# Simple cheap method of measuring mass distribution of object

• B
• davekm
In summary, the conversation discusses methods for measuring the weight and mass distribution of objects, specifically a tennis racket and a human arm. The participants mention using balance scales, pivot points, and moments to calculate the weight and pivot point of an object. They also discuss the challenges of measuring the mass distribution of a human arm due to its attachment to the body and suggest using MRI scans or fixed positions on a table as alternative methods. The conversation also touches on the importance of understanding weight and mass distribution in sports, specifically in relation to a tennis stroke.

#### davekm

I'm looking to measure the weight of an tennis racket at different lengths. Obviously If I stick it on the scale it will give roughly the sam,e weight no matter if I put the whole racket on, or a third of the racket. Would there be a way I could measure the weight, of say the first third of an object, without chopping it up into thirds. I'm also looking to measure portions of my own forearm and see the mass distribution at different points of the forearm, like at the wrist etc

davekm said:
I'm looking to measure the weight of an tennis racket at different lengths. Obviously If I stick it on the scale it will give roughly the sam,e weight no matter if I put the whole racket on, or a third of the racket. Would there be a way I could measure the weight, of say the first third of an object, without chopping it up into thirds. I'm also looking to measure portions of my own forearm and see the mass distribution at different points of the forearm, like at the wrist etc
do you understand the concepts of balance scale and pivot points? You could do what you want with a solid rod very easily with those concepts. Given the irregular mass distribution of a tennis racket, it would definitely be harder but you could likely get a good approximation. As for your arm, that technique would require that you detach it from your body, so you probably will want to find some other way for that.

davekm
I'm an computing major who's trying to take an interest and learn all about physics as a hobby. So I don't know much yet. I'm jumping the gun a bit doing this experiment, but it's very doable forva physics novice. To be honest, the more important one was knowing the mass distrubution of an arm, specifically the forearm. I don't know much about pivot points no. I have the equipment to measure the centre of balance for a tennis racket

I'm looking to create a double pendulum like the one in this link- LINK Though I'd like the plank of wood that represents the forearm to have mass distribution similar to a human forearm, by attaching weights to it.

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I think I would use Moments. With a number of different additional masses, placed at different points on the racket, you could easily find the pivot point for equilibrium in each case. Divide the racket up into lengths and assume their individual CMs are at each mid point, you could then write a set of simultaneous equations with m1 to mn, for the masses of the n sections and the positions and values of the added masses at balance. n unknowns and n equations; Bob's your Uncle.

davekm
Thanks all. I don't actually need the mass distribution of an object, it turns out. I just need to find out the mass distribution of a human arm.. Then I'll just place weights on the plank of wood that represents the forearm pendulum. Would there be a way of doing this? I've read that putting your forearm on a scale does actually give approximate weight measurement for the forearm. I had thought if you put your forearm onto a scale, you would get a measurement for more then just your forearm.

I have found a few websites with info on the average forearm mass. It seems the average mass of forearms is between 1.7-1.8% percentage of total body mass, with the centre of mass being at 43% of the length of the arm. This isn't really much help in finding the mass distribution though.

To work out the mass distribution of a piece of human anatomy is pretty problematic as it's mechanically 'joined on' to the rest of you. If you could arrange for a muscle relaxant, you could eliminate the effect of the arm's servo mechanisms and the elbow hinge is always attached and would upset the experiment I suggested. An MRI scan would show the distribution of the different tissues and the knowledge of tissue density (plenty of info available about that) would give a good idea of the masses of the different parts. Expensive if you wanted data about a specific arm but there are many MRI images available, that you could look at. I googled MRI scan arm image and found a lot of possible stuff. You could divide the image into slices and assign each slice a mass of estimated volume times tissue density. The finer you chop up the image, the more accurate would be your estimate.

davekm
Alternatively, and accepting the limitations of being 'joined on, you could keep the elbow in a fixed position on a table and measure the share of the weight force on a kitchen scales at various distances from the elbow. Again, dividing the arm into n sections and using n positions for the scale would give you the n Moments equations that' you'd need to find the masses of all the sections. You would need a very relaxed subject to do the measurements on (or lots of repeat measurements).

davekm
Okay, I've just tried measuring my arm on my kitchen scales, which are calibrated to measure within 0.1g, whilst keeping my arm deadly still. The readings were jumping all over the place, so I'll have to give up on that method. I feel I should explain the experiment I'm doing and why I need to know the mass distribution. I'm down on sleep but hopefully I explain this clearly.

