Question

he 2140-N block shown is in contact with 45° incline. The coefficient of static friction is 0.25. Compute the value of the horizontal force P necessary to just start the block up the incline. Write your answer in two decimal places.

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 $P$:

$$Pcos\theta-\mu_s(Wcos\theta+Psin\theta)-Wsin\theta=0,$$ $$P=\dfrac{W(sin\theta+\mu_scos\theta)}{cos\theta-\mu_ssin\theta},$$ $$P=\dfrac{2140\ N\cdot(sin45^{\circ}+0.25\cdot cos45^{\circ})}{cos45^{\circ}-0.25\cdot sin45^{\circ}}=3566.67\ N.$$