Question

Compute the position and diameter of the image of the Moon in a polished sphere of diameter 20
cm. The diameter of the Moon is 3500 km, and its distance from the Earth is 384 000 km,
approximately.

Answer 1

Answer:

Correct option is

C

15 mm

Here, angle subtended by moon

=

3⋅5×10

5

3500

radians

Also, focal length of the mirror,

f=R/2=1⋅5m

=1⋅5×10

3

mm

Required diameter of image

∴D=fθ=1⋅5×10

3

×

3⋅5×10

5

3500

=15mm

Answer 2

Given:

Diameter of a polished sphere, $d=20cm=0.2m$

Diameter of the moon $=3500km=3.5*10^6m$

Distance between earth and moon $=38300km=3.84*10^{8}m$

To Find:

Position & diameter of the image of the moon on the sphere, $h'$.

Solution:

Here, the radius of curvature of the convex mirror $=$ Radius of the sphere

                                                                                $R=\frac{d}{2} =0.1m$

Focal length, $f=\frac{R}{2} =0.05m$

Distance of the object, $u=-3.84*10^8m$

We first need to find out the distance of the image, $v$.

Using the mirror formula,

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

⇒ $\frac{1}{-3.84*10^8} +\frac{1}{v} =\frac{1}{0.05}$

⇒ $v=0.05m$

Using the definition of magnification,

$m=\frac{h'}{h} =-\frac{v}{u}$  

  ⇒ $\frac{h'}{3.5*10^6} =-\frac{5*10^{-2}}{-3.84*10^8}$

  ⇒ $h'=4.6*10^{-4}m$

  ⇒ $h'=0.46mm$

Hence, the diameter of the image of the moon is equal to 0.46mm formed 5 cm behind the mirror.