Answer:
[tex]r_k=0.5r_e[/tex]
Explanation:
[tex]M_e[/tex] = Mass of Earth
[tex]M_k[/tex] = Mass of Kardashia = [tex]4M_e[/tex]
[tex]R_e[/tex] = Radius of Earth
[tex]R_k[/tex] = Radius of Kardashia
Gravitational force of Earth on object
[tex]F_e=\frac{GM_em}{r_e^2}[/tex]
Gravitational force of Kardashia on object
[tex]F_k=\frac{GM_km}{r_k^2}\\\Rightarrow F_k=\frac{G4M_e}{r_k^2}[/tex]
The gravitational force i.e., the weight of the body is same on both the planets
[tex]F_e=F_k[/tex]
If the same particle is used then the mass will also be equal
Dividing the forces
[tex]\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[/tex]
The radius of the planet Kardashia is half of the radius of Earth
