I am having a bar and surface (made of aluminum). I need to attach some material to the end of the bar so that, if I apply any force (25 N-sideways) to bar it has to stick with the surface.
Force is applied normally to the flatter surface of the bar. I need to select a material at the end of bar so that the force applied on the bar is minimal.
What methodology should I choose to find an answer?
Aim is lowest necessary force to keep the rod intact (with a material added to rod end)
I'm trying to approach the problem at the high school level.
The vertical force is proportional to the tangential force acting at the other end of the rod, the ratio of these two is called the static coefficient of friction.
Here the applied forces are due to robotic arm and the weight of the rod: $$(F+mg)\mu_s=25\ [N]$$ or: $$F\mu_s=25\ [N] - mg\mu_s$$
As you can see we have an equation with two unknowns, $F$ the vertical force of robots arm and the material dependent parameter $\mu_s$.
If you look at the plot: 
you observe if you increase the $\mu_s$, the applied force will decrease, but it makes it difficult for the rod to move freely, if you increase the force the coefficient of friction will decrease but you want to keep the applied force minimum so this is not an option. So we need to define a range for the force $F$ to solve the problem with bounded extrema, or simply choose a heavier rod, but make sure the force $25\ [N]$ can pull it back.
