# Math Help - [SOLVED] Show that no force can open the door?

1. ## [SOLVED] Show that no force can open the door?

There is a small interval between the bottom of a door and the floor, and a wedge of no appreciable weight has been thrust into this interval, the coefficient of friction between its base and the floor being known. If the angle of the wedge is smaller than a certain amount, show that no force can open the door, the slant edge of the wedge being supposed smooth.

2. ## Solution

Here's how to do it:

Spoiler:

Let the force exerted by the door on the wedge be of magnitude $F_0$ . This force must act perpendicular to the surface of the wedge as it is smooth.

The wedge has a reaction force (from the ground) acting vertically upwards on it and a friction force $F_{\mu}$ acting horizontally on it.

In order for the wedge to remain where it is we must have (resolving vertically)

$R = F_0 \cos \alpha$

and (resolving horizontally)

$F_{\mu} \ge F_0 \sin \alpha$

but since

$F_{\mu} = \mu R$ then

$\mu F_0 \cos \alpha \ge F_0 \sin \alpha$,

$\color[rgb]{0,0,1} \boxed{\Rightarrow \alpha \le \arctan \mu}$.