Chapter 17, Problem 81A

To determine

The focal length of the convex mirror used.

Answer to Problem 81A

  $-72\text{cm}$

Explanation of Solution

Given:

The height of the image, $h_i=\frac{1}{2}\left(\text{height of the object}\right)=\frac{h_o}{2}$

The image distance from the mirror, $d_i=-36.0\text{cm}$ (negative sign denotes that the image is formed behind the mirror)

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 formula of magnification, the object distance will be,

  $\begin{array}{c}\frac{h_i}{h_o}=-\frac{d_i}{d_o}\\ \frac{\left(\frac{h_o}{2}\right)}{h_o}=-\frac{\left(-36\right)}{d_o}\\ d_o=72\text{cm}\end{array}$

Then, using the given values in Mirror equation, the focal length of the convex mirror will be,

  $\begin{array}{c}\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}\\ \frac{1}{f}=\frac{1}{\left(-36\right)}+\frac{1}{72}\\ f=\frac{72}{-2+1}\\ =-72\text{cm}\end{array}$

Conclusion:

Therefore, the focal length of the required convex mirror is $-72\text{cm}$ .