i thought that the force of static friction exerted on an object is always going in the opposite direction of any other force exerted on the same object. however, this problem seems to disregard that fact
The coefficient of static friction between your coffee cup and the
horizontal dashboard of your car is μ= 0.800. (a) How fast can you
drive on a horizontal roadway around a right turn of radius 30.0 m
before the cup starts to slide?
to solve this problem, we need to find the centripetal acceleration and then use Newton's second and third laws. i.e if we say that
$F[c] = \frac{mv^2}r$ centripetal force
that means that the car have to give back that same force in the opposite direction (third law) and that is the force that will cause the cup to start sliding. so to actually measure the speed it takes to move the cup, I said that
$-F[s] = F[c]$
where the negative indicates that the force of friction is in the opposite direction. so
$$-μ(n) = \frac{m*v^2}r => -μ(mg) = \frac{mv^2}r => -μ(g) = v^2/r => v = \sqrt{-μ(g)(r)}$$
this answer doesn't make sense because of the negative under the square root. and the only way to solve this problem, is by saying that F[s] = F[c] (i.e. get rid of the negative) which does not make any sense to me. can somebody please explain to me why should the negative be neglected in this problem. thanks
Best Answer
You write:
$$ -F_s = F_c $$
and your reasoning is that if you have a centripetal force $F_c$ inwards then there must be a static force $F_s$ outwards that resists this force. The trouble is that you are starting with the observation that the cup is stationary with respect to the car, but the car is a non-inertial reference frame and Newton's laws do not apply to it. While physicists sometimes use non-inertial frames they are full of traps for the unwary and are best avoided if at all possible.
In this case, suppose you are standing by the road watching as the car races past. Newton's first law tell us that if no net force acts on an object then it will move in a straight line at a constant speed. So if the net force on the cup was zero the cup would move in a straight line and fly out of the car window.
What we actually observe is that the cup is moving in a curve, so there must be a non-zero force acting on it. The only thing acting on the cup is the dashboard surface, so the dashboard must be applying the force that is making the cup accelerate, and the only force between the dashboard and the cup is the frictional force $F_s$.
So the point is that the frictional force is the centripetal force, that is:
$$ F_s = F_c $$
That's why the centripetal and frictional forces have the same sign, because they are the same thing.