rise a bit or float like a golf ball but it doesn't accelerate up as my simulation would suggest. This doesn't seem right.

I have attached my work. It shows all the calculations and at last two pages I do the units check to makes sure all the units are consistent.
Can anybody find what is wrong or right with my simulation? The trajectory looks good but I don't believe the high forces due to the Magnus effect.

#### Attachments

• Mathcad - TT Trajectory 10 deg.pdf
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pnachtwey said:
Summary: From viewing golf and table tennis balls with back spin, you can see the balls float or even rise against the force of gravity. This means the force due the Magnus effect is relatively equal to that of gravity but in my simulation, the force due to the Magnus effect is much more. Why, What did I do wrong or are my calculations correct?

I have made a simulation of a table tennis ball being hit and landing on the table. There are 5 differential equations that are integrated to compute the horizontal position, horizontal velocity, vertical position, vertical speed and spin. by integrating 5 differential equations simultaneously I have checked the units so when I divide the forces due to gravity, viscous friction, damping and the Magnus effect by the mass of the table tennis ball, I get units of meters per second squared as expected.
What I didn't expect is that the acceleration, actually deceleration, due to drag and Magnus effect would be many times that of gravity. The calculations look correct but it bothers me that the Magnus effect should be so high.

Often a golfer will drive a ball and you can see it lift during the flight. In this case the upwards force due to the Magnus effect is a little greater than that caused by gravity. However, in my simulation the force due to the Magnus effect is 10 times that due to gravity. A table tennis ball "chopped" with heavy back spin will appear to rise a bit or float like a golf ball but it doesn't accelerate up as my simulation would suggest. This doesn't seem right.

I have attached my work. It shows all the calculations and at last two pages I do the units check to makes sure all the units are consistent.
Can anybody find what is wrong or right with my simulation? The trajectory looks good but I don't believe the high forces due to the Magnus effect.
I get F~1N. I am not sure how you got ω0 to be so high though. Check the magnitude of your velocities. If accurate then the force is correct. The acceleration you compute is high but it would not stay that high because drag forces would keep it from accelerating to much. There is no requirement that the Magnus force has to be in some relation to gravity. It is totally independent.

You might find some information of relative velocities in the supplemental informations of this video;

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I agree with your approximation of the Magnus force. I get similar numbers depending on whose approximation I am using. I get 0.886 N which doesn't seem big. Here is the where everything goes wrong. Divide the approximate 1N force by the mass, 2.7gm and the acceleration rate is over 350 meter/second^2. That is over 35 g! The ball would fly around in circles if not obstructed. ω0 is not unreasonable for spin on a table tennis ball. I have a video where a chopper's back spin is measured at 137 rev/sec. 150 rev/sec seems to be a good maximum. I am only using 75 rev/sec or 471.239 rad/sec. If I used twice the spin, the Fm would be twice as high. I don't think this is right. A chopped ball would move in loops if the formula is right.

pnachtwey said:
I agree with your approximation of the Magnus force. I get similar numbers depending on whose approximation I am using. I get 0.886 N which doesn't seem big. Here is the where everything goes wrong. Divide the approximate 1N force by the mass, 2.7gm and the acceleration rate is over 350 meter/second^2. That is over 35 g! The ball would fly around in circles if not obstructed. ω0 is not unreasonable for spin on a table tennis ball. I have a video where a chopper's back spin is measured at 137 rev/sec. 150 rev/sec seems to be a good maximum. I am only using 75 rev/sec or 471.239 rad/sec. If I used twice the spin, the Fm would be twice as high. I don't think this is right. A chopped ball would move in loops if the formula is right.
I sent you a video with some information. You might check that out and look at the speeds. The equations of motion involve all the forces together, not just the Magnus force so it's "gee" is meaningless by itself.

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I see the video. I gave it a thumbs down. The units don't work out to be a force.
What is R in the video? The diameter of a TT ball is about 0.04m, not the radius.
You can see that I break the acceleration into horizontal and vertical calculations. It makes no difference if I sum the forces and then divide by the mass or divide each force by the mass and then sum the accelerations.

pnachtwey said:
I see the video. I gave it a thumbs down. The units don't work out to be a force.
What is R in the video? The diameter of a TT ball is about 0.04m, not the radius.
You can see that I break the acceleration into horizontal and vertical calculations. It makes no difference if I sum the forces and then divide by the mass or divide each force by the mass and then sum the accelerations.
Are you including all the forces or just calculating them separately and estimating things? The equation of motion is complicated with gravity, drag, Magnus, and a rotational drag torque. It is the net force that governs the motion. This has likely been solved many times and there should be some papers with all the details available somewhere.

