I want to bring up a question that I had asked in the middle of another thread: http://www.sciforums.com/showthread.php?t=96044&highlight=golf&page=3 because it was never answered. Is there an equation relating the moment of inertia of a golf club (MOI of shaft + MOI of clubhead) to the eventual speed of the golf ball? Take 2 clubs identical in every way except differing MOI. To simplfy matters, the experiment is performed in a vacuum so there is no air resistance. Does anyone have an equation?
The speed of the golf ball should not depend on the MOI of the club but only on its speed. Since the golf club is in any case much more massive than the ball, the latter will move away with twice the velocity of the club (in the reference frame of the club, the ball will just be bouncing off it (i.e. reversing its velocity) as if hitting a massive wall). Thomas
Yes but you just approximated away the entire effect of the MOI. For two clubs travelling at the same speed the one with higher MOI will certainly result in a higher velocity for the golf ball. This is why people keep making these drivers with ridiculously large heads. Admittedly it is a diminishing returns type situation, once you get to a certain head mass then you don't gain much by making it bigger, just makes the club more awkward to swing.
Kurros: Moment of inertia is a measure of resistance to torque force, making the following incorrect. Those drivers with very large heads are lighter than their predecessors with smaller heads. Club designers try to reduce Moment of Inertia to make it easier to obtain higher club head velocity. The larger heads allow for a larger sweet spot, making it less harmful to have the ball impact off the exact center of the club face. It is more correct to say that for two clubs traveling at the same speed, the one with the higher MOI requires the golfer to apply more torque force. If ball/club interaction was an inelastic collision with no further torque force at impact & zero time of ball/Club face contact, the speed could be calculated as a simple momentum exchange. Momentum lost by the club head would be equivalent to momentum gained by the ball. The ball/club interaction is not analogous to billiard ball collisions. It is more analogous to bouncing a ball off the floor or off of a wall. The ball/club interaction is an elastic collision & the golfer continues to apply torque force at impact & for the brief time that the ball is in contact with the club face. This creates a situation which is very difficult to analyze using the laws of physics. At impact the ball is deformed considerably, resulting in the storage of energy which is converted into ball speed when the ball regains its original shape. The ball stays in contact with the club face for a measurable amount of time during which the golfer continues to apply torque force. Note that compression is a significant factor in the choice of a golf ball. The male touring pros use higher compression balls than lessor mortals. The higher handicap golfer cannot deform a high compression ball as much as can be done by a touring pro. The duffer should use a mushier ball, allowing him to deform it with his lower club head speed at impact. To start an analysis, one would have to consider the club head velocity which would have occurred in the absence of impact with a golf ball. Then compare with the velocity immediately after the ball loses contact with the club face. The difference would be used to calculate either energy transfer to the golf ball or momentum transfer to the golf ball. I suspect that an energy computation is required rather than a momentum transfer. In either case, Moment of Inertia is not involved in the calculation. MOI is resistance to torque force. The higher the MOI, the more difficult it is for a golfer to obtain a given club head speed at impact & during the time the ball is in contact with the club face.
The large heads are there to prevent the club head from rotating in case the ball hits the club face off center (i.e. if the contact point with the ball is not exactly in line with the center of the mass of the head). Both a higher mass and a center of mass location as far as possible behind the contact surface will reduce the rotation. The point is that a rotating club head during impact would introduce a spin on the ball, and the latter would go off to the right or left (it would obviously lose also some of the velocity if a fraction of the kinetic energy goes into the spin, but this is probably not the main aspect here). In any case, if the contact point is exactly in line with the center of mass, the mass of the club head should be completely irrelevant for the speed of the ball. Only the club speed during the impact matters. Thomas
So are you guys telling me that you expect that a golf ball, hit with a super light club which has the same rigidity as a club with some mass to it, will have just as much energy transferred to it as it would have had you used the latter club? I guess this makes sense if you believe that a) at the moment of impact the energy is in fact transferred from the golfers body to the ball, rather than b) the energy having been accumulated during the course of the swing in the club itself. My argument was that a light club does not carry as much energy as a heavier club (for a given velocity), although it can still transfer energy, leaving scenario (a) as a possibility. I'm no expert golfer but I generally feel that when I swing the club the majority of the energy is coming from the swing, not from my body at the moment of impact. Of course, kinetic energy is mass*velocity^2 so the velocity is the more important factor, but you cannot say that the mass doesn't matter at all unless you agree with scenario (a). I'm open to either situation here, I just always expected it was scenario (b) that was the case.
