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I'm studying the dynamics of a sphere in a horizontal plane driven by a force. The situation is the following: I have a stationary sphere of mass $m$ and radius $r$ in a pool table whose lining has coefficients of friction $\mu_k$ and $\mu_s$. The question i have is: Knowing the data mentioned above, despising the rolling resistance, can i calculate the force needed to move the sphere some distance $d$?

If is not possible, Is there any way to get this data without experimenting?.

PD: to contextualize, this study is about an elliptical pool table, where there is one pocket in one of the focus of the ellipse, and the other focus is used as a reference, since each time a ball passes through it with the appropriate force, it will reach the other focus where the pocket is.

Excuse me in advance if there is an error, I am new in physics.

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  • $\begingroup$ Once you get the ball rolling it will travel indefinitely if you assume a surface without rolling friction or energy loses from hitting walls of the table $\endgroup$
    – BioPhysicist
    Commented Oct 2, 2018 at 2:52
  • $\begingroup$ But the coefficients of friction aren't enough? $\endgroup$
    – Trobeli
    Commented Oct 2, 2018 at 2:57
  • $\begingroup$ Aren't enough for what? $\endgroup$
    – BioPhysicist
    Commented Oct 2, 2018 at 2:57
  • $\begingroup$ Stopping the ball from travelling indefinitely $\endgroup$
    – Trobeli
    Commented Oct 2, 2018 at 3:00

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Knowing the data mentioned above, discarding(?) the rolling resistance, can i calculate the force needed to move the sphere some distance d?

As already explained by Aron Stevens, the rolling resistance is essential here, so you should not ignore it (if that's what you meant).

The force, by itself, would not be sufficient: you would also need to define either a time interval the force is acting on the ball - the impulse (to get the ball's initial momentum) or a distance over which the ball is pushed by the cue - the work (to get the ball's initial kinetic energy).

Once you know the ball's initial momentum or kinetic energy (if you know one, you know the other) as well as basic physical characteristics of all the involved objects, you could estimate how far the ball will move.

As an example, if you know the initial kinetic energy of the ball, $E_k$, and the rolling friction coefficient, $\mu_r$, you can estimate $d$ from the work-energy formula $E_k=\mu_r mg d$.

If is not possible, Is there any way to get this data without experimenting?

You can start with this page, which lists typical values for all the parameters you'll need for your initial estimates, e.g., ball mass, coefficient of rolling resistance, etc. Eventually, though, you would want to perform some experiments to see how accurate your estimates are and to make necessary adjustments.

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  • $\begingroup$ Thanks, I understood almost everything. My doubt is: when you menction the work-energy formula $E_k=\mu_r mg d$, Why gravity is involved here, where the movement is started by an external force (the hit with the cue)? $\endgroup$
    – Trobeli
    Commented Oct 3, 2018 at 23:20
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    $\begingroup$ @RodrigoPizarro A general expression for friction force is $\mu N$, where N is normal reaction. On a horizontal surface, normal reaction is equal to the weight of an object, $N=mg$. So the expression is a product of the friction force, $\mu_k mg$ by distance, $d$, which is work performed by table on a ball to reduce its kinetic energy to zero. $\endgroup$
    – V.F.
    Commented Oct 3, 2018 at 23:41
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If you neglect rolling friction or energy losses from collisions with the table walls then the ball will travel indefinitely. This is because the only force that could stop the ball is either static or kinetic friction.

Now if the ball is rolling without slipping, then the point of the ball that touches the table is at rest with respect to the table at any instant in time. Therefore, kinetic friction is not at play because the two surfaces are not sliding past each other (or by definition of rolling without slipping).

What about static friction then? Well static friction is only present when you are trying to slide the two surfaces past each other, but the force that is being applied is not large enough to overcome the static friction force. The problem is that there is no other force acting on the ball trying to slide the surfaces past each other. Therefore there is no static friction force at play either.

If you want the ball to slow down, you have to consider rolling friction that arises from deformation of the surface and the ball. This causes normal forces that are not perpendicular to the plane, so horizontal forces and torques come into play that actually slow the ball down.

Of course you could argue that there is some slipping between the ball and the table, but that is usually included with effects from rolling friction since this probably won't follow the simple $f_k=\mu_k N$ model.

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