# Springboard physics questions for diving coach at YMCA

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• HighFlyer
In summary, the diving coach at the local YMCA wants to give a lesson on diving off the board with maximum efficiency. The diving board with an adjustable fulcrum is used, and according to the diving coach's analysis, the torque on the fulcrum is maximized when the fulcrum is positioned in the middle of the board. Practical experience tells the diving coach that the position of the fulcrum changes the stiffness of the board, and that Hooke's law can be applied to model this. The stiffness of the board also affects the force necessary to displace the free end of the diving board, and the amount of energy stored in the board is determined by the equation Euler-
HighFlyer
I'm a diving coach at the local YMCA and I want to give a lesson regarding the physics of diving off the board with maximum efficiency. This is the type of diving board with an adjustable fulcrum, basically a lever with one end fixed with bolts. Its been almost 15 years since I've taken a physics course. I've been able to refresh my knowledge of the torque at various point of the board (at the fixed end, at the free end, at the fulcrum) and according to my analysis, when the fulcrum is positioned at the midpoint of the diving board the torque on the fulcrum is maximized.

Practical experience tells me that the position of the fulcrum changes the stiffness of the board, i.e. how much force the board pushes against the weight + downward force of a jumping diver to return the board to equilibrium position. My instinct tells me Hooke's law can be applied to model this, but I'm not too sure how to apply it because as the position of the fulcrum is changed, the "spring constant" is not constant but variable. The stiffness of the board also affects the force necessary to displace the free end of the diving board, i.e. the diver needs more force with a stiffer board (fulcrum closer to free end) to achieve a displacement of equal value vs the force applied to a less stiff board (fulcrum closer to fixed end.

final four entries under divers weight are the additional force from falling from ~2ft, trying to take into account a jumping diver

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• springboard physics.xlsx
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The theory you need to look at is the Euler-Bernoulli beam theory.

I found your case already solved in this figure (although the end is not bolted, but free to pivot):

Your diver exerting a force ##P## at the end of the board will create a deflection ##\Delta_{max}## on the board. Therefore, the equivalent stiffness ##k## for the board is:
$$k = \frac{P}{\Delta_{max}} = \frac{3EI}{a^2 (l+a)}$$

##E## is the Young's modulus of the material for the board;
##I## is the second moment of area of the board, which represents the geometry of the cross-section of the board.

If you go to the extreme case of a cantilever beam, the equation is:

$$k = \frac{F}{\Delta_{max}} = \frac{3EI}{L^3}$$
The amount of energy ##E## stored in the board - which will be returned to push the diver back up - is:
$$E= \frac{1}{2}kx^2= \frac{1}{2}k\left(\frac{F}{k}\right)^2= \frac{F^2}{2k}$$
So for the cantilever beam:
$$E = \frac{F^2}{6EI}L^3$$
And for the beam with a fulcrum:
$$E = \frac{F^2}{6EI}a^2 (l+a)$$
So a cantilever beam will offer the maximum returned energy for a given force ##F## applied to it (assuming ##l+a = L##). That is because it offers the maximum deflection possible. For the first case, the further the fulcrum is from the force (##l \rightarrow 0##), the more deflection you get (i.e. the more your board looks like a full cantilever beam).

erobz and HighFlyer
I agree with the beam calculations. I did it the long way by integrating ##\int \frac{M^2}{2EI}dx## same results for this single applied load. (however, ignoring shear strain energy for this wide/shallow beam could be an error), but I'm not sure of its "overall factor" in setting up the dive?

Watching this video,

The first one she jumps and on the landing seems to be converting horizontal translational energy into a change in vertical potential, then a sequence of two jumps that seem to be focusing adding energy vertically with the legs on each jump.

So what is the mechanism that aids the increase in potential. Is a stiff spring less efficient in returning the energy - Obviously hoping on the floor ( a very stiff spring) like that isn't going to get you anywhere. Is it the persons physiology that is less efficient at generating the energy input on a stiff spring?

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erobz said:
So what is the mechanism that aids the increase in potential. Is a stiff spring less efficient in returning the energy
You pump energy into the system by pushing the spring farther from its equilibrium position (of course). For a fixed amount of available displacement (how far can you extend your legs) you provide more energy by pushing against a stiffer spring. However, there is a limit to how hard you can push.