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Did you see the link to my original document? There are no worked examples except the one in the link. If you can find something wrong with it I would be grateful. The only thing I can find wrong in my document so far is that my initial velocity is too high but even if I reduce it the Magnus force is still too high. Have you thought about the acceleration causes by 1N force on 2.7 gm ? That acceleration is too high.

pnachtwey said:
Did you see the link to my original document? There are no worked examples except the one in the link. If you can find something wrong with it I would be grateful. The only thing I can find wrong in my document so far is that my initial velocity is too high but even if I reduce it the Magnus force is still too high. Have you thought about the acceleration causes by 1N force on 2.7 gm ? That acceleration is too high.
I read it. Honestly, in the format it is in I cannot tell exactly what you are doing. It looks like you calculated the trajectory and then calculated different forces. Is that correct? Why don't you enter the equation of motion in Latex format so we can see what you are doing.

As far as what I said earlier about F~1N, that was based on your numbers but I did state the spin seemed high. And yes, that seems way too high but I have no information of what the other forces such as drag are. The important thing is what is the total force on the ball just after it is hit and what velocity and spin does it have? That will determine the motion.

The pdf is generated by Mathcad. If the ball is traveling horizontally, the drag force opposes the horizontal motion. It doesn't add to the Magnus force which is down. When the ball is traveling horizontally I add the acceleration due to the downward Magnus force. If the ball is traveling up or down there is a drag force that opposes vertical motion. There are enough comments that explain what I am doing. The key are the formulas for x'' and y'' which are the horizontal and vertical accelerations. The accelerations get integrated to velocities and the velocities into positions using Runge-Kutta. So the 5 states are horizontal and vertical position, horizontal and vertical velocities and angular frequency. The angular frequency decays by 3% per second.
The Runge-Kutta routine, rkfixed, doesn't like units but the last two page I include units just to make sure they are right and consistent.

pnachtwey said:
The pdf is generated by Mathcad. If the ball is traveling horizontally, the drag force opposes the horizontal motion. It doesn't add to the Magnus force which is down. When the ball is traveling horizontally I add the acceleration due to the downward Magnus force. If the ball is traveling up or down there is a drag force that opposes vertical motion. There are enough comments that explain what I am doing. The key are the formulas for x'' and y'' which are the horizontal and vertical accelerations. The accelerations get integrated to velocities and the velocities into positions using Runge-Kutta. So the 5 states are horizontal and vertical position, horizontal and vertical velocities and angular frequency. The angular frequency decays by 3% per second.
The Runge-Kutta routine, rkfixed, doesn't like units but the last two page I include units just to make sure they are right and consistent.
So, if I understand, it takes about 0.05s for the ball to travel to the far end of the table after it is hit? The acceleration in x and y are functions of time, not single values though there will be an average value over the trip.

When you hit the ball, how long is it in contact with the paddle? I read the force is about 7.5N.

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40 m/s is too high for the initial speed of the ball. 15 m/s to 20 m/s is better. The maximum speed is about 30 m/s. I am integrating the accelerations using Runge-Kutta. I am assuming the positions, velocities, acceleration and spin are changing every 250 microseconds.
This document is what I am mainly using for the formula for the Magnus effect. See equation 1
https://deltamotion.com/peter/TableTennis/2012-GM-B-414%20Magnus%20Effect.PDF
I also looked up the formula for the Magnus effect on Wikipedia.
https://en.wikipedia.org/wiki/Magnus_effect
it is almost the same except it divides by 3 instead of multiplying by Cm which the previous document calls the Magnus coefficient. Basically they differ a little in scaling but both have the correct units.
Even if I slow the ball down to 15 m/s the Magnus coefficient seems to be too high. If I reduce the Magnus coefficient to 0.05, I get reasonable results. What is interesting is that 0.05 is almost 2PI less than the 0.3333 from Wikipedia or 0.29 from the
https://deltamotion.com/peter/TableTennis/2012-GM-B-414%20Magnus%20Effect.PDF
I wonder if these papers used rev/sec instead of rad/sec for ω.
See this video of a looper and chopper playing and being recorded with high speed cameras. The chopper returns a ball with a back spin of 137 rev/sec.
https://deltamotion.com/peter/TableTennis/High%20Speed%20Table%20Tennis%20Video.flv
Notice that the ball may rise a little like a golf ball does but not much. This means the upwards force due to the Magnus effect offsets the downwards force due to gravity by a little, not 10+ times. To get a similar trajectory as the chopper I must reduce Cm to about 0.05.

Try computing the trajectory without Magnus and then without drag and compare predicted results to see what differences they make in principle.

It is easy to do what you ask. I can just set Cd and Cm to 0. The ball flies past the end of the able.
The Cm, Magnus coefficient, plays a much bigger part in landing the ball than the drag. I see nothing unexpected as long as I set Cm to 0.05 instead of 0.29 as in the document. Wikipedia has a 4/3 term were the 1/3 is about 0.3333 like the Cm
The drag force will reduce the ball speed by about 1/2 for every 5 meters of travel. The terminal velocity of a TT ball is about 8m/s.
Did you figure out my differential equations?
I will try to get entered in LaTeX. I also have a website that uses LaTeX.

