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## Sliding Block on Inclined Plane

### Introduction

The Inclined Plane motion problem is a classic problem in demonstrating the laws of motion. It necessitates
the use of force vectors. It also requires integration of the acceleration induced by the force vector sum,
to obtain the block displacement, `s`. Here we will use shorten the phrase "inclined plane" to the term "ramp".
The gravity force component that is perpendicular to the ramp surface
results in a sliding friction force that reduces the impact of the gravity force component that is parallel to
the ramp. For small ramp angles the friction force can be large enough that the block does not start to slide.

#### Graphic Summing of Vectors

This page shows how the parallel and perpendicular force vectors sum together to be equivalent to the downward
gravity force vector, `mg`. To sum two vectors graphically, we place the tail of an arrow representing the
second vector at the tip of the arrow representing the first vector.
Here the parallel and perpendicular force vectors are blue and they add up to the red downward gravity force vector
as they must.

### Equations

The forces on the block are:

`F_("parallel")=mg sin theta \ \ \ \("gravity force along ramp")`

`F_("perpendicular")=mg cos theta \ \ \ \("gravity force perpendicular to ramp")`

`mu=C_("friction") mgcos theta\ \ \ \ \("friction force")`

where `m` is the mass of the block,
`g` is the acceleration of gravity, `mu` is the frictional force,
and `C_("friction")` is a factor that determines the effectiveness of friction relative to the force applied
along the perpendicular to the ramp.

The equations of motion are:

`m (d^2s)/dt^2=mg sin theta-C_("friction") mgcos theta\ \ \ \ ("equation of motion")`

`s=("gt"^2)/2(sin theta-C_("friction")cos theta)) \ \ \ \ \("solution for s")`

where `s` is the block's distance along the ramp, `(d^2s)/dt^2` is the acceleration of the block and `t` is time.
The "equation of motion" is `"mass times acceleration=force"` which is Newton's law of motion. I've never been able to
figure out why teachers try to separate Newton's laws of motion into 3 statements when this one statement above
is not only much more concise but more precise.