Answer
(a) The magnitude of $\vec{P}$ is $52.35N$.
(b) The magnitude of $\vec{P}$ is $39.2N$.
Work Step by Step
Before taking a look at each situation, we consider the forces involved individually:
- Force $P$ has 2 components here: horizontal $P_x=P\sin\theta=0.5P$ and vertical $P_y=P\cos\theta=0.87P$
- Weight of the block $mg=39N$. $mg$ and $P_y$ are in opposite direction.
- Kinetic frictional force: $f_k=\mu_kF_N$
- Normal force $F_N$ is directed outward and perpendicular to the wall, as shown in the image below. As the block does not move horizontally, the horizontal forces must balance each other. Therefore, $$F_N=P_x=0.5P$$
So, $f_k=\mu_k\times0.5P=0.125P$
(a) When the block slides up the wall, $f_k$, opposing the motion, is directed down the wall, supporting $mg$ (image a). The constant velocity signifies that the net vertical force is $0$. So, $$P_y=mg+f_k$$ $$0.87P=39+0.125P$$ $$P=52.35N$$
(b) When the block slides down the wall, $f_k$, opposing the motion, is directed up the wall in reverse, supporting $P_y$ (image b). The constant velocity signifies that the net vertical force is $0$. So, $$P_y+f_k=mg$$ $$0.87P+0.125P=39$$ $$P=39.2N$$
