December 6, 2013

Rod Moment of Inertia

Although I am not a physicist or an engineer I like to try to classify rods, particularity tenkara rods. As you are aware, the "standard" way to classify a tenkara rod is by where in the rod it begins to flex, or its point of flexure. This is often given in a ratio such as 5:5, 6:4, 7:3, etc. We've all seen this and we've all had issues with this system.

Other methods of trying to quantify rod action have been proposed. One such method is the Common Cents System (CCS). This uses simple tools, all readily available, that are then used to measure the flex of a rod. Step #1: The butt of the rod lays exactly horizontal and the length of the rod is measured. Step #2: This number is divided by 3. Step #3: A small, very lightweight bag is attached to the end of the rod. To this bag is added US pennies (specifically those minted after 1996) until the tip of the rod reaches the number calculated in step #2. The Common Cents System is easy to do, reproducible to within an acceptable error, and can be done quickly.  Chris Stewart of Tenkara Bum has a nice CCS database of common fixed-line rods already calculated.

One thing the CCS doesn't do is adjust for rod length. If one rod has a CCS of 18 pennies but its overall length is 270 cm, and another rod has a CCS of 18 pennies but its overall length is 400 cm I guarantee that do not feel the same when being cast. The 270 cm rod will feel pretty stiff, while the 400 cm rod will feel softer. So, to get length out of rod comparison, take the CCS (in pennies) and divide it by the length of the rod (in meters). This will give you (what I call) the Rod Flex Index (RFI). The smaller the RFI, the softer the rod. In our example above, the 270 cm rod will have an RFI of 6.67, while the 400 cm rod will have an RFI of 4.5. As you can readily see 6.67 and 4.5 are not the same number. Thus the RFI tells you that the 270 cm rod will be stiffer (faster) than the 400 cm rod (this is irregardless of the rod's 5:5, 6:4. 7:3 rating given by the company/manufacturer). The RFI will give you a better idea of how the rod will feel and how it compares to other rods than the CCS or the 5:5-6:4-7:3 system.

Although both the CCS and RFI give the user some idea of the rod's overall action and comparison to other rods they don't let you know how much force will be required to stop that rod in mid-air and change its direction (as in casting the rod). This is where Moment of Inertia (MOI) comes in. A rod that has a larger MOI will be more fatiguing over hundreds of casts than a rod that has a smaller MOI. The formula for MOI is: MOI=rod mass X radius^2 or I=m*r^2. Here is an easy online calculator to use. I use (g)m^2 as my units, where g=grams and m=meters.


Another way to look at MOI as it pertains to fly rods is by what is called the "swingweight". Many sports use swingweight. These included tennis, baseball (the bat), squash, golf, etc. Basically any sport that uses a swinging tool uses swingweight. In these sports, the swingweight is the MOI around a predetermined point. For tenkara, the swingweight can be easily estimated by the MOI around the rod's center of gravity (or balance point from the rod butt) when the rod is fully extended.

Take for instance my Daiwa LT36S-F. This is a beefy 360 cm tenkara rod that is not as delicate as other rods. It has a mass of 82.4 g. When fully extended, it has a center of gravity (balance point) of 76.5 cm from the rod butt. This gives it a MOI or swingweight of 48.2 (g)m^2.  Now compare the Nissin Zerosum 360 7:3 which has a mass of 67 g and a center of gravity point at 72 cm. This give the Zerosum a MOI of 34.7. Therefore, the Zerosum has much less MOI than the Daiwa and will be less fatiguing over the course of fishing many hours. BTW, this is exactly what my personal experience with these two rods has shown!

Here is a better definition of MOI. This is from the calculator page: "The moment of inertia calculates the rotational inertia of an object rotating around a given axis. It represents how difficult it overcomed to change its angular motion about that axis. Moment of inertia of a same object will change against different axis. The more far away from the axis, the more moment of inertia the object has."

MOI doesn't take into account the overall length of a rod, but other formulas do. Personally, I find the other formulas too complicated. MOI is easy to calculate and rods can be easily compared to each other, particularly if they are approximately the same length (as in the LT36S-F and the Zerosum 360 7:3).

Here are some other rods that compare to those provided above:

Tenkara Times Try 360 6:4...............................MOI 32.9 (g)m^2
Nissin Zerosum 360 7:3.....................................MOI 34.7
Nissin Pro Spec 2-way 7:3.................................MOI 39.1 (at 360 cm)
Tenkara Times NEXT 360 5:5...........................MOI 42.1
Tenkara Rod Co Sawtooth.................................MOI 44.8
Daiwa LT36S-F................................................. Moi 48.2

Here's another 360 cm rod I have on hand, but I suppose it doesn't count when compared to those above -- because I wrapped the butt section with a tennis racket grip!! Its MOI is low however.

