Hey there, folks! Last time, we talked about the start of the ball’s roll. Now, let’s just sit back and watch whether it’ll sink into the cup or get close to an OK putt. But before that, let’s take a moment to calculate how the ball rolls.
For today’s discussion, we’ll be focusing on putting on a flat surface on the green. The ball experiences a force from gravity \(g\) and initial velocity \(v_0\), as well as a force from the friction between the ball and the green.
If we illustrate it in a diagram, it’ll look like this:
The new force in motion is the frictional force \(\mu{}mg\). Here, \(\mu\) is called the friction coefficient, which acts as a force opposing the ball’s movement. The force from the putter head gradually decreases due to friction until it comes to a stop. But what the heck is a friction coefficient, right?
Let’s break it down a bit more. The force that restrains the ball from rolling is from the grass on the green, commonly referred to as “green speed.” Do you know how to quantify this speed? Well, there’s a tool called a “Stimp meter” that measures it. It’s apparently the de facto standard for measuring green speed and is available for purchase.
The catch is, Stimp meters can be pretty expensive. So, even if you want to buy one, it may not be affordable for many folks. But here’s the good news: some golf courses display their Stimp meter readings. If you come across one of these courses, you might want to use that info to your advantage and put it to good use!
Have you ever heard of the Stimp meter? It’s a tool used to measure the distance a golf ball rolls on a green, and it gives an indication of how fast the green is. Typically, on most golf courses, the Stimp meter value ranges between 8 to 9 feet. So, let’s do some calculations using three equations of motion for constant acceleration.
$$F=ma$$
$$x=\frac{1}{2}at^2+v_0t$$
$$v=at+v_0$$
First, we need to know that the force of friction is μmg, where μ is the coefficient of friction, m is the mass of the ball, and g is the acceleration due to gravity. When the ball is rolling in the positive direction, the force on the ball is given by \(ma=F=-\mu{}mg\), which means that the acceleration of the ball is \(a=-\mu{}g\).
When the ball comes to a stop, its final velocity is zero \(v=0\). Using the third equation of motion, we can calculate the time it takes for the ball to come to a stop:
$$0=at+v_0=-\mu{}gt+v_0$$
$$t=\frac{v_0}{\mu{}g}$$
Now, let’s assume that the Stimp meter value is \(s=9\), the coefficient of friction is \(\mu=0.1\), and the initial velocity of the ball is \(v_0=2m/s\). Using the equation we derived earlier, we can calculate that the ball will stop after 1.96 seconds. Using the second equation of motion, we can calculate that the ball will travel a distance of 2 meters on a green with a Stimp meter value of \(s=9\) when it is hit with an initial velocity of 2 m/s.
But wait! Before you go and start applying these calculations to your golf game, there are a few things to keep in mind. First of all, we have ignored the effects of air resistance and wind on the ball. Additionally, we have assumed that the ball is a perfect sphere, which means we have neglected the effect of dimples on the ball’s surface. Lastly, the calculation does not take into account the energy lost due to friction, which may be included empirically in the conversion from Stimp meter values to coefficients of friction. If you find that your ball rolls significantly differently than expected, let me know, and we can adjust the calculations accordingly.
So, there you have it! The physics of a ball rolling on a flat surface.
