If you press an apple crate against a wall so hard that the crate cannot slide down the wall, what is the direction of (a) the static frictional force${\overrightarrow{\mathbf{f}}}_{\mathbf{s}}$on the crate from the wall and (b) the normal force ${\overrightarrow{\mathbf{F}}}_{\mathbf{N}}$on the crate from the wall? If you increase your push, what happens to (c)${\mathit{f}}_{\mathbf{s}}$, (d)${\overrightarrow{\mathbf{F}}}_{\mathbf{N}}$, and (e)${\mathit{f}}_{\mathbf{s}\mathbf{.}\mathbf{m}\mathbf{a}\mathbf{x}}$ ?

The press applied on the crate against a wall is so hard that the crate does not slide down.

To show all the forces, we have to draw a free-body diagram. Using this body diagram, we can observe the directions of the forces acting on it and their behavior if the applied press is increased on the crate.

Formula:

The static frictional force acting on a body,

${\mathbf{f}}_{\mathbf{s}\mathbf{.}\mathbf{max}}\mathbf{=}{\mathbf{\mu }}_{\mathbf{s}}{\mathbf{F}}_{\mathbf{N}}$ (1)

Free body diagram of blocks

From the above free body diagram acting on the crate, the static frictional force${\overrightarrow{\mathrm{f}}}_{\mathrm{s}}$ on the crate from the wall is directed to the up.

From the above free body diagram acting on the crate, the normal force${\overrightarrow{\mathrm{F}}}_{\mathrm{N}}=\mathrm{N}$ on the crate from the wall is to the left of the wall.

If you increase push P on the crate then,${\overrightarrow{\mathrm{f}}}_{\mathrm{s}}$ does not depend on the push P and so it remains the same.

The normal force${\overrightarrow{\mathrm{F}}}_{\mathrm{N}}$ is also increased if we increase push P on the crate.

As we increase push P, the normal force${\overrightarrow{\mathrm{F}}}_{\mathrm{N}}$ is also increased and the relation between${\overrightarrow{\mathrm{f}}}_{\mathrm{s}.\max }$ and normal force${\overrightarrow{\mathrm{F}}}_{\mathrm{N}}$ considering the equation (1).

${\overrightarrow{\mathrm{f}}}_{\mathrm{s}}$is proportional to the normal force${\overrightarrow{\mathrm{F}}}_{\mathrm{N}}$ , thus as${\overrightarrow{\mathrm{F}}}_{\mathrm{N}}$ increases,${\overrightarrow{\mathrm{f}}}_{\mathrm{s}.\max }$ it also increases.