Question

A newly-discovered planet ("Kardashia") has a mass exactly 4.0 times that of Earth. A particle on the surface of Kardashia has the same weight that it would have on the surface of Earth. Determine the radius of Kardashia, in terms of the radius of Earth (RE ).A) .5 REB) 2 REC) 2.5 RED) 3 REE) 3.5 RE

Answer 1

Answer:

$r_k=0.5r_e$

Explanation:

$M_e$ = Mass of Earth

$M_k$ = Mass of Kardashia = $4M_e$

$R_e$ = Radius of Earth

$R_k$ = Radius of Kardashia

Gravitational force of Earth on object

$F_e=\frac{GM_em}{r_e^2}$

Gravitational force of Kardashia on object

$F_k=\frac{GM_km}{r_k^2}\\\Rightarrow F_k=\frac{G4M_e}{r_k^2}$

The gravitational force i.e., the weight of the body is same on both the planets

$F_e=F_k$

If the same particle is used then the mass will also be equal

Dividing the forces

$\frac{F_e}{F_k}=\frac{\frac{GM_em}{r_e^2}}{\frac{G4M_e}{r_k^2}}\\\Rightarrow 1=4\frac{r_k^2}{r_e^2}\\\Rightarrow r_k^2=\frac{1}{4}r_1^2\\\Rightarrow r_k=\frac{1}{2}r_e\\\Rightarrow r_k=0.5r_e$

The radius of the planet Kardashia is half of the radius of Earth