Suntech Suikei GM 39 Keiryu Special.................MOI 33.6 (at 360 cm)

So MOI is another rod comparison tool. Now you can get an idea of a rod's flex score (CCS), overall action and action comparison to other rods (RFI), and how fatiguing it will be (MOI) all without ever seeing or casting the rod! It's not perfect but it should help a lot when you want to know whether to spend your hard earned cash on that shiny new tenkara rod!! But remember: If you really want to know how a rod feels, casts, fights a fish you have to feel it, cast it and fight fish with that particular rod YOURSELF.

I'm sure I'll hear from you math brainiacs out there that I'm wrong and not to quit my day job, but what do you think? Would adding the MOI to my RFI rod comparison table be helpful?

Special thanks to David Walker for piquing my interest in MOI.


  1. Tom,
    The Moment of Inertia formula and calculator shown above is for a "point mass".
    The MOI for a "uniform thin rod" is 1/3ML².
    The MOI for a "tapering thin rod" is best found with process outlined in the "Swingweight" article.


    1. Greg,

      You are correct. But not being an engineer, I was just trying to find a simple and approximate way of showing differences between rods. What I didn't say above in my narrative is that for using the formula shown the rod must be assumed to be a point mass and not a tapered flexible rod with mass. It's cheating, I know, but then again its only an estimation and not hyper accurate. Also what I failed to say above was that the number generated is only for comparison to other rods and not calculated to be the the actual MOI for the rod in the true sense of what that means.

      You are correct and illustrate that that it is quite complicated to calculate the actual MOI for a tenkara rod. It can be done but more complicated math and science is needed. I was just going for a simple comparison between rods. Once again I failed!

      Thanks for your input Greg, I appreciate it!

  2. Replies
    1. Me too, Jason! Me too!! I think I'll just stick to fishing!

  3. If this is angling. I'm practicing another outdoor sport. Please, whoever wrote this article, can you give me a break? I've got lost after the first sentence.

    1. Sorry, I'm having some cold weather down time. Can't get out and fish, but as you can tell, I need to!

  4. Tom,

    I have been interested for some time in a way to measure swing weight or moment of inertia. However, for me to follow your formula, you will need to walk me through it (in excruciating detail).

    It has been a long time since I have had a math class or had to use more math than required to make change or calculate a tip. So, starting from the beginning, how did you get from I=m*r^2 to g-m^2?

    Also, what is g? Is it grams or gravity?

    Walk me through a calculation, leaving nothing out.

    1. Chris,

      Measure your rod in grams (g). Extend the rod fully and find the balance point (this acts as our simple center of mass or gravity point). Measure (in centimeters) from the butt of the rod to the balance point. Open the link to the MOI calculator ( Change the units for "Mass of the Object" to g (grams). Input the mass (weigh in grams) of your rod. Change the units for "Distance from the axis" to cm. Input your measured balance point in centimeters. Click "calculate". This will give you a simple MOI for the rod.

      Mind you, this is just a simple, none hyper accurate, way of measuring MOI. I would not build a bridge with this method. But this gives you a simple easy way to compare the MOI between rods. The larger the number, the more force it will take to change the directions of the rod (like casting back and forth). I rod with a smaller MOI will be less fatiguing over time than a rod with a larger MOI.

      Others have suggested just using torque. Which is simply weight (in kg) times distance to the balance point. They both give you numbers you can compare.

      The best way to measure MOI for a rod is to take the rod apart and measure the MOI for each section then add it together, as is suggested in the Sexyloops article. I'm too lazy for that. The other way is to use integral calculus -- like I'm going to do that!

  5. Still, is there a typo when you went from mass TIMES radius squared to grams MINUS meters squared? If not, tell me how to do it in excel rather than on the web formula.

    1. Sorry, it's mass TIMES radius squared. That's not a minus sign. I'll change it to (g)m^2, where g is grams and m is meters.

  6. Here are my figures as measured by me to the nearest cm or g. I've included both the Moment/Torque and the simplified MOI calculations. Sorry for the formatting (or lack thereof.)

    Rod Length Weight CG Moment MOI
    Nissin Air Stage 390 396 42 105 4.41 46.305
    Nissin Air Stage 450 Honryu 450 84 100 8.4 84
    Daiwa Kiyose 43MF 384 86 81 6.966 56.4246
    Daiwa Kiyose 43MF (zoomed) 430 86 111 9.546 105.9606
    Nissin SP 450 441 57 111 6.327 70.2297
    Kitoyaki 27 271 38 64 2.432 15.5648

    1. Hey great!! I don't have some of these rods so its really good to know their numbers!! Wow, the 43MF at 430cm is 105gm^2. That's something! I need to see how that compares to my LT44SF.

      Thanks again!

