# The radius of planet A is half the radius of planet B. If the mass of A is MA, what must be the mass of B so that the value of g on B is half that of its value on A? - Science and Technology 1

Numerical

The radius of planet A is half the radius of planet B. If the mass of A is MA, what must be the mass of B so that the value of g on B is half that of its value on A?

#### Solution 1

The acceleration due to gravity of a planet is given as

$\text{g} = \frac{\text{GM}}{\text{r}^2}$

For planet A:

$\text{g}_\text{A} = \frac{\text{GM}_A}{\text{r}_\text{A}^2}$

For planet B:

$\text{g}_{B} = \frac{\text{GM}_\text{B}}{\text{r}_\text{B}^2}$

Now,

$\text{g}_\text{B} = \frac{1}{2} \text{g}_\text{A}$ ...(Given) or,

$\frac{\text{GM}_\text{B}}{\text{r}_\text{B}^2} = \frac{\text{G M}_\text{A}}{2 \text{r}_\text{A}^2}$

$\Rightarrow \text{M}_\text{B} = \frac{\text{M}_\text{A} \text{r}_\text{B}^2}{2 \text{r}_\text{A}^2}$

Given:
$\text{r}_\text{A} = \frac{1}{2} \text{r}_\text{B}$

$\Rightarrow \text{M}_\text{B} = \frac{\text{M}_\text{A} \text{r}_\text{B}^2}{2(\frac{1}{2} \text{r}_\text{B})^2} = 2 \text{M}_\text{A}$

Thus, the mass of planet B should be twice that of planet A.

#### Solution 2

radius of planet ‘A’ = RA, radius of planet ‘B’ = R
Mass of planet ‘A’ = MA, mass of planet ‘B’ = MB = ?
From given...

"R"_"A" = ("R"_"B")/2; "g"_"B" = 1/2 "g"_"A"

"g" = ("GM")/("R"^2);

∴ "g"_"A" = ("GM"_"A")/("R"_"A"^2);

∴ "g"_"B" = ("GM"_"B")/("R"_"B"^2)

("GM"_"B")/("R"_"B"^2)

("M"_"B")/("R"_"B"^2)  = 1/2(("GM"_"A")/(("RB"/2)^2))

("M"_"B")/("R"_"B"^2)  = 1/2 (4("GM"_"A")/(("R"_"B")^2))

"M"_"B" = 2 "M"_"A"

Concept: Acceleration Due to Gravity (Earth’s Gravitational Acceleration)
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