I simulated a golf drive using data I got off the internet but using the same Mathcad equations and a Cm of 0.29 which seems too high for a table tennis ball but about right for a golf ball.
I am simulating a pro drive. A pro can hit the ball at 180 mph. The optimal angle is 16 degrees. The resulting spin is about 3000 rpm or about 50 rev per second. The flight path looks reasonable if not a little far. What I need to do is calculate a Reynold's number every interval because the Reynold's number changes as a function of velocity and this changes the Cd but nothing is said about Cm. I think the problem is getting the right drag and Magnus coefficients.

hutchphd
bob012345 said:
More ping pong specific Magnus modeling here;

Formula looks correct except it doesn't say what S0, the Magnus coefficient, is. This coefficient is the key. This document is like all the others. They have no idea what the Magnus coefficient is.

pnachtwey said:
Formula looks correct except it doesn't say what S0, the Magnus coefficient, is. This coefficient is the key. This document is like all the others. They have no idea what the Magnus coefficient is.
Here is the formula for an ideal spinning ball, i.e. the theoretical coefficient is 1.

https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/ideal-lift-of-a-spinning-ball/

Perhaps you can compare the ideal force to the measured force in those papers under the same conditions to get an estimate of what the coefficient should be.

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bob012345 said:
Here is the formula for an ideal spinning ball, i.e. the theoretical coefficient is 1.

https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/ideal-lift-of-a-spinning-ball/

Perhaps you can compare the ideal force to the measured force in those papers under the same conditions to get an estimate of what the coefficient should be.
This still doesn't work. Although the formula results in units of force, the magnitude of the lift is too great.
I use b=0.0205 m for the radius of a 40+ TT ball. I use 25 rev/sec for s ( spin ) and 17 m/sec for the velocity V.
Notice that NASA specifies the spin in revolution per second as oppose to radians per second.
I get a lift force of 1.483 which is even higher than what was estimated above. Divide the lift force by the mass of the table tennis ball, 0.0027 kg, the acceleration due to lift is 549 m/s^2 which is WAY TOO HIGH.
A chopper can back spin the ball at 137 rev/sec which is much greater than the 25 rev/sec I used for my calculations.

This shouldn't be that complicated but is seems that no one knows anything. Do you EVER see an example with real numbers and simulation that work? I haven't.
If I use the orginal formula where there is a Magnus coefficient of Cm=0.29. I will get reasonable results if the spin is entered in revolutions per second but the formula shows ω which is the symbol for angular velocity in radians per second. This effectively reduces the Magnus effect by a factor of 2π

pnachtwey said:
This still doesn't work. Although the formula results in units of force, the magnitude of the lift is too great.
I use b=0.0205 m for the radius of a 40+ TT ball. I use 25 rev/sec for s ( spin ) and 17 m/sec for the velocity V.
Notice that NASA specifies the spin in revolution per second as oppose to radians per second.
I get a lift force of 1.483 which is even higher than what was estimated above. Divide the lift force by the mass of the table tennis ball, 0.0027 kg, the acceleration due to lift is 549 m/s^2 which is WAY TOO HIGH.
A chopper can back spin the ball at 137 rev/sec which is much greater than the 25 rev/sec I used for my calculations.

This shouldn't be that complicated but is seems that no one knows anything. Do you EVER see an example with real numbers and simulation that work? I haven't.
If I use the orginal formula where there is a Magnus coefficient of Cm=0.29. I will get reasonable results if the spin is entered in revolutions per second but the formula shows ω which is the symbol for angular velocity in radians per second. This effectively reduces the Magnus effect by a factor of 2π
I don't have an issue with high acceleration since it only lasts for a short fraction of a second. When the ball is hit there is a force around 7.5N which I got from another problem which is ~2778 m/s^2. This a very very high acceleration for a brief instant to go from zero to ~50m/s in ~0.018s Then the ball goes the length of the table in ~0.05s or so. Under those conditions a Magnus acceleration of 550 m/s^2 would move the ball a fraction of a meter.

My point was to compare the NASA ideal value with some of the data of real measured forces to estimate the coefficient.

I don't have an issue with high acceleration since it only lasts for a short fraction of a second.
What do you mean by a short fraction of a second? If the acceleration due to the Magnus effect is 1.48N and mass is only 2.7gm the ball will take off. This It will accelerate up or down exceesively depending on the direction of spin.

