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Understanding of Field

Find the rela...

Question

Find the relation between the gravitational field on the surface of two planets A and B of masses $m_{A}, m_{B}$ and radii $R_{A}$ and $R_{B}$ , respectively, if
a. they have equal mass,
b. they have equal (uniform) density.

a. For

m_{A} = m_{B}, \frac{E_{A}}{E_{B}} = \frac{R_{B}^{2}}{R_{A}^{2}}

b. For

ρ_{A} = ρ_{B}, \frac{E_{A}}{E_{B}} = \frac{R_{A}}{3 R_{B}}

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a. For

m_{A} = m_{B}, \frac{E_{A}}{E_{B}} = \frac{R_{B}^{2}}{R_{A}^{2}}

b. For

ρ_{A} = ρ_{B}, \frac{E_{A}}{E_{B}} = \frac{R_{A}}{R_{B}}

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a. For

m_{A} = m_{B}, \frac{E_{A}}{E_{B}} = \frac{R_{B}^{2}}{R_{A}^{2}}

b. For

ρ_{A} = ρ_{B}, \frac{E_{A}}{E_{B}} = \frac{2 R_{A}}{R_{B}}

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a. For

m_{A} = m_{B}, \frac{E_{A}}{E_{B}} = \frac{R_{B}^{2}}{R_{A}^{2}}

b. For

ρ_{A} = ρ_{B}, \frac{E_{A}}{E_{B}} = \frac{2 R_{A}}{2 R_{B}}

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Solution

The correct option is A a. For $m_{A} = m_{B}, \frac{E_{A}}{E_{B}} = \frac{R_{B}^{2}}{R_{A}^{2}}$
b. For $ρ_{A} = ρ_{B}, \frac{E_{A}}{E_{B}} = \frac{R_{A}}{R_{B}}$ .
Let $E_{A}$ & $E_{B}$ be tje gravitational field intensity on surface of A and B, then
$E_{A} = \frac{G M_{A}}{R_{A}^{2}} = \frac{G \frac{4}{3} π R_{A}^{3} ρ_{A}}{R_{A}^{2}} = \frac{4 G π ρ_{A} R_{A}}{3}$
Similarly, $E_{B} = \frac{G M_{B}}{R_{B}^{2}} = \frac{4 G π ρ_{B} R_{B}}{3}$
a) If $m_{A} = m_{B}$
$\frac{E_{A}}{E_{B}} = \frac{R_{B}^{2}}{R_{A}^{2}}$
b) For $ρ_{A} = ρ_{B}$
$\frac{E_{A}}{E_{B}} = \frac{R_{A}}{R_{B}}$

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Find the relation between the gravitational field on the surface of two planets A and B of masses mA,mB and radii RA and RB, respectively, ifa. they have equal mass,b. they have equal (uniform) density.

The correct option is A a. For mA=mB,EAEB=R2BR2Ab. For ρA=ρB,EAEB=RARB.Let EA & EB be tje gravitational field intensity on surface of A and B, then EA=GMAR2A=G43πR3AρAR2A=4GπρARA3Similarly, EB=GMBR2B=4GπρBRB3a) If mA=mBEAEB=R2BR2Ab) For ρA=ρBEAEB=RARB

Find the relation between the gravitational field on the surface of two planets A and B of masses $m_{A}, m_{B}$ and radii $R_{A}$ and $R_{B}$ , respectively, if
a. they have equal mass,
b. they have equal (uniform) density.