A guy on a tennis forum I post on created a formula called MgR/I, which is inspired by past research on a tennis stroke being viewed as a double pendulum. It gives a measure of the racquet's natural swing frequency as it pivots about the wrist axis on a forehand. If either pendulum lags behind the other, then the racket will not strike the ball at a straight angle, leading to timing errors on shots. If MgR/I is perfectly tuned neither pendulum will lage behind the other. It's a fascinating concept that could be relevant to other sports too. I'll provide the link that explains the formula in more detail - http://tt.tennis-warehouse.com/index.php?threads/optimum-racquet-balance-for-performance-ii-mgr-i-data-for-atp-pros.387805/#post-5817042

The OP has established that his own optimal Mgr/I is 21 but it varies from person to person, based on their arm length, racket swingweight, forehand grip and whether they wear a wristband. So at present, only the OP and a few others have fully benefited, which is a real shame. I've tried to establish the influence of these four factors but I haven't been successful in finding my personal MgR/I tuning against a wall, or hitting against a ball machine. I was thinking I would experiment to find out these factors by creating this double pendulum model, with the forearm pendulum being weighted close to a human arm.

Hopefully this clears up my goals here. I'll add more later. Thanks.

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davekm said:
The readings were jumping all over the place,
You must have been too excited! haha
Reading a digital display that's jumping about can be difficult. A simple analogue / pointer type would probably be easier to read an average off. Alternatively, a datalogger (school / college type) could do the job for you by plotting a series of samples and giving you the mean.
The 'biological' approach of looking at the existing MRI scans or even measuring the thickness of you arm at different points could give you a clue about muscle distribution. You could estimate the cross section of the bones by 'feel' and then scale up or down, using MRI images.
I have just read that link and I can't find any mention of arm measurement apart from length (??), which would be very easy.
Interestingly, the optimum MgR/I values seem to be different for different player standards. Which group are you?
I remember, tens of years ago, reading a New Scientist article about the choices of baseball bat weights. I think it all depended on strength. You go for the heaviest one you can handle easily because you need to be able to get bat (/ racket) speed as high as possible and, of course, a heavy bat at the same speed as a lighter one will transfer more momentum to the ball.
It's harder in tennis. of course. because of the range of different shots the racket needs to cope with.
With the kitchen scales, you should be able to measure the MI of the racket by the measurement system I suggested at the top. (There are many variations on the same method)

Thanks Sophie. Haha. I'll try again. I have no hand tremors and my scale doesn't usually jump around at all. Otherwise, I'll try the MRI images off google. I did a bit more research and found that a number of studies have shown that the average forearm mass is 1.85% of total body mass. Plus the centre of mass is, on average 44% up from the proximal end. Which may help me in estimating the mass distribution.

The MgR/I is influenced by many things but the main difference would be arm length, with the forearm pendulum being naturally slower for people with longer arms. The only way to calculate the influence of peoples arm lengths is to create a double pendulum model, and keep adjusting the length of the forearm pendulum. It would be great to tune it for myself but I'm more looking at creating a personal MgR/I calculator that many can use.

I've also learn't that you can calculate the moment of inertia of the forearm pendulum, if you know the length and mass of the arm. Here's the link for that - LINK . I don't know if you can then use this to find out how different arm lengths affect MgR/I. I understand the concept of a double pendulum and how it affects a tennis groundstroke, but once we start going into the formula, sadly I'm over my head. I've only just taken an interest in learning physics as a hobby. I flunked it at school. I've also found a calculator for working out the forearm moment of inertia - http://www.clbme.bas.bg/projects/motco/data/massinertial.html [Broken] Though, it looks off to me as the arm lengths in section 1 are too short.

Actually, that link also seems to throw in a few doubts, like the real length of the arm pendulum is somewhere between the elbow & shoulder. This matches a line from one of the big tennis physicists who first investigated double pendulums, that "candidates are the wrist, the elbow, the shoulder, or the center of the body. In reality, different parts of the swing are dominated by different axes, and the net result is an axis that lies somewhere in-between them all." Again, apologies if this is all a bit muddled. This is all an ongoing learning process for me. This MgR/I formula has been about for 5 years, yet it's still not clear and the OP is pretty non compliant when you try to quiz him.

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So, I've managed to master the scales and I can now measure the mass of my arm and the mass distribution of my arm. I'm learning a lot along the way here. I've achieved my sub goal of finding mass distribution of the arm but I still need to find the link between a persons arm length and their personal optimal MgR/I. I'm still not sure if it would be necessary or worthwhile to build my own double pendulum, or if I can work this out the cheap way, via a formula.

