To remove a cork stuck in the neck of a thermos, you poke a thin awl into the neck of the thermos. At what angle $\theta$ to the axis of the thermos can the awl be poked in without fear that the cork will fall through the thermos? The coefficient of friction of the cork against the walls of the neck is $\mu = 0.5$.
Decision:
Suppose that when we poke the awl in, we act on the cork with a force $F$. This force can be decomposed into two components: along the axis of the neck, equal to $F \cos \theta $, and perpendicular to it, equal to $F \sin \theta $. If the cork remains in equilibrium, then $F \cos \theta = F_{T}$, where $F_{T}$ is the frictional force. It's clear that $F_{T} \geq \mu N$, where $N$ is the force of the normal pressure of the cork on the walls of the neck. Note that $N \leq F \sin \theta$, since the cork is in compression and presses on the neck of the bottle even at $F = 0$. The condition that guarantees that the cork won't fail is $F \cos \theta \geq \mu F \sin \theta$, i.e., $ \theta \leq arcctg 0.5 \approx 63^{\circ}$.
Decision:
Suppose that when we poke the awl in, we act on the cork with a force $F$. This force can be decomposed into two components: along the axis of the neck, equal to $F \cos \theta $, and perpendicular to it, equal to $F \sin \theta $. If the cork remains in equilibrium, then $F \cos \theta = F_{T}$, where $F_{T}$ is the frictional force. It's clear that $F_{T} \geq \mu N$, where $N$ is the force of the normal pressure of the cork on the walls of the neck. Note that $N \leq F \sin \theta$, since the cork is in compression and presses on the neck of the bottle even at $F = 0$. The condition that guarantees that the cork won't fail is $F \cos \theta \geq \mu F \sin \theta$, i.e., $ \theta \leq arcctg 0.5 \approx 63^{\circ}$.