A body of mass $m$ is pressed against the ceiling with force $F$ directed at an angle $\alpha$ to the horizon. At what values of the coefficient of friction $\mu$ between the body and the ceiling will the body remain stationary?
Decision:
$Q$ is the ceiling reaction force, $F_{T}$ is the rest friction force. Since the body is assumed to be stationary, then
$F + mg + Q + F_{T} = 0$.
In this case it is obligatory
$|F_{T}| \geq \mu N$ (1)
Let $N$ be the force of normal pressure.
Considering for certainty that $0 < \alpha < \frac{ \pi }{2}$, we find that
$F \sin \alpha - mg - Q = 0$ (2)
In these equations, $F, Q$ and $F_{T}$ are positive. Solving equations (2) with respect to $Q$ and $F_{T}$, we obtain
$Q = F \sin \alpha - mg, F_{T} = F \cos \alpha$. (3)
Substituting expressions (3) for $|F_{T}|$ and $N = Q$ into inequality (1), we have
$F \cos \alpha < \mu (F \sin \alpha - mg).$
Let us solve this inequality with respect to $\mu$ given that
$F \sin \alpha \leq mg$
(otherwise the body is not pressed against the ceiling):
$\mu \leq \frac{F \cos \alpha}{F \sin \alpha - mg}$
Decision:
$Q$ is the ceiling reaction force, $F_{T}$ is the rest friction force. Since the body is assumed to be stationary, then
$F + mg + Q + F_{T} = 0$.
In this case it is obligatory
$|F_{T}| \geq \mu N$ (1)
Let $N$ be the force of normal pressure.
Considering for certainty that $0 < \alpha < \frac{ \pi }{2}$, we find that
$F \sin \alpha - mg - Q = 0$ (2)
In these equations, $F, Q$ and $F_{T}$ are positive. Solving equations (2) with respect to $Q$ and $F_{T}$, we obtain
$Q = F \sin \alpha - mg, F_{T} = F \cos \alpha$. (3)
Substituting expressions (3) for $|F_{T}|$ and $N = Q$ into inequality (1), we have
$F \cos \alpha < \mu (F \sin \alpha - mg).$
Let us solve this inequality with respect to $\mu$ given that
$F \sin \alpha \leq mg$
(otherwise the body is not pressed against the ceiling):
$\mu \leq \frac{F \cos \alpha}{F \sin \alpha - mg}$