I should have added the formula for MgR/I is (M/1000)*g*R)/Io
M = Mass. G=g is 980.5cm/s^2 (acceleration of gravity).
R= balance point
I = moment of inertia Io about the wrist axis Io = Swingweight + 20MR - 100M

Now that I know my arm length and mass, I can work out the MOI of my forearm. To calculate the moment of Inertia of the forehand rotating around the elbow, treating it as a uniform bar, I used the formula - Ia = 1/12 * Ma * La^2, where Ma=mass of arm and La=length of the arm . So my MOI is 1/12 * 1.770 * 26.5^2 = 103.581875 . Does anyone know how to do this calculation, including the varying mass distributions of my arm, to get a more accurate MOI value?

Assuming that the balance point of my forearm is based on the average centre of mass of 45.74% up from the proximal end, it would be 12.1211cm. From this I can work out that the MgR/I of my forearm pendulum is (1770/1000g)*980.5*12.1211/103.581875 = 204.59. MgR/I of tennis rackets tend to fall in between 19.5 and 21.5. I think if the two match, then that would create a perfectly timed tennis groundstroke. I'm not sure why my forearm MgR/I is so much higher then rackets. I'll add my thoughts on this later or tomorrow.

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@davekm: As far as I can see, that article is not adequate for you to get a proper result. It is useful in as far as it produces a parameter for a tennis racket that can be matched to a particular player but I could't see where the players dimensions could actually be used to choose an optimum.
I reckon the best way of getting the best racket for you is to have a pile of samples (without knowing their 'number') and choose the best ones for you. If the article is valid, all the ones you have chosen should have similar Numbers. You would need to borrow a lot, I guess. I expect the other factors like the stringing of the rackets could make a big difference, too.
This is the sort of work that's best suited to a manufacturer with a source of many different rackets and a budget for many weeks of research time.

Thanks again. I'm not sure which article you mean, I've linked quite a few. I've done demoing rackets to death and I ended up picking a decent racket that I play well with. I'm not overly fussed about improving my own tennis, I'm more interested in the MgR/I formula, as it could be pretty revolutionary for the game of tennis.

I believe all you would need is a players arm dimensions, then you could give them an optimal MgR/I that they could alter their racket specs towards, till it meets the optimal value. It's also not easy to tell which racket performs better, as if you play at a decent level, you can play well with any racket. Plus, some rackets to do some things well and some perform better in other areas. This is where the game of tennis can drive you mad :)

Would you weighing in your expertise on a few more questions please? I'll put them on here tomorrow or tonight. Then I'll take my findings to the creator of the formula, who is usually extremely hard to get hold of.

This will be another long post. Thanks for sifting through these scattered thoughts and replying so quickly. This should be the last post. Then I will take it all to the formula creator. :)

1) My first question is, what's the formula to work out the balance point of human arm, if I know the mass distribution in portions of 1/5? I could measure my arm in even shorter segments, if it helps improve accuracy.

2) To calculate the moment of Inertia of the forehand rotating around the elbow, treating it as a uniform bar, I used the formula - Ia = 1/12 * Ma * La^2, where Ma=mass of arm and La=length of the arm . So my MOI is 1/12 * 1.770 * 26.5^2 = 103.581875 . Does anyone know how to do this calculation, including the varying mass distributions of my arm, to get a more accurate MOI value?

3) Does anyone know of a simulator, that allows you to create a double pendulum and input the mass, balance point and swingweight of an object. So I can test this all visually, which will perhaps be easiest.

Here is the main article on it from the creator, by the way. - http://tt.tennis-warehouse.com/index.php?threads/optimum-racquet-balance-for-performance-ii-mgr-i-data-for-atp-pros.387805/I wasn't able to tune my MgR/I using the steps he provided, which is I why I'm looking to do it via a formula or building a double pendulum rig.

I'll put a another qoute from travler that I've dig up, that explains MgR/I further -
"The arm+hand+handle constitutes the first pendulum, and the hand+racquet is the second pendulum. For maximum control on a groundstroke, the racquet face should remain at a constant angle through the hitting zone. For this to occur, the forward component of the velocity vector of the racquethead must have equal magnitude to the forward component of the velocity vector of the hand. In other words, if the racquethead moves forward faster (or slower) than the hand during the moment of impact, then small errors in timing will result in changes in racquetface angle, leading to less accuracy of the shot. But if the racquethead moves at the same speed as the hand, then small errors in timing are inconsequential and the ball will still go toward the target."