    2. Happy to contribute.

      Experimenting with the 450 Honryu, if I add 140g, it has the same MOI as the Zerosum 7:3 360. :)

  7. Had a thought... isn't it the distance from hand to CG that is really important and not butt to CG? Some rods are obviously made to be cast while gripping the butt. While others, like the 450 Honryu are fished most comfortably choking up on the handle, a finger on the shaft.

    1. Good thought. That makes it even more complicated! My head is starting to hurt!! ;o)

  8. Denovich,
    Your observations Are valid, but they reflect "user variation". When I was using an Amago, I always gripped at the butt. Many others move to the middle of the grip, or top, for "finger on the rod".

    As it stands, the "axis of rotation" is at the end of the rod. An argument could be made that it should be at a "standardized" grip location, but good luck in setting that standard.

    For comparison purposes, when looking at a table of data, "end of rod" is OK, as long as "end of rod" is always used.


  9. I would love to see a separate chart for MOI of the same rods in your RFI chart.
    Just to see how different the two measures are. I can see ratings based on stars = user preference, the RFI to get a sense for ACTION, and then MOI to compare the perceived weight of a rod or FATIGUE/est. Swing Weight.
    I'd especially like to see the numbers for the ONI and is there truly a similar rod for less that has the same RFI and perceived weight

  10. A thank you to Tom and Chris for applying a little science to rod measurements. The reality is niche products like fixed-line rods are going to bought online the majority of the time and measurable metrics are a great tool for the consumer.

  11. Tom, I think it would be better to promote the measurement of the Moment rather than MOI. Measuring the Moment of the rod takes 5 minutes. It only requires two measurements: rod weight and distance to the balance point, or center of gravity, from the butt end of the rod. Followed by the simple multiplication of 2 numbers. All that is needed is an inexpensive scale, a tape measure, something to balance the rod across and a basic 4-function calculator. I think measuring Moment would be more widely accepted. And the numbers from more rods more easily compared by more people.

    From measuring the Moment of my own rods I know the ones with a Moment < 5 feel very light. And ones with M closer to 9 feel heavier. Looking at the Gamakatsu page I see several Seiryu or Keiryu rods with M values ranging from 8 to 32.

    I fear my posting of the link to the sexyloops MOI article has sent people onto a path into a complex way of measuring rods. That quickly confused or scared people off. But when I did a google search for "Moment of a fly rod". How to measure rod MOI web pages are what turned up. Not web pages about the Moment of rods. The sexyloops page seemed to be the clearest explanation I found. But to measure MOI correctly for a 9 section tenkara rod requires making over 20 measurements of weight and CG of each section plus other measurements of just the base or handle. And the use of complex math. I’m of the opinion that if you don’t calculate the MOI by following the full procedure it’s like saying you tuned up your car after only changing the fuel filter and one spark plug. If some people measure MOI one way and others another way or if some people measure CG from the butt end and others measure CG from a favored hold position then numbers measured will not be consistent. The Moment numbers will be as varied as the ratio numbers: 5:5, 6:4, 7:3, from one company to another.

    Gamakatsu calls it the モーメント指数 (Moment Index) or the formula itself is listed as モーメント指数計算式 ( Moment Index Formula or Moment Index Calculation Formula). M = the weight of the rod (g) × to the center of gravity from the rod butt length (cm) ÷ 1000

    You can find the formula listed that way on this web page,
    Where Gamakatsu calls them self the company of Moment notation, the revolutionary way by numbers to get a sense of the weight of the rod not understood by its weight or length alone. (or something close to that)

    I do not think Gamakatsu developed this idea alone. Though they seem to be its sole promoter. A 2007 document found on Fuji Hiromichi's webpage mentions the Moment of a rod as being important. But does not say how it is measured. Maxrod seems to be involved with designing bamboo rods. I do not find Moment mentioned on their English language webpage. But on their Japanese language page, under the topic Feel of the Cast, they talk about rod Moment, the Moment graph and the dynamic aspects of Moment, etc. It seems the measurement of the Moment of a rod has been used for some time in Japan. Though perhaps not known here.

  12. David W,

    I received your comment but it will not publish for some reason. Yes, moment or torque is likely the better way to go than MOI for rods since true MOI is so complex to calculate. It makes no difference to me, moment or MOI. Both are just numbers that are not overly scientifically accurate (the way we calculate them), but moment (torque) is likely the better of the two. All we need the numbers for is to show comparison between rods and help further classify rods. In my mind absolute values are of relatively little benefit.

    From now on I'll go with moment (torque) instead of MOI. Moment (torque) = wt X r (where r is the distance from the rod butt to the center of mass or balance point).

  13. Excellent decision.
    I will sleep soooooo much better tonight.
    Thank you.


  14. Terrific. In an hour you could put up the Moment values of 12 well know rods and create quite a buzz. I don't think I would go to a fly fishing show without a scale, tape measure and balance t-stand in my backpack. DW