Kurros: The mass of the club head probably has some effect. I think your intuition is influenced by not taking into account the difference between elastic & inelastic collisions. For inelastic collisions, there is a simple transfer of momentum & the masses of the two objects is highly significant. For elastic collisions, deformation of the objects results in storage of energy much of which can be released when each object returns to its original shape. Imagine various objects a club head held motionless in a vise. A golf ball or a steel ball bearing would would bounce back with a high percentage of the impact velocity. Now consider a ball made of putty or modeling clay. The putty or clay would deform & not regain its original shape. Almost all of the energy would be lost in the form of heat and the work required to deform the putty or clay.
Let me jump in and assume that there is a difference with heavier heads. Why? Isn't there some short of energy transfer between the head and the ball? I assume a heavier mass has more energy to pass on than a lighter head...
Dinosaur: Nobody doubts that there is a difference between elastic and inelastic collisions. Nobody doubts that it would be easier to propel a steel ball than a ball made of putty. All of this is off topic. Let's only talk about golf balls. Having said that, can you explan why you don't think that a clubhead of mass 300g would not propel the ball further than a clubhead of mass 200g both swung at 100 mph?
Ok. let's see: according to http://farside.ph.utexas.edu/teaching/301/lectures/node76.html, the velocity of the ball in a perfectly elastic collision should be (Eq.230; if the ball is initially resting): v2' = 2*(m1+m2)/m1 *v1 This yields for m1=200g , m2=46g , v1= 100 mph v2' = 162 mph and fir m1=300g v2' = 173 mph As I said earlier, in case the mass m2 is infinite, the ball should be twice as fast as the club head, so the maximum would be 200 mph for a 100 mph swing. I think in fact that it is incorrect to use just the mass of the club head here, because the latter is rigidly connected to the shaft, and to a certain degree to the arms and body as well. So I think if you model this through a 2-body collision, the effective mass of m2 should be way higher, and thus the speed much closer to 200 mph. The fact that in practice the ball speeds are even smaller than derived above (see e.g. http://www.instantgolflesson.com/2008/03/23/golf-ball-speed/ and http://www.probablegolfinstruction.com/golf_club_longest_golf_ball.htm ) has to be attributed to the fact that a) one is actually not dealing with a perfectly elastic collision and b) in reality there will usually be a spin of the ball that takes up some of the kinetic energy). In any case, it is obvious that the mass/force with which you hit the ball is only of secondary importance compared to the speed. This holds not only for golf but for most ball sports. You can also see this from the fact that golf-,tennis-, soccer players etc. do not exactly have a body builder's physique but are generally quite slim. Increased body and muscle mass is not only of no advantage here, but of disadvantage as it tends to slow you down. And the speed with which you can move your limbs, club, racket etc. is all important here. Thomas
Obviously, swing speed is a limiting factor as you say. Your ball speed can only ever be nearly twice as fast as your swing speed. However, I believe that in golf, mass is probably just as important for the simple reason that swing speed does not increase linearly with reduced mass. It's a result of the mechanics of our bodies. You can swing a club of negligible weight only slighly faster than an ordinary club. As a result, mass becomes quite important in the ball speed equation (as can be seen in the different ball speeds you calculated). You also correctly mentioned that the shaft and even arms contribute to the mass term. However, these terms get progressively less important the further the mass is from the point of impact (i.e. the arm mass is less important than the shaft mass which is less important than the head mass).
