Question

if radius of planet r is increased to 3r, how must density d change to keep the acceleration due to gravity g constant

Answer 1

$$\begin{gathered} accelerartionduetogravity \\ g=\frac{GM}{R^2} \\ g\text{'}=g \\ \frac{GM\text{'}}{R{\text{'}}^2}=\frac{GM}{R^2} \\ mass=voulme\times density \\ \frac{v\text{'}\times d\text{'}}{R{\text{'}}^2}=\frac{v\times d}{R^2} \\ \frac{{\displaystyle \frac{4}{3}}\pi R{\text{'}}^3\times d\text{'}}{R{\text{'}}^2}=\frac{{\displaystyle \frac{4}{3}}\pi R^3\times d}{R^2} \\ R\text{'}d\text{'}=Rd \\ 3rd\text{'}=rd \\ \frac{d\text{'}}{d}=\frac{1}{3} \\ densitymustdecreasestothe1/3oftheinitialvalue. \end{gathered}$$