There have been reported cases where a golf ball rolls into the hole and then pops out again. This blog post creates a simple model to explore this behaviour.
Introduction
This strange behaviour is well documented in this youtube video The Golf Ball Paradox (youtube.com) by Steve Mould and this video included a link to a paper that describes the dynamics of the system (golfer.pdf (ulisboa.pt)). From this paper a simple Modelica model is created.
The equations
There are two different dynamic equations, when the ball is in the air and when the ball is rolling on the inside of the hole/tube.
The equations of motion under a constant acceleration, used to describe the ball when it is outside of the hole are:
The other equations to describe the motion are from this paper golfer.pdf (ulisboa.pt), and the main equation of importance is this one:
This equation describes the vertical motion of the centre of mass of the ball while rolling in inside a tube without slip.
Where:
z is the vertical height of the centre of mass of the ball
Ω is the angular velocity of the centre of mass of the ball spinning around the centre of the tube
g is the gravitational constant
K is a dimensionless number that reflects the geometry of the density distribution of the ball
Combining the equations
A model was created that used the equations of motions and the equations of the ball rolling in a hole without slipping. When the centre of mass of the ball z was positive it is assumed to be outside of the tube and when it is less than z is it assumed to be in the tube. The initial velocity of when the ball comes into contact with the tube is given as u_start and this is used to determine an initial starting position where ż=0. The resulting equations used are:
Where:
The variables with the _eject suffix store the motion and of the ball at initialisation and at the time when the ball leaves the tube. The initial calculations not shown above.
pos_ball is the position of the centre of the ball in the x,y and z axes.
v_ball is the velocity of the centre of the ball in the x,y and z axes.
The phi_ prefix is to describe angles, these are either the angle of the ball or the angle of the entre of mass of the ball relative to the tube.
The _s suffix is used to describe the angle of the ball or centre of mass of the ball relative to the tube, in the top/arial view.
The _z suffix is the angle of the ball when describing the ball rolling up or down the tube in the vertical plane.
Results
Animation 1 is of a solid ball being thrown into a tube
In Animation 1, the simulation is running at 1/20 of real-time and the ball being used is uniformly solid and has K=2/5.
In Animation 2 two, balls are thrown. The blue ball is solid and has K=2/5 and the green ball is hollow and has K=2/3.
The results for animation 2 are as expected where the hollow ball is ejected before the solid ball.
Discussion
A model was recreated to simulate a ball being thrown into tube such that the ball rolls around the side of the tube without slip. This model gave the expected results. Get the Modelica code from here.
In a follow up blog post I’d be modelling this behaviour using a contact model so that the effect of slip can also be included in the model.
Written by: Garron Fish – Chief Engineer
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