Question

What is the solid angle subtended
by the moon at any point of the Earth, given
the diameter of the moon is 3474 km and its
distance from the Earth 3.84x108m.
​

Answer 1

Answer:

See below.

Explanation:

Diameter of moon  = 3474 km.

Therefore, radius is 1737 km. The area will be,

$A_m = \pi r^2 = (3.14)(1737)^2 = 9.473 \times 10^6 \ m^2$

The distance from the Earth is given as,

$= 3.84 \times 10^8 \ m$

Squaring this value,

$r^2 = (3.84 \times 10^8)^2 = 14.74 \times 10^{16} \ m^2$

Now the solid angle in steradians is,

$\Omega = \frac{9.473 \times 10^6}{14.74 \times 10^{16}} = 0.642 \times 10^{-10}$

And that should be the answer.

Answer 2

The angle subtended by the moon on the earth is 0.52° (approx).

Given:
Diameter of the moon $=3474km$

Distance of the moon from the earth $=3.84$ × $10^{8}m$

To find:

Angle subtended by the moon at the point on earth.

Solution:

We know, the relation between the subtended angle $(\alpha )$, distance from the object $(D)$ and the diameter $(d)$ of the object is given by

$d=2Dsin(\frac{\alpha }{2} )$

We have,

$d=3474km=3474$ × $10^{3}m$

$D=3.84$ × $10^{8}m$

Substituting the known values, we get

$3474*10^{3}=2(3.84*10^{8} ) sin(\frac{\alpha }{2} )$

$452.34$ × $10^{-5}=sin(\frac{\alpha }{2} )$

$sin(\frac{\alpha }{2} )=0.0045234$

$\frac{\alpha }{2} =sin^{-1}(0.0045234)$

$\frac{\alpha }{2}=0.26^{o}$

$\alpha =0.52^{o}$

Final answer:

Hence, the angle subtended by the moon on the earth is 0.52° (approx).