There is also a trade-off. The amount of force that the legs can deliver depends on how close they are to being fully extended. With the knees flexed greatly, the available force is severely curtailed. With the knees only slightly bent, one can provide a much larger force. The ankles can do quite well also, albeit over a smaller range of extension.

With a strong spring, you get too little displacement to do any good.
With a weak spring, you get too little force to do much good.

Ideally, one would also like to tune things so that the cycle speed is low enough that you are not fighting muscle ramp-up time. A strong spring will leave you with little time to tense the muscles. But there are not enough parameters to adjust. You cannot optimize for everything.

The mechanism... You are increasing spring energy from ##\frac{1}{2}kx^2## to ##\frac{1}{2}k(x + \Delta x)^2##. Spring stiffness is in the ##k##. Leg extension is in the ##\Delta x##. Peak leg force will be given by ##k(x + \Delta x)##.

Trampolines, half-pipes and swings being pumped from a standing posture are situations very similar to diving springboards. Swings max out at 3 g's when you get up to a 180 degree arc. That is easily do-able. I have no personal experience with half-pipes. Nor have I ever had the pleasure of doing a swing with rigid supports replacing the chains -- that would have been fun.

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HighFlyer, hutchphd and erobz
jbriggs444 said:
With a strong spring, you get too little displacement to do any good.
With a weak spring, you get too little force to do much good.
I figured time had to be a factor. With a stiff spring our physiology limits due to the short timescales by which the energy is returned (not allowing us time to maximize our jump force), and with a soft spring the diving boards elastic response limits because we would tend to separate from the before the all the stored energy is returned when optimizing the jump.

Ideally you want the optimum jump mechanics to be timed with the "full" return of energy from the fall? Is that approximately, correct?

erobz said:
Ideally you want the optimum jump mechanics to be timed with the "full" return of energy from the fall? Is that approximately, correct?
You need to push off at the very bottom of the stroke -- at the moment when the board is flexed most. I suppose that this is what you mean by "full return of energy", with all of the kinetic energy from the fall having been stored as potential energy in the springboard.

This same rule of thumb applies to pumping swings, jumping on trampolines and working the half-pipe. You want to go into the bottom of the stroke crouched. Then at the very bottom, you extend and hold.

The heuristic is that you push when you will feel the greatest resistance to your push. Which makes sense because that is when your push will be doing the most work.

Edit: To be perfectly clear, it is not really the moment of maximum potential energy that is relevant for optimization purposes. Rather it is the moment when the gradient of potential energy with respect to position is maximized. For springs that roughly obey Hooke's law, that position is also the position where potential energy is maximized. What I had called a "heuristic" is actually the proper metric.

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HighFlyer and erobz
jbriggs444 said:
To be perfectly clear, it is not really the moment of maximum potential energy that is relevant for optimization purposes. Rather it is the moment when the gradient of potential energy with respect to position is maximized. For springs that roughly obey Hooke's law, that position is also the position where potential energy is maximized. What I had called a "heuristic" is actually the proper metric.
One should also mention that biomechanics and physiology complicate all this. For example, you might not have enough leg strength to extend them at the bottom of the half-pipe, or at least not in a manner were the muscles work efficiently, as they are not ideal force generators. And you obviously cannot do it instantaneously at the most optimal point.

jbriggs444 and erobz
A.T. said:
One should also mention that biomechanics and physiology complicate all this.
Also the diving board is tapered if I recall correctly, so it is not a simple beam..

A.T. said:
And you obviously cannot do it instantaneously at the most optimal point.
This might also be one reason why many half-pipes aren't geometrically half-pipes anymore, but have a flat section in the middle.

hutchphd said:
Also the diving board is tapered if I recall correctly, so it is not a simple beam..
The bending stress is greatest at the anchored end and least at the diver's end. It would be a waste to put excess material on the diver's end.

Consider that from the point of view of a shear plane at the anchor point (over the adjustment roller) you have 12 feet of board (for instance) being supported. But from the point of view of a shear plane halfway out, you only have 6 feet of board being supported. Half the torque through the shear plane.

For the same reason, if you use a lever that is too weak for the load, it will tend to break at the fulcrum. [For class 1 levers -- fulcrum in the middle]

hutchphd
hutchphd said:
Also the diving board is tapered if I recall correctly, so it is not a simple beam..
If you can write ##I(x)##, then that can be handled (in theory). To me the dynamics/biomechanics seem much more complicated.

thanks to everyone who has responded, its been very helpful

berkeman and erobz

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