My point was to compare the NASA ideal value with some of the data of real measured forces to estimate the coefficient.
Are you sure? Have you EVER seen this equation used in a simulation where all the work is shown? I haven't.
We agree on the approximate Magnus force. What you can't seem to understand is that this force is too much for a 2.7 gm ping pong ball.

pnachtwey said:
What do you mean by a short fraction of a second? If the acceleration due to the Magnus effect is 1.48N and mass is only 2.7gm the ball will take off. This It will accelerate up or down exceesively depending on the direction of spin.Are you sure? Have you EVER seen this equation used in a simulation where all the work is shown? I haven't.
We agree on the approximate Magnus force. What you can't seem to understand is that this force is too much for a 2.7 gm ping pong ball.
The time of play of the ball is only fractions of a second from the hit to the bounce during a serve. Gentle play back and forth doesn't do much for speed of spin so there aren't great forces involved. I was referring to high velocity serves which also may have a lot of spin. In those cases when one hits the ball the acceleration is far greater than the Magnus force. The balls can take it. Physics shows the force is not too much. I just showed you why the ball doesn't go crazy... because the time of flight is too short.

You somehow got the idea the force is too great and the ball would fly away out of control but that is just not the case here. These forces only act for a short time. It all works out fine.

This is basically the same simulation as before using the NASA formula. The acceleration due to the Magnus effect is still way too high.
I am not simulating a serve. I am simulating a loop where the ball is it at a height 50mm about the table so the ball must be hit up to go over the net and the Magnus effect makes the ball drop instead of keep going up. However, the NASA version of the Magnus effect is still 10 times too high. The ball accelerates downwards so fast that it doesn't even clear the net. If I reduce the Magnus effect by a factor of 10, the ball will sail off the end of the table. A ball hit with 25 rev/sec top spin and speed of 17m/s is fairly typical

#### Attachments

• Mathcad - TT Top Spin Trajectory NASA.pdf
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pnachtwey said:
This is basically the same simulation as before using the NASA formula. The acceleration due to the Magnus effect is still way too high.
I am not simulating a serve. I am simulating a loop where the ball is it at a height 50mm about the table so the ball must be hit up to go over the net and the Magnus effect makes the ball drop instead of keep going up. However, the NASA version of the Magnus effect is still 10 times too high. The ball accelerates downwards so fast that it doesn't even clear the net. If I reduce the Magnus effect by a factor of 10, the ball will sail off the end of the table. A ball hit with 25 rev/sec top spin and speed of 17m/s is fairly typical
The NASA version is for an ideal ball not an actual ball. I pointed you towards papers with measured forces so you can compare and adjust the Magnus coefficient.

So what ball is ideal? I would think a table tennis ball is pretty close to ideal. They are relatively smooth and seamless. A table tennis ball wouldn't be so far off from ideal that the Magnus force is 10+ times higher than it should be.

All the equations that are valid ( the units are correct ) depend on the Magnus coefficent which is too high. I can reduce the Magnus coefficient so the trajectory looks more realistic but this is "fudging it".

No one seems to know anything for certain.
We are getting no where. I have a high speed camera but converting video to a .csv file is a lot of work.

pnachtwey said:
So what ball is ideal? I would think a table tennis ball is pretty close to ideal. They are relatively smooth and seamless. A table tennis ball wouldn't be so far off from ideal that the Magnus force is 10+ times higher than it should be.

All the equations that are valid ( the units are correct ) depend on the Magnus coefficent which is too high. I can reduce the Magnus coefficient so the trajectory looks more realistic but this is "fudging it".

No one seems to know anything for certain.
We are getting no where. I have a high speed camera but converting video to a .csv file is a lot of work.
I think you are going to have to do that, record the trajectory data and fit the coefficients to that data. Sorry I can't be more help.

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## 1. What is the Magnus effect?

The Magnus effect is the phenomenon where a spinning object experiences a force perpendicular to the direction of its motion, resulting in a curved path.

## 2. How does the Magnus effect relate to gravity?

The Magnus effect is a result of the interaction between the object's spin and the surrounding air, and is independent of gravity. However, the Magnus effect can affect the trajectory of an object in the presence of gravity.

## 3. What factors affect the strength of the Magnus effect?

The strength of the Magnus effect is affected by the velocity of the object, the rate of spin, the density and viscosity of the surrounding air, and the shape and surface characteristics of the object.

## 4. How does the relative strength of the Magnus effect compare to gravity?

The strength of the Magnus effect can vary greatly depending on the factors mentioned above. In some cases, it can be strong enough to overcome the force of gravity, while in others it may have little to no effect on the object's trajectory.

## 5. What are some real-world applications of the Magnus effect?

The Magnus effect has various applications in sports, such as in the spin of a soccer ball or the curve of a baseball pitch. It is also used in the design of aerodynamic vehicles, such as airplanes and race cars, to improve their performance. Additionally, the Magnus effect is utilized in wind turbines to generate electricity.

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