Chapter 23, Problem 18P

Some rearview mirrors produce images of cars to your rear that are smaller than they would be if the mirror were flat. Are the mirrors concave or convex? What is a mirror’s radius of curvature if cars 16.0 m away appear 0.33 their normal size?

To determine

The radius of curvature of the mirror.

Answer to Problem 18P

Solution:

The nature of mirror is convex mirror as focal length is negative and its radius of curvature will be $15.8\text{m}$.

Explanation of Solution

Given Info:

Image is upright.

Let height of object be $+h_0$

Height of image $h_i=+0.33h_o$

Object distance $u=+16.0\text{m}$

Image distance = ‘v’

Let the focal length of mirror be ‘f’

Formula Used:

(i)Apply focal length formula for mirror

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

(ii)Magnification of mirror $m=\frac{h_i}{h_0}=-\frac{v}{u}$

Here, all alphabets are in their usual meanings.

Calculation:

Apply magnification of mirror $m=\frac{h_i}{h_0}=-\frac{v}{u}$

$\begin{array}{l}\frac{+0.33h_o}{+h_0}=-\frac{v}{+16.\text{0}\text{m}}\\ or,v=-5.28\text{m}\end{array}$

Apply focal length formula for mirror

$\begin{array}{l}\frac{1}{f}=\frac{1}{v}+\frac{1}{u}\\ \frac{1}{f}=\frac{1}{\text{-5}\text{.28}\text{m}}\text{+}\frac{\text{1}}{\text{+16}\text{.0}\text{m}}\\ f=-7.88\text{m}\end{array}$

Since the radius of curvature R = 2f

So, the radius of curvature R = $2\times f=2\times \left(\text{-7}\text{.88}\text{m}\right)\text{=-15}\text{.8}\text{m}$