# Putt Physics Calculation 01: Swinging motion of putt

Here are some of the standard precautions you will often hear when putting.

・Place the ball directly under your eyes.
・Do not move your head.
・Align your backswing and follow-through
・Hold the ball loosely and do not use your wrists.
・Swing as straight as possible.

No matter how you think about it, you want to be a pendulum, don’t you? That’s all I can think. I decided to think so. I’ve made up my mind, so I calculate the putter swing as a pendulum motion in the application “Putt Physics”.

Pendulum motion, so-called single pendulum. If you studied physics in high school, you may know this. The components of a single pendulum are a string and a weight attached to the end of the string. It looks like this.

So, here’s the thing.

If the length of the string is L, the weight of the weight is m, and everyone’s favorite gravitational acceleration is g, the formula follows from the law of conservation of mechanical energy.

$$mgL(1-cos\theta)=\frac{1}{2}mv^2$$
Potential energy at the highest point of the weight = Kinetic energy at the lowest point of the weight

We will replace this with the putter swing. First, the length of the string, L. The tip is without hesitation the putter head. The question is where to place the fulcrum. I was advised not to move the head, so I could think of the head as the fulcrum, but I thought it might be a little lower, so I decided to use the center of the shoulder.

In “Putt Physics” the height is entered, but in the actual calculation, the height minus 30 cm is used as the length of the thread (L). The distance from the top of the head to the center of the shoulder and the distance from the center of the shoulder to the center of the shoulder are taken into account. 30 cm is honestly appropriate, so players should decide the length of the string length L taking into account their own form and other factors.

Now that the length of the string L has been determined, the next step is to determine the mass of the weight m. Since there is one weight on the right side and one on the left side of the string, we divide the weight by the mass of the weight. Yes, the weight of the weight = the weight of the putter does not affect the pendulum motion. No matter what kind of putter you buy, the ball’s motion will not change. Any putter is the same. Yes, that’s a lie. That is not true. The mass m of the weight has no effect on the single pendulum equation of motion. As will appear later, the club characteristics of the putter have an effect on the collision motion between the putter and the ball.

Angle θ is the angle of the highest point of the pendulum, so it corresponds to the swing angle of the putter. Velocity v is the speed of the weight, so it is the speed at which the putter head moves. Now that all the substitutions are done, we have the values and can do the calculations. Now that we know the values and can calculate them, let’s solve the equation of motion.

It seems good if we know the velocity v when the putter head reaches the lowest point. After that, the putter head and the ball collide and the energy is transferred to the next motion.

$$v=\sqrt{2gl(1-cos\theta)}$$

Let us calculate the case of a 170cm tall person swinging at a putter swing up angle of 30 degrees. Since the length of the string is 170 cm – 30 cm = 140 cm
$$v=\sqrt{2\times9.8\times140/100\times(1-cos30^\circ)}=0.606321938$$

This concludes the calculation of putter head speed. This calculation is actually done in “Putt Physics”.

To digress a bit as an aside, you are sometimes advised to swing in the same rhythm for putting. This is actually closely related to the pendulum motion. In the pendulum motion, if the length of the string and the mass of the weight are the same, the period of the motion will remain the same even if the highest point is different. It is interesting to see that this is true when you actually make a pendulum and move it.

Thinking about this in reverse, if you change the rhythm of the putter swing according to the width of the swing, the physical law will be different from the pendulum motion. For example, if the rhythm is slowed down when the backswing is large, the head speed is slowed down compared to the pendulum motion. Please note that if the rhythm changes according to the swing width in this way, it will not fit in the “snap” swing width calculation.

Next, I will note what is not calculated in the swing motion. First, air resistance. I think putters are also subject to wind resistance, but this is ignored. Second, human swing friction. The human swing cannot be in an ideal state like a fixed fulcrum of a string. There are many factors that interfere with the swing, such as shoulders, hips, and muscles in various places, but we ignore them all. Then there is the human force (acceleration) during the downswing. This is a force that cannot be ignored depending on the swing method, so it is included in the calculation factor and will be discussed later.

This completes the calculation of putter swing motion.

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