Physics Dev Q - Centripetal Force / Golf Cup

[nirgaul]

I'm working on a physical simulation that would involve something like a golf ball rolling toward a cup. If it hits the rim of the cup, it will take on an elliptical path, and depending on its approach velocity prior to contact with the cup, will either spiral down into the cup, or be partially altered in its course.

If someone has addressed this problem before, could they summarize all the Forces needed to be taken into account, and possibly refer me to some code examples that implement this?

Thanks!

[Smada]

The following solution is simplified by the fact, that:
1. The ball cannot leave the surface, it cannot jump
2. the radius used to calculate the centripetal force is approximated by the comparison of the momentary centripetal force and the reacting surface force.
3. Rotational energy is not taken into account


i) Energy in the system is constant

Ekinetic+Erotational+Epotential=C ||Erot=>0 ||C= initial energy= Ekin(t=0)+Epot(t=0)=1/2*m*v0²+mgr

1/2*m*v²+mgh=C
v=sqr(2*(C-mgh)/m) (1)
h=r*sin(alpha) (2)
d=r*cos(alpha) (3)


R',R''= momentary radius of the motion

Forces:
Fg=m*g <-gravity
Fz=m*v²/R' <-centripetal force
Flx=Fg/sin(alpha) <- support of surface x-direction
Fly=Fz/cos(alpha) <- same y-direction

This is an approximation[2] as it states that the sum of all forces is 0



the acceleration of the ball is now calculated by geometrical interpretation of its position
Example:
at frame0 the ball is not touching the cup, therefor alpha=90, and R'=inf (there is no centripetal force at this moment)
frame1: the ball enters the cup, alpha<90, the surface accelerates the ball towards the center,
at this moment Fz is still 0 (this is the approximation)
We can now calculate the triangle between R',R'' and AB, R'' is the distance of our new rotational center (R''=>R')
frame3: Fl and Fz now both act and accelerate the ball with the force Fr
again a new triangle can be calculated a new center of movement is found and the ball can move on...

if the ball was shot in a way, that it will exit the cup again, at one frame Fz will be bigger than Fl and the ball will start moving outwards again.
The finer the steps of the calculation the better the movement will become

at approximation [1]: The ball does never leave the surface, it is basically like a magnetic ball "glued" to the surface (pic 2), if this was not simplified, the ball would start a ballistic trajectory if it was shot to steep towards the hole, I don't have enough coffee for that at the moment ;)


http://img84.imageshack.us/img84/5966/zentriuh8.th.jpg (http://img84.imageshack.us/my.php?image=zentriuh8.jpg)http://img84.imageshack.us/images/thpix.gif (http://g.imageshack.us/thpix.php)

[Smada]

it didn't let me include all images so here the rest:

http://img74.imageshack.us/img74/1317/zentri2ez0.th.jpg (http://img74.imageshack.us/my.php?image=zentri2ez0.jpg)http://img74.imageshack.us/images/thpix.gif (http://g.imageshack.us/thpix.php)

http://img74.imageshack.us/img74/3508/zentri3kq5.th.jpg (http://img74.imageshack.us/my.php?image=zentri3kq5.jpg)http://img74.imageshack.us/images/thpix.gif (http://g.imageshack.us/thpix.php)

i haven't tested it and i never will, but it should work : ) (<- this is the smiley that was still to much X_x)

[nirgaul]

Thanks very much. I'm not a physics expert, but I think if I follow your notes I'll be able to work this out.

[lafe]

The following solution is simplified by the fact, that:
1. The ball cannot leave the surface, it cannot jump
2. the radius used to calculate the centripetal force is approximated by the comparison of the momentary centripetal force and the reacting surface force.
3. Rotational energy is not taken into account


i) Energy in the system is constant

Ekinetic+Erotational+Epotential=C ||Erot=>0 ||C= initial energy= Ekin(t=0)+Epot(t=0)=1/2*m*v0²+mgr

1/2*m*v²+mgh=C
v=sqr(2*(C-mgh)/m) (1)
h=r*sin(alpha) (2)
d=r*cos(alpha) (3)


R',R''= momentary radius of the motion

Forces:
Fg=m*g <-gravity
Fz=m*v²/R' <-centripetal force
Flx=Fg/sin(alpha) <- support of surface x-direction
Fly=Fz/cos(alpha) <- same y-direction

This is an approximation[2] as it states that the sum of all forces is 0



the acceleration of the ball is now calculated by geometrical interpretation of its position
Example:
at frame0 the ball is not touching the cup, therefor alpha=90, and R'=inf (there is no centripetal force at this moment)
frame1: the ball enters the cup, alpha<90, the surface accelerates the ball towards the center,
at this moment Fz is still 0 (this is the approximation)
We can now calculate the triangle between R',R'' and AB, R'' is the distance of our new rotational center (R''=>R')
frame3: Fl and Fz now both act and accelerate the ball with the force Fr
again a new triangle can be calculated a new center of movement is found and the ball can move on...

if the ball was shot in a way, that it will exit the cup again, at one frame Fz will be bigger than Fl and the ball will start moving outwards again.
The finer the steps of the calculation the better the movement will become

at approximation [1]: The ball does never leave the surface, it is basically like a magnetic ball "glued" to the surface (pic 2), if this was not simplified, the ball would start a ballistic trajectory if it was shot to steep towards the hole, I don't have enough coffee for that at the moment ;)


http://img84.imageshack.us/img84/5966/zentriuh8.th.jpg (http://img84.imageshack.us/my.php?image=zentriuh8.jpg)http://img84.imageshack.us/images/thpix.gif (http://g.imageshack.us/thpix.php)

and your a junior member

[SataMaxx]

And one's mathematical and physical abilities aren't proportional to his postcount on fun-motion forum....

[lafe]

And one's mathematical and physical abilities aren't proportional to his postcount on fun-motion forum....
oh,
i thought it ment age.
not time on forum

[GreyMario]

and your a junior member

This post is worthy of a ban.

[lafe]

This post is worthy of a ban.
how does that get me baned,
i was prasing him for being smart and (i thought for his young age).
it was an attempt at being nice, something i havent seen you do

[GreyMario]

In the right eyes it can look like an insult based on postcount.

[Colescheffler]

In the right eyes it can look like an insult based on postcount.
Wow, he didn't know, when something says junior member you'd think hes young.

[GreyMario]

If that were the case, then why would lafe have the title of Senior Member?

He should have really taken the time to apply some logic. Or perhaps use his brain.


But this is getting offtopic, and this question's been answered. :V

[UberGimp]

In the right eyes it can look like an insult based on postcount.

I think you mean with the wrong eyes.
Stop being such a cunt GreyMario.
I see nothing from you but MEANIE posts.