Answer to Question #167481 in Physics for Froilan

Question

The 2197-N block shown is in contact with 45° incline. The coefficient of static friction is 0.3. Compute the value of the horizontal force P necessary to just prevent motion down the incline.

Answer 1

Let's apply the Newton's Second Law of Motion in projections on axis $x$ and $y$:

$$Pcos\theta+F_{fr}-Wsin\theta=0,$$ $$N-Wcos\theta-Psin\theta=0.$$

Let's rewrite our equations:

$$Pcos\theta+\mu_sN-Wsin\theta=0$$ $$N=Wcos\theta+Psin\theta.$$

Let's substitute $N$ into the previous equation and solve for $P$

$$Pcos\theta+\mu_s(Wcos\theta+Psin\theta)-Wsin\theta=0,$$ $$P=\dfrac{W(sin\theta-\mu_scos\theta)}{cos\theta+\mu_ssin\theta},$$ $$P=\dfrac{2197\ N\cdot(sin45^{\circ}-0.3\cdot cos45^{\circ})}{cos45^{\circ}+0.3\cdot sin45^{\circ}}=1183\ N.$$