Chapter 9.2, Problem 51E

To determine

To find: The angle formed between the inclined plane and the horizontal axis.

Answer to Problem 51E

The angle of inclination of the plane is $23.6°$.

Explanation of Solution

Given:

The weight of the package is 200 lb.

The force required to keep the package from sliding is 80 lb.

Calculation:

The Figure 1 which shown below the package exerts a force of 200 lb in downward direction.

From Figure 1, it can be observed that the box exerts force into two components u and v. The angle of inclination is $\theta$.

The angle formed between u and w is,

$\theta =90°-\theta$

The magnitude of the part of force that requires to keeps the box rolling down.

$\begin{array}{c}F=\left|u\right|\\ =\text{component}\text{of}w\text{along}v\\ =w\cos \theta \end{array}$

Substitute 200 for F and $90°-\theta$ for $\theta$ in the above formula.

$\begin{array}{l}80=200\times \cos \left(90°-\theta \right)\\ \cos \left(90°-\theta \right)=\frac{80}{200}\\ \sin \theta =\frac{4}{10}\end{array}$

Simplify further,

$\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{4}{10}\right)\\ =23.57°\\ \approx 23.6°\end{array}$

Thus, the angle of inclination of the plane is $23.6°$.