# Crate on a Frictionless Incline

Example 5.6 (Serway and Beichner 126) makes the geometric
assumption that the angle of elevation equals the angle
between the crate's normal force and the force of gravity
exerted on the crate. A geometric proof of this fact follows.
In figure 1, $\ \overline{AB}$ corresponds to the inclined plane and
$\angle BAC$ the angle of elevation. The force of gravity on
the crate points in the direction $\overrightarrow{BC}$ and the normal
force in the direction $\overrightarrow{DB}.$
Fig. 1

i Definition. If two triangles $x$ and $y$ have equal angles,
they are similar and we write $x\sim y.$

ii Lemma. If two triangles $x$ and $y$ have two equal angles,
then their third angles are equal.

iii Lemma. If $x\sim y$ and $y\sim z$ then $x\sim z.$

iv Theorem. Let $\Delta ABC$ and $\Delta BCD$ be given, as in
figure 1, such that $\angle ACB=\angle BCD=\angle ABD=90^\circ.$
Then $\Delta ABC\sim\Delta BCD.$

Proof of iv.

0. Suppose the hypothesis.

1. $\angle BDC=\angle BDA$ since they are congruent.

2. $\angle CBD=\angle BAD$ by ii on 0 and 1.

3. $\Delta ABD\sim\Delta BCD$ by i on 0, 1, and 2.

4. $\angle BAC=\angle BAD$ since they are congruent.

5. $\angle ABC=\angle BDA$ by ii on 0 and 4.

6. $\Delta ABD\sim\Delta ABC$ by i on 0, 4, and 5.

7. $\Delta ABC\sim\Delta BCD$ by iii on 3 and 6.

\[\tag*{$\blacksquare$}\]

## Works Cited

Serway, Raymond A. and Robert J. Beichner. Physics for Scientists and Engineers. 5th Edition. Brooks/Cole, 2000. Print.