inclined surface and speed

[Fulvio Romano]

Okay, I need help, but I'm ashamed.

we have a ball on a sloping surface. If the plan is straight I'll take some time. If the surface is curved I will put on? More? less? and friction with the air how does it affect the thing?

Intuitively I assumed that, considering friction with air, the tilted plane is the shortest route because the one that maintains a lower average speed and therefore globally less wasted energy.
I tried to mathematically derive the hourly equation according to the surface, but I lost myself in the calculations.
I tried to mathematically derive the equation of energy, but nothing comes out.
In the end, as a mechanical hub, I did a simulation in matlab. I confirm that by high viscous coefficient the best way is straight, but with viscosity nothing exists faster when the surface begins to curve, and then returns disadvantageous.

attached you see the simulations with sigma equal to 0, 5 and 10. in each diagram you can see the sliding surface, acceleration, speed and position.

who helps me better understand this phenomenon?

 

[ceschi1959]

ciao @fulvio Roman ,

lost in this then.. .

brachistocrona

 

[Fulvio Romano]

ciao @fulvio Roman ,

lost in this then.. .

brachistocrona

Look, man, I told you to solve my problem, not create another one!! !

:

 

[Stan9411]

It is clear that the sloped plane is the shortest road, but it is also the one that offers “less angle” to the only source of acceleration or gravity. on the contrary a very steep surface initially that straightens horizontally very close to the ground, will allow the ball to reach great peak speed but will imporre a longer road and therefore will come anyway after. The trade-off is in the middle. there will be a fairly short trajectory but at the same time quite inclined to minimize the travel time.

It seems quite intuitive.. and if you simulated it, you must have understood it. otherwise what mathematical model did you write?

 

[cacciatorino]

perhaps the variable not considered is the kinetic energy stored by the sphere under the sign of rotation around its baricentric axis.

 

[Fulvio Romano]

It is clear that the sloped plane is the shortest road, but it is also the one that offers “less angle” to the only source of acceleration or gravity. on the contrary a very steep surface initially that straightens horizontally very close to the ground, will allow the ball to reach great peak speed but will imporre a longer road and therefore will come anyway after. The trade-off is in the middle. there will be a fairly short trajectory but at the same time quite inclined to minimize the travel time.

It seems quite intuitive.. and if you simulated it, you must have understood it. otherwise what mathematical model did you write?

a simulation is not a mathematical model. It's a simulation. I made parabolic trajectories, what happens with circular trajectories? and elliptical? a mathematical model answers this question, a simulation no.

It is not intuitive anyway. the inclined plane does not offer "less angle". the average angle is the same, otherwise it would not come at the same point. also in a conservative field what you say is highly counterintuitive, nothing else!

perhaps the variable not considered is the kinetic energy stored by the sphere under the sign of rotation around its baricentric axis.

My simulation considers a material point. You're right. but the phenomenon is observed the same.

 

[cacciatorino]

My simulation considers a material point. You're right. but the phenomenon is observed the same.

consider that at the end of the plan you have that the initial potential energy is equal to the final kinetic energy, kinetic energy that is the sum of that linear and that of rotation. Does your software require the diameter of the ball as a parameter, or do you consider the dot object?

 

[mamobono]

"if the plan is straight I'll take some time. If the surface is curved I will put on? More? less? "
to these questions the answer is:
the property of cycloid brachistochrin
... johann bernoulli proposed a challenge to all European mathematicians, including the following
problem: are given two points in a vertical plane a and b, with higher than b, but not on
vertical of b, and a thin iron wire on which a beads can flow without friction. to find what form to give to the iron wire so that the pearl uses as little time as possible to go from to b under the action of gravity.
in other words the “form to give to the iron wire” corresponds to the trajectory that the body (to this bound without friction) must follow in order to employ the least time between a and b under the action of the
gravity. to this trajectory we give the name of brachistocrona. . see the pdf

then if we consider friction by making the energy budget we can assess the output speed in straight and cycloid cases

but I do not get rid of + I am too old and rusty

Hi.

 

[Fulvio Romano]

consider that at the end of the plan you have that the initial potential energy is equal to the final kinetic energy, kinetic energy that is the sum of that linear and that of rotation. Does your software require the diameter of the ball as a parameter, or do you consider the dot object?

I wrote it, it's a material point. dots.

... johann bernoulli proposed a challenge to all European mathematicians, including the following
problem:

You took my day. So I'm not the jerk, it's the problem that's complex.

smack!

 

[Fulvio Romano]

I'm reading pdf with passion. It's beautiful.

and incidentally demonstrates one of the definitions of the engineer as "he who has forgotten more mathematics than you have ever known"

 

[Stan9411]

a simulation is not a mathematical model. It's a simulation. I made parabolic trajectories, what happens with circular trajectories? and elliptical? a mathematical model answers this question, a simulation no.

Sorry, but how did you get the position charts, speed and acceleration for different tracks if you didn't write the mathematical model?

It is not intuitive anyway. the inclined plane does not offer "less angle". the average angle is the same, otherwise it would not come at the same point

but the allegation that the ball receives goes with the little thing of the angle of inclination ... if you give it an almost vertical plane throughout the first part of the trajectory changes all the speed profile ... that the average angle is the same has nothing to do with it. I suspect you left the trajectory and went back to speed and acceleration, but that's not true. is an acceleration system (variable) imposed.

If I can, tonight I try to write in matlab the equations that, in my opinion, govern this phenomenon.

I'm sorry I got caught up as a guy who writes shit... maybe I'm really telling you... when I have a moment I try to self-support myself with numbers.

 

[Fulvio Romano]

Sorry, but how did you get the position charts, speed and acceleration for different tracks if you didn't write the mathematical model?

the mathematical model is a formula, an equation, a functional one that describes the phenomenon. a simulation is made by means of an iterative process that for every moment of time calculates the value of the next indwelling. at the end of the simulation you do not have the mathematical model of the phenomenon, but only its graphic representation in the conditions under which the simulation was launched.

but the allegation that the ball receives goes with the little thing of the angle of inclination ... if you give it an almost vertical plane for the whole first part of the trajectory changes all the profile of speed

Look at my charts. Clearly, it changes the speed profile.