Question

A straight ladder is leaning against the wall of a house. The ladder has rails 4.90 m long, joined by rungs 0.410 m long. Its bottom end is on solid but sloping ground so that the top of the ladder is 0.690 m to the left of where it should be, and the ladder is unsafe to climb. You want to put a flat rock under one foot of the ladder to compensate for the slope of the ground. (a) What should be the thickness of the rock? (b) Does using ideas from this chapter make it easier to explain the solution to part (a)? Explain your answer.

Short answer

The thickness of rock is 0.0574 m.

Step-by-step solution

Identification of given data

The length of rails of ladder is $L=4.90m$

The length of rungs joined is $l=0.410m$

The position for the top of ladder is $d=0.690m$

Conceptual Explanation

For small angles, the angular displacement is described as the ratio of arc length to the radius of the circular path.

Determination of thickness of rock

The angle for the top of the ladder is given as:

$\theta =\frac{d}{L}$

For$d=0.690m$ and L=4.90 m , the above equation gives-

$$\begin{gathered} \theta =\frac{0.690m}{4.90m} \\ \theta =0.14rad \end{gathered}$$

The thickness of rock is given as:

$t=l·\theta$

For $\theta =0.14rad$ and $l=0.410m$, the above equation becomes-

$$\begin{gathered} t=\left(0.410m\right)\left(0.14rad\right) \\ t=0.0574m \end{gathered}$$

Therefore, the thickness of rock is 0.0574 m .