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re: .-1
The assumption that the shaft is an extension of the right forearm
at the point of impact is false. This would be true if the right
wrist was cocked downward and fully straightened at the point of
impact, but this is not the position that professional golfers
assume.
The angle between the left and right forearms is still related to
theta, though, under the assumption that there is a preferred
or natural orientation of the hands. For example, Ben Hogan talked
about 'supinating' the left wrist (toward the right wrist) at the
point of impact, which according to some has the effect of maintaining
club alignment, thereby reducing the tendancy to 'hook' the ball.
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Here's an update to .0; with a few modifictions to the maple script.
Using some approximate measurements, my optimal 'theta' works out to
about 70�. If the left wrist is allowed to loosen up so that theta is
only 35�, then the clubhead speed declines by about 5%, or about 12
yards on a 250-yard drive.
note: it may be more intuitive to think about the complement of theta,
which is the interior angle formed between the left forearm and
the clubshaft.
#;
#; The velocity of a golf clubhead is determined by the angular velocities
#; and the radiuses involved in the golf swing. There are several axes of
#; rotation, but obviously there cannot be more than three orthogonal axes:
#;
#; 1. The backbone. The radius is equal to the perpendicular distance
#; between the backbone and the ball. This distance is the
#; hypoteneuse of the right triangle formed between the ball, the
#; left shoulder socket, and the closest point on the backbone to
#; the left shoulder socket.
#;
#; 2. The left wrist. The radius is equal to the club length. Action
#; of the right forearm (about the right elbow) and the right wrist
#; rotates the club around this axis. Since the right wrists moves
#; directly away from the right forearm, it follows that maximum
#; leverage is obtained when the right forearm is aligned directly
#; at the ball at the point of impact.
#;
#; 3. A point that is in line with the left arm, but at a distance
#; club_length * cos(theta) from the left wrist, where theta is
#; the angle formed between the left arm and the shaft. Rotation
#; of the left forearm accelerates the clubhead by this radius.
#; Rotation of the right forearm around the left forearm provides
#; leverage relative to this axis.
#;
#; theta is the complement of the angle formed between the left arm
#; and the shaft. Theta can be increased by cupping the left wrist
#; toward the head, subject to the constraint that the right forearm
#; and the shaft must lie in the same plane when viewed from behind,
#; toward the target. These constraints can be mutually satisfied
#; by lifting the left shoulder and dropping the right shoulder,
#; thereby increasing the radius club_length * sin(theta).
#;
#;
radius_wrist := club_length;
radius_theta := club_length * sin(theta);
radius_arm := sqrt(radius_theta^2 +
(arm_length + club_length * cos(theta))^2
);
radius_backbone := sqrt((shoulder_width/2)^2 + radius_arm^2);
clubhead_speed := ang_velocity_backbone * radius_backbone +
ang_velocity_axial * radius_theta +
ang_velocity_wrist * radius_wrist;
dSpeed_dTheta := diff(clubhead_speed,theta);
ang_velocity_backbone := ang_velocity_arm + ang_velocity_shoulders;
ang_velocity_arm := ang_velocity_shoulders / 2; #; approx
ang_velocity_wrist := ang_velocity_backbone * 2; #; approx
ang_velocity_axial := ang_velocity_backbone; #; approx
shoulder_width := 17;
arm_length := 23;
club_length := 53;
dSpeed_dTheta := diff(clubhead_speed,theta);
#; opt_theta := solve(dSpeed_dTheta, theta);
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a better measurement for the club shaft length is 39", which results in
a theoretical optimum value of theta = 66.5�. However, this means that
that your hands would be only about 15.5" off the ground at the point of
impact. Unless you hang out in trees, it's unlikely that you'll be
able to get your arms that low.
So the result simply says that you position your left hand as close to
your body as is reasonable; say, one hand-width away from your leg,
and lower your right shoulder relative to your left so that the right
forearm is in the same plane as the club shaft when viewed from behind,
toward the flag.
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