Question

Consider a heavenly body whose mass is 3x10^24 kg (half that of the earth) and radius is 3200km
(half that of the earth). What is the acceleration due to gravity of surface of the body?
(Take ge = 9.8 m/s?)
​

Answer 1

Let the radius and mass of Earth be R and M respectively.

We have,

Mass of Earth, $M_{e} = M$

Radius of Earth, $R_{e} = R$

Mass of the Heavenly body, $M_{hb} = \frac{M_{e}}{2}$ ( Given in the question that the mass of the Heavenly body is half of that of Earth. )

Radius of the Heavenly body, $R_{hb} = \frac{R_{e}}{2}$ ( Given in the question that the radius of the Heavenly body is half of that of Earth. )

We know that,

$\boxed{\bold{g \; = \; \frac{Gravitational \: constant \times Mass \: of \: the \: body}{radius^{2}}}}$

Where,

Gravitational constant = G,

Mass of the body = m,

Radius of the body = r

Ratio of Earth's g to that of Heavenly body's g is:

$\Rightarrow \qquad \frac{\frac{\cancel{G} \cdot M_{e}}{{R_{e}}^{2}}}{\frac{\cancel{G} \cdot \frac{M_{e}}{2}}{{\left( \frac{R_{e}}{2} \right)}^{2}}}$

$\Rightarrow \qquad \frac{\frac{M_{e}}{(R_{e})^{2}}}{\frac{4M_{e}}{2(R_{e})^{2}}}$

$\Rightarrow \qquad \frac{\frac{M_{e}}{{(R_{e})}^{2}}}{ \frac{2M_{e}}{{(R_{e})}^{2}}}$

$\Rightarrow \qquad \frac{\cancel{M_{e}}}{\cancel{{(R_{e})}^{2}}} \cdot \frac{\cancel{{(R_{e})}^{2}}}{2 \times \cancel{M_{e}}}$

$\Rightarrow \qquad \frac{1}{2}$

Hence, The acceleration due to gravity of the Heavenly body is two times to that of Earth's.

Therefore,

$g_{heavenly body} = 2 \times g_{earth}$

$g_{heavenly body} = 2 \times 9.8$ ( Since the acceleration due to gravity of Earth is given as 9.8 m/s² )

$g_{heavenly body} = 19.6$

Answer:

$\huge{\bold{19.6 \; m/ s^2}}$

Answer 2

$\huge\bold\red{\underline{\underline{\mathfrak{Solution:}}}}$

As we know g = Gm/r²

g on earth = 9.8 m/s²

Mass of heavenly body is half of earth

Radius of heavenly body is half of earth

Therfore: 1/2(m)(G)/1/2(r²) = 9.8

1/2{(m)(G)/(r²) = 9.8

m(G)/(r²) = 9.8 × 2

Since Gm/r2 = g

g = 19.6 m/s²

Therefore the Acceleration due to gravity on the heavenly body is 19.8 m/s²