Golf swing and angular momentum

2016 April 19
by Daniel Lakeland

Thanks to Richard Kennaway for getting me thinking a little more carefully about the angular momentum issues in the golf swing. Here's another fantastic slow motion shot of Tiger Woods hitting a driver. It's slow enough that we can mention time points and see what's going on. I'm going to try to analyze how energy is pumped into the club, and how it's transferred to the ball.

Let's ignore the backswing, because although it's important to develop a good feel and rhythm, in other words, to help you control, it's not involved in adding power to the club since the club is stationary at the top. So starting at 0:33 the downswing begins. The wrists are cocked so that the club is at an approximately 90 degree angle to the left arm. By 0:39 we can see the hands are at the belt line, and the wrists are just beginning to uncock and the angle of the club to the left arm is starting to increase. Between these two time points he's put a lot of torque into turning his upper body, arms and torso but it seems mostly arms. A lot of energy is now stored in the motion of the club and the motion of his upper body and arms, like the energy stored in a gyroscope.

At 0:41 Tiger's whole upper body has nearly zero angular velocity relative to the large angular velocity at say 0:39. His hands become nearly stationary at a point near his belt buckle. Well, perhaps that's an exaggeration, but I'd estimate maybe half the peak linear velocity of the hands. I'd love to have access to one of those little video drawing and measuring tools. But, the club continues to swing thanks to the unhinging of the wrists.

During the period from 0:40 to 0:42 the club is unhinging. Let's approximate the linear velocity of the hands when they reach the belt buckle area as zero. The club head is still going quite fast, so we have as an approximation, the club purely rotating around a point where his left thumb is, and his upper body is stationary. This is a pretty rough approximation, clearly he is still turning, but not nearly as fast as at the middle of the downswing. This is the period of time when energy is being transferred from the upper body into the club. The club actually bends forward.

In fact, I think he is feeling a force couple applying torque to the club. His right hand will feel like it's "pushing forward" in the direction of the swing, and the left hand is pulling backward. Those forces are responsible for pumping energy into the club. The key in the swing is to do this late enough that the club doesn't "get ahead" of your arms. At impact the club should still form an angle of less than or just about equal to 180 degrees to the wrist. At this point, your left arm is pulling upward, and you'll see Tiger's shoulder rising post impact.

EDIT: Something is definitely going on at that point in the video, the hands do seem to be slowing, but I don't think that the above description is accurate having looked and thought more carefully about it.

Now, here's a simple mechanical analogy experiment you can try at home. Take a normal desk ruler, preferably a flat metal one. Hold it flat between your right thumb and forefinger on one end of the ruler. Cock the ruler so it's trailing at 90 degrees to your arm. Move your right arm from level out at your right side around in an arc towards your front, holding tight until you get about halfway through this right arc, and then partially release the pressure on your thumb so that the ruler is free to rotate. Decelerate your arm so that it's stationary when it gets in front of you. You will see the ruler rapidly begin turning around your thumb, and whack you in the forearm... watch out that the ruler doesn't fly out of your hand, or hit you in the face or arm in a painful way.

I think there's a flavor of this type of energy transfer in the golf swing. It's similar to a trebuchet which has a stiff arm, and a rope-sling attached to the end. Towards the end of the stiff arm's swing it slows down and strong forces are exerted on the rope to accelerate the rock around a pivot point with arm length equal only to the rope length. In essence, the orbital radius changes rapidly from something approximating the length of the trebuchet arm, to something approximating the length of the rope.

Tutleman analyzes the whole thing as a double physical pendulum, and concludes that the wrists shouldn't add positive torque until very late in the swing, tens of milliseconds before impact... But Hogan who advocated the "swing with the hands" concept probably is feeling something real even if he doesn't understand how to turn it into physics.

But what Tutleman doesn't analyze and I'd be interested to see, is what happens when the body adds negative torque at an appropriate point. That is, slow the upper body rotation speed while allowing the wrists to unhinge. That will induce a backwards force on the top of the club and explain the forward bend of the club that he mentions. It is the same basic motion as stopping your arm while letting the ruler swing freely.

I'll write some maxima code to integrate the ODEs and see what I get.


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