I came across this question in one of my books and don't completely understand the answer: An object is resting on a board. One side of the board is slowly raised, up to an angle of 60 degrees. The object does not slide. The questions asks if, as the board is being raised, the frictional force increases or decreases. My reasoning for the force of friction decreasing is that as the angle between the horizontal and the board increases, the normal force decreases (Force of friction = μmgcos(θ).
The answer/explanation in the book, however, states that since the object does not move, the frictional force is equal to the component of gravity parallel to the board (which obviously increases as the board is raised). Is this correct? And if so, how can one reconcile this with the normal force decreasing?
Best Answer
As long as the object does not slide, the static frictional force is equal to $mg\sin \theta$. Once the frictional force also becomes equal to the normal force $mg\cos \theta$ times the coefficient of static friction, the object begins to slide. But the static friction law is an inequality, not an equality. $$F\leq \mu_s N$$ As long as the F is less than $\mu_sN$, the object won't slide. The static friction coefficient only tells you the value of the friction force at the point where the object is just on the verge of sliding.