That's nice. Right? But again, we should check this equation. Does it have the correct units? Yes—the denominator of this expression just has μ s and trig functions which do not have units. This means that overall the expression has units of Newtons. That's good. What would happen if I push with an angle of zero degrees? Putting in zero for θ in this expression I get the same answer as the first question (pushing horizontal)—so that makes sense. Finally, what if I push straight up with θ equal to 90 degrees? This would give a force magnitude equal to the weight of the block—again, that seems reasonable.

But wait! What about the answer to the question about pushing horizontal compared to pushing at an angle? Which requires more force? As long as the angle is between 0 and 90 degrees, the stuff in the denominator will be bigger than just μ s such that the force will be smaller at an angle. Is there a "best" angle that requires the least amount of force? Yes! Let's find it.

Of course you could take the equation for the force as a function of θ above and take the derivative with respect to θ and then set it equal to zero. This would be your standard max-min problem from calculus class.

What if I just sort of use brute force to solve this problem? I can calculate the force required and a bunch of different angles and then just find the angle with the smallest force. Since I don't actually want to do this by hand—I will use Python. And here is that program (so that you can play with it too).

Just push the Run button to run it and then you can go back and edit the code if you want to play with it. Don't worry, you can't really break anything. This is a fairly straightforward program. But the cool part is that there is a minimum pushing force at some angle. It's easier to see with larger values for the coefficient of friction, so in the code above, I have it set at 0.6. With this value, a push angle of about 59.9° gives the lowest possible force. This force is lower than pushing it horizontal and lower than pushing straight up. In fact, every angle gives a lower force than pushing it horizontal.