Question

Consider a planet in some solar system which has a mass double the mass of the earth and density equal to the average density of the earth. An object weighing W on the earth will weight

विकल्प

• W

• 2 W

• W/2

• 2^1/3 W at the planet.

Answer 1

2^1/3 W at the planet 

The weight of the object on the Earth is \[W = m\frac{G M_e}{{R_e}^2}\]

Here, m is the actual mass of the object; M_e is the mass of the earth and R_e is the radius of the earth.
Let R_p be the radius of the planet.
Mass of the planet, \[M_p = 2 M_e\]

If \[\rho\] is the average density of the planet then

\[\frac{4}{3}\pi {R_p}^3 \times \rho = 2 \times \left( \frac{4}{3}\pi {R_e}^3 \times \rho \right)\]

\[ \Rightarrow R_p = \left( 2 \right)^\frac{1}{3} R_e\] 

Now, weight of the body on the planet is given by

\[W_p = m\left( \frac{G M_p}{{R_p}^2} \right) = m\left( \frac{2G M_e}{2^\frac{2}{3} {R_e}^2} \right)\]

\[ \Rightarrow W_p = 2^\frac{1}{3} \times m\left( \frac{G M_e}{{R_e}^2} \right)\]

\[ \Rightarrow W_p = 2^\frac{1}{3} \times W\]