Chapter 17.2, Problem 17PP

To determine

The object position and object height.

Answer to Problem 17PP

  $26.67\text{}\text{cm}$ from the mirror, and $1.86\text{cm}$ , respectively.

Explanation of Solution

Given:

The focal length of the concave mirror, $f=10.0\text{cm}$

The height of the image, $h_i=-3.0\text{cm}$ (negative sign denotes that the image is inverted)

The image distance from the mirror, $d_i=16.0\text{cm}$

Formula used:

Mirror equation,

  $\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}$

Where,

  $\begin{array}{l}f=\text{focal length}\\ d_i=\text{ image distance}\\ d_o=\text{object distance}\end{array}$

And,

Magnification, $\frac{h_i}{h_o}=-\frac{d_i}{d_o}$

Where,

  $\begin{array}{l}{h}_i=\text{image height}\\ h_o=\text{object height}\end{array}$

Calculation:

Using the given values in Mirror equation, the object distance will be,

  $\begin{array}{c}\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}\\ \frac{1}{10}=\frac{1}{16}+\frac{1}{d_o}\\ d_o=\frac{16\times 10}{16-10}\\ =26.67\text{cm}\end{array}$

Then, the object height will be,

  $\begin{array}{c}\frac{h_i}{h_o}=-\frac{d_i}{d_o}\\ \frac{-3.0}{h_o}=-\frac{16}{26.67}\\ h_o=5.0\text{cm}\end{array}$

Conclusion:

Hence, the object position and the object height is $26.67\text{}\text{cm}$ and $5.0\text{cm}$ , respectively.