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CSIR-NET 2015 (22)

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CSIR-NET 2015

 

 

22. In order to have the same value of the surface acceleration due to gravity as on earth, a planet twice its radius must have a mean relative density of.

 

1)2.75                      2)5.5                    3)1.4                 4)2.25


Solution:

Formula,

Acceleration due to gravity, g=\frac{Gm}{r^2}---(1)

From the above problem, we need a relation between the radius and density.

So,

Density= \frac{mass}{volume}

mass=Density\times volume

m=\rho \times v ---(2)

Substitute equation (2) in equation (1)

g= \frac{G\times \rho\times v}{r^2}

then,

\frac{G\times \rho_1\times v_1}{r_1^2}= \frac{G\times \rho_2\times v_2}{r_2^2}---(3)

Volume of the Earth and planets are, v=\frac{4}{3}\pi r^3

Substitute the above volume information in equation (3)

\frac{G\times \rho_1\times \frac{4}{3}\pi r_1^3}{r_1^2}= \frac{G\times \rho_2\times \frac{4}{3}\pi r_2^3}{r_2^2}

Therefore simplifying the above equation we get,

\rho_1\times r_1=\rho_2\times r_2

\frac{\rho_2}{\rho_1}=\frac{r_1}{r_2}---(4)

From above problem

r_2=2\times r_1

\rho_1=5.5 gm/cc Earth density

Substitute above values in equation (4)

\frac{\rho_2}{\rho_1}=\frac{r_1}{2 \times r_1}

\frac{\rho_2}{\rho_1}=\frac{1}{2}

\rho_2=\frac{5.5}{2}

\rho_2=2.75 gm/cc

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