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प्रश्न
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?
उत्तर १
The acceleration due to gravity of a planet is given as
\[\text{g} = \frac{\text{GM}}{\text{r}^2}\]
For planet A:
For planet B:
\[\text{g}_{B} = \frac{\text{GM}_\text{B}}{\text{r}_\text{B}^2}\]
\[\text{g}_\text{B} = \frac{1}{2} \text{g}_\text{A}\] ...(Given) or,
\[\frac{\text{GM}_\text{B}}{\text{r}_\text{B}^2} = \frac{\text{G M}_\text{A}}{2 \text{r}_\text{A}^2}\]
\[\Rightarrow \text{M}_\text{B} = \frac{\text{M}_\text{A} \text{r}_\text{B}^2}{2 \text{r}_\text{A}^2}\]
\[\Rightarrow \text{M}_\text{B} = \frac{\text{M}_\text{A} \text{r}_\text{B}^2}{2(\frac{1}{2} \text{r}_\text{B})^2} = 2 \text{M}_\text{A}\]
उत्तर २
radius of planet ‘A’ = RA, radius of planet ‘B’ = RB
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"`
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