John Connellan:It could very well be that the heavier club head would cause more ball velocity given the same impact speed. I am not certain that weight of the club has any effect if speed at impact is the same, although it might have an effect. Perhaps extra weight would result in the club head not slowing down after impact. The real issue is what parameters are most important, not whether one particular one has an effect. It is my belief that club head speed & deformation/restoration of ball shape are the most important factors, with club head weight being less important. My most modern driver is large & hollow. It is much lighter than the drivers I used years ago. Even a duffer like I am could get almost (if not the same) same speed with a slightly heavier club head. If extra weight would help, I would expect club manufacturers to provide heavier, not lighter drivers in the past 10-20 years. BTW: I think modern hollow drivers have a slight trampoline effect which might be more important than extra weight. As stated in an earlier post, it is extremely difficult to apply the laws of physics to the effects of hitting a golf ball. Rocket science is probably easier. I am almost certain that club & ball manufacturers rely more on experimentation than analysis.
I think it is obvious that the weight counts. Consider: Being hit by a bullet vs. being hit by a canonball (same speed). The bullet might takes you down but the ball will throw your body far away....
Syzygys: Can you indicate how the following applies to hitting a golf ball? Do you happen to know the modulus of elasticity of a golf ball compared to that of a human being? How long does it take a human body to regain its original shape after being hit by a bullet or cannon ball? As indicated by my above commnents, I do not think your analogy has any relevance to golf ball dynamnics.
Try this: Throw a few pieces of sand at a golf ball. It hardly moves. Now throw a pool ball at it, it will knock it a few feet away. The pool ball's mass is probably 10 000 times bigger than a piece of sand, thus the difference...
Syzygys: Your analogies indicate that you have no clue relating to the effects of ball deformation. You are also clueless relating to the effects of the golfer continuing to apply torque force while the ball is in contact with the club face. Your analogies are not applicable to the dynamics of hitting a golf ball. As I mentioned in previous posts, for the same club head speed at impact, club head mass might have some effect. It is not a major effect. As also mentioned in a previous post, applying the laws of physics to hitting golf balls is a formidable task & I do not think an accurate analysis has ever been done. Club manufacturers have done all sorts of experiments in order to provide the best performance possible without violating USGA rules. Modern clubs are noticeably lighter than they were 10-20 years ago. This surely indicates that club head mass is not an important factor. Since neither of us is capable of applying the laws of physics to the problem, my opinion will be influenced by the equipment provided by club manufacturers. I wonder if any manufacturer has ever performed experiments to determine the effect of club head mass for a fixed speed at impact. Unlike a human, a machine could easily obtain the same club head speed at impact with either a very heavy or a very light club. Since you believe that club head mass is significant, I suppose you have thought about it. What insights do you have relating to the effect? Given the same speed at impact, would a 25% increase in club head mass result in less than a 25% increase, a 25% increase, or more than a 25% increase in ball speed? Would the percentage effect of club head mass be independent of club face speed at impact?
It is nice you asked my opinion, although you think I am clueless. :0 The connection is most likely not linear. Once the head reaches a certain weight it is getting harder and harder to swing it, thus there is no point in unlimited increase for the head mass. I assume the headmakers and the association both want the same, the possible farest distance. Since we are approaching it, I guess the current weight is pretty close to the ideal. Now it is just down to the person doing the swinging and to its style and power. The head size is limited to 460 cc. Why exactly that? I am just assuming testing came up with this as the most ideal, and increase beyond that wasn't giving better distance....
Actually even if you had a robot which could swing at the exact same speed for any head weight, the returns you get from increasing the weight get very small (approaching zero) when clubhead is much heavier than golf ball. I believe this is how the equation behaves. So when you also factor in the fact that humans can't swing heavy clubs very fast, you can see why we generally use very light titanium clubheads.
Syzygys:USGA rules relating to equipment are intended to prevent extreme increases in performance due to advanced technology. I think that drivers with an extreme trampoline effect could add a lot of yardage. I think there are golf ball designs which could increase distance quite a bit.
