Say for instance you have two tennis balls. Placing one on the table, hold the other in your hand and have it touch somewhere between the side and top of the other tennis ball. Now, when you start to firmly press down on the ball in your hand, the other ball begins to move out of the way.

When you try this with smoother balls, like ping-pong balls, the force that can be applied before it moves is quite low. Move to a rougher surface, like a paper towel, and it becomes incredibly difficult to have the other ball move. Therefore, the maximum applied force before static equilibrium is broken must be dependent on the coefficients of static friction (CoSF) between the balls and the balls with the surface.

My question is this: What is the maximum applied force $F$ that I can apply to the top tennis ball before the bottom one moves, in terms of $\mu_g$ (CoSF with the ground) and $\mu_b$ (CoSF with between the balls)? For simplicity's sake, let's just assume that I'm applying $F$ at angle of $60^{\circ}$ from the vertical. I have created a free body diagram shown below that I think models the bottom ball:

enter image description here

Initially, I thought this would be a straight forward and simple process: simply find the net force equations in both $x$ and $y$, along with the net torque equation and solve for the coefficients of friction in terms of the applied. I created these equations:

$$\Sigma F_x = \mu_g N - F \sin(60^{\circ}) + \mu_b F \sin(30^{\circ}) = 0$$

$$\Sigma F_y = N - mg - F \cos(60^{\circ}) - \mu_b F \cos(30^{\circ}) = 0$$

$$\Sigma \tau = r(\mu_b F - \mu_g N) = 0 \rightarrow \mu_b F = \mu_g N$$

However, when I thought more about exactly what I was doing, I realized that there might be some key issues with this way of solving it. When I plugged in the equations to my calculator, as $F$ increased, $\mu_b = 2 - \sqrt{3}$ always and $\mu_g$ approached that value. What this insinuates is that as $F$ increases, as long as both CoSF are at least $2 - \sqrt{3}$, the bottom ball will never move... which is obviously very wrong.

I'm at my wits end, as I have no idea how to attack this problem differently. Now, I understand that $f \leq \mu_{\mathrm{static}} N$, and that most likely factors into the situation somehow. I also understand that the solution to the problem will most likely not be a nice numerical answer, and instead depend on the relationship of $\mu_b$ and $\mu_g$.

Any help at all would be appreciated, thank you!


closed as off-topic by sammy gerbil, ACuriousMind Mar 9 at 2:20

This question appears to be off-topic. The users who voted to close gave this specific reason:

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – ACuriousMind Mar 9 at 2:20
  • $\begingroup$ Please note that homework-like questions and check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual questions over those just asking for a specific computation. $\endgroup$ – ACuriousMind Mar 9 at 2:20
  • $\begingroup$ @Inothernews1 I'm sure this could get reopened if you focused your question on whether or not it is possible to have things not move no matter how large the applied force $\endgroup$ – Aaron Stevens Mar 9 at 3:16