3) Can You mathematically work out the MOI of a double pendulum, and how the two interact, to get the two to have the same MOI or MgR/I? In other words, is there a formula where I can see how the forearm MOI if I add a racket with known mass, balance point and inertia to the end of the forearm pendulum at the wrist? Which should slow down and thus lower the MOI of the forearm pendulum. Then I can see how the MOI of the racket pendulum is lowered by the forearm pendulum, then keep changing the the weighting of the racket till the two match.

davekm said:
Can You mathematically work out the MOI of a double pendulum,
MI is a characteristic of a rigid body so you are dealing with two separate MIs and the way they interact - just like two masses colliding in the translational world.
Your problem is a hard one because there is muscle force being applied during the time of impact and it's not just a 'collision' process. I guess most of the muscle effort during contact is used to keep the racket head facing in the right direction (as you mention).
I am way out of my pay grade here, I'm afraid. I would need to resort to making a simplified model and trying different combinations of dimensions in a bench-top experiment. Or you could buy a very light racket and try putting masses on it to turn it into a heavier one. You could choose the best one for you, on court. But what would you do in the shop when trying to find a good match? You'd need to take some scales and all the measuring gear in with you. It is possible that the shop would actually approve highly of that, in fact.

Thanks. I'm curious, what was the bench top experiment you we're thinking of? Yes, it would be best to consult the MgR/I creator, who will know more about the axis of the pendulums and the formula. Tennis websites list the specs of rackets and most rackets include them on throat area. I like to buy cheap, light rackets then pile lead tape on them anyway. Would you mind taking a look at some of my other questions pleaase? It will help me to approach the creator with as much info as possible. I'll repost them.

1) My first question is, what's the formula to work out the balance point of human arm, if I know the mass distribution in portions of 1/5? I could measure my arm in even shorter segments, if it helps improve accuracy.

2) Does anyone know the calculation for MI, including the varying mass distributions of my arm, to get a more accurate OI value? Without including the balance point, or mass distribution, the formula I used for the forearm, as a uniform bar, rotating around the elbow would be sketch, I believe.

3) Does anyone know of a simulator, that allows you to create a double pendulum and input the mass, balance point and swingweight of an object. So I can test this all visually, which will perhaps be easiest.

4) Is there a formula where I can see the MI of the forearm+Hand+Racket . I know how to calculate the MI of the forearm + Hand. The sticking point is, I don't have the racket mass distribution but I do have the balnce point and exact MI.

I was thinking of using two linked pendulums (the model using the arm info you have or could gain) and a projectile hitting the end of one (working backwards, which is fair enough) the 'muscle' at the other end could be a spring or an energy absorbing piston.
I don't know any formula but it would need some homework to find how to derive one. Like I said, I don't know anything like that off hand.

Thanks. I'll put something on this thread if I hear back from the MgR/I creator about this, I don't know if your pendulum model would be similar to the one in this link. LINK Appreciate that your interested in experimenting to try and help. I'd wait until I've heard back from the creator. I'm going to ask him about creating a pendulum model.

Just revisiting this. I'm a bit further on with the simulation and have rough estimations of the balance point and mass distribution of my arm.

I asked about methods of measuring the mass and moment of inertia of my forearm elsewhere. I received a detailed answer, but I couldn't grasp some of the instructions and equations.

Here is the question - http://physics.stackexchange.com/qu...the-mass-and-balance-point-of-a-human-forearm

I'm just having issues with steps 1 and 3, if anyone can help. This is beyond my basic comprehension. .

For step 1, I couldn't find anything about lying on the plank, so I wasn't sure of the equation for the measurements I took. I lied on a plank, balanced by a scale and books on either end, and measured my forearm and hand when vertical. The readings on the digital scale jumped a lot, so I took an average from ten measurements. The weight rose by 0.05kg for the hand, and 0.52kg for the forearm. I know that the length of my forearm is 26.3cm and the centre of mass is 10.2 cm from the elbow.

For step 3, I think I understand the formula for step 3. It's just Mr2. With R being the rotation of axis/gyration. Do I have to subtract the MOI of the arm from the MOI of the entire body?

I'm also wondering how to estimate the Moment of inertia about the centre of mass, using the measurements I've taken for the mass distribution in 1cm segments. Is it (M1L12+M2L22...)(MtotalH2) Sorry for the layout. Having issues with Latex at the minute. M being mass, L=Length and H= Balance point.

