Question

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

Answer 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.

Answer 2

radius of planet ‘A’ = R_A, radius of planet ‘B’ = R_B 
Mass of planet ‘A’ = M_A, mass of planet ‘B’ = M_B = ? 
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"`