Thanks

I think I see where that link is going and there are a lot of possible methods for estimating the mass and CM position of a limb. Measuring the difference in torque (tilted table and 'steelyard' balance (look it up) arrangement can tell you the position of the arm's CM and its mass by having the arm horizontal and vertical (also doing the same with the elbow). You could do a similar thing for MOI by measuring the angular acceleration with the arm against the body and with it outstretched. A bit fiddly to do because you'd need a good pivoted table and an accurate position sensor to measure the motion under a know torque. But it's all there in principle (a lovely cop-out statement I know).
It's useful to you that MOIs add together and you can also use the parallel axis theorem to refer MOI to an axis other than through the CM. You need to get familiar with this stuff before you move on to measuring the Body, though. I have no suggestions for a suitable textbook (last time I read about it was in the 60s) but some intensive searching will surely get you somewhere.
There is a problem here because MOI is far more commonly used in the context of structural engineering, where the same formulae for MOI will also tell you about the strength of beams etc. - nothing to do with spinning anything. That could be confusing for someone who hasn't done Mechanical Engineering.

I've been too busy to do this, but I thought I'd revisit it over the next few days and hopefully finally get a decent estimate.

@sophiecentaur I have read a good few articles on tilted tables and steelyard balances. I have a better grasp of it now and how you can use torque to measure mass and COM. A mechanical scale is less likely to suffer the problem of readings jumping all over the place, like my digital scale was. Though I still feel, in hindsight, it was ridiculous I took this on as a computing student, with plenty of curiosity but minimal relevant knowledge. I am willing to swallow my pride and scrap trying to measure my forearm. Before I do that, do you know what would be the cheapest item I could use as a pivot. I have a plank, scale and all other equipment but my budget is thin. Thanks again. for your help so far.

If the density of your arm is fairly constant, you can measure the shape of your arm and use that to estimate the mass distribution. My suggestion is to get a big tub of water and dip your arm into it down to your wrist, then to your elbow, then to your shoulder, etc. You measure the displacement volume at various arm depths, and you can get a volume of each part of your arm. Then you can equate the volume with the mass. You can even submerge your arm very gradually, taking measurements every small step. (Obviously, you have the problem that bone isn't the same density as muscle, but I don't think they are THAT different.)

Since you want to know the MOI of your arm (or wohever's), why try to calculate it from things which are difficult to measure? Why not simply measure it directly. If you release your arm from a non-vertical position and let it swing freely, it behaves, as you say, like a pendulum, albeit heavily damped.
At a first approx you'd get an estimate from the period of oscillation, but if you could digitise the angular displacement vs time, I'd have thought the DSP software around these days could use this step response to give as accurate an estimate of MoI and damping factor as you can get.

Thanks. The MOI is more of a secondary goal. The mass and COM is the main aim. I have measurements for the volume of my arm, in 1cm segments, measured using the Archimedes method. Though, according to this link, there is quite a big difference between the density of bone and muscle, so I think I will need to try the complicated method. I will try and find a cheap pivot and a mechanical scale, then try the torque method.

davekm said:
Thanks. The MOI is more of a secondary goal. The mass and COM is the main aim. I have measurements for the volume of my arm, in 1cm segments, measured using the Archimedes method. Though, according to this link, there is quite a big difference between the density of bone and muscle, so I think I will need to try the complicated method. I will try and find a cheap pivot and a mechanical scale, then try the torque method.
Are you allowed to get a set of x-rays.? If so, then you have the computer construct a 3d computer aided design model of your arm bones. Then you take a cast of your arm shape (dipping into plaster), and use computer measurement tools to construct a 3d computer aided design model of your external arm shape. Then you find out the density of human flesh and human bone (typical). Then you integrate using computer aided design software to get any distribution information you desire, including center of mass and moment of inertial.

davekm
Chestermiller said:
Are you allowed to get a set of x-rays.? If so, then you have the computer construct a 3d computer aided design model of your arm bones. Then you take a cast of your arm shape (dipping into plaster), and use computer measurement tools to construct a 3d computer aided design model of your external arm shape. Then you find out the density of human flesh and human bone (typical). Then you integrate using computer aided design software to get any distribution information you desire, including center of mass and moment of inertial.

That would be great! But its way outside of my budget.

A bit late, but I just revisited this. I's been on my to do list for months. I tried the plank method from this link, using a mechanical scale. I used an old detached wooden door for the plank. Same issue as the digital scale, the scale measurement jumped all over the place, despite my attempt to keep deadly still. I researched the steelyard balance method mentioned by sophiecentaur. I couldn't understand it, or come up with a method using a steelyard balance. I don't have access to a tilted table, and can't find anywhere I can buy one. I have the measurements from the archimedes method, for the volume of my forearm, in 5mm segments. Just not the mass of forearm, which is obviously the crucial component. I could then estimate the mass distribution, using historical data on the density of forearm muscle/bone mass. I think this is one I will have to give up on, unless anyone has any suggestions.