Question

If a planet consists of a satellite whose mass and radius were both half that of the earth, then the acceleration due to gravity at the surface of the planet would be
$\begin{align}
  & \text{A}\text{. 5}\text{.0m}{{\text{s}}^{-2}} \\
 & \text{B}\text{. 6}\text{.5m}{{\text{s}}^{-2}} \\
 & \text{C}\text{. 7}\text{.9m}{{\text{s}}^{-2}} \\
 & \text{D}\text{. 19}\text{.6m}{{\text{s}}^{-2}} \\
\end{align}$

Answer 1

Hint: Obtain the expression for acceleration due to gravity for a planet. Find it for earth and then find it for the other planet by using the given conditions in the question. Compare these two to find the required answer.

Complete step-by-step answer:
Let the mass of earth is M and the radius of earth is R.
The acceleration due to gravity on the surface of earth can be given by the mathematical expression,
$g=\dfrac{GM}{{{R}^{2}}}$
Where, g is the acceleration due to gravity and G is the universal gravitational constant with value, $G=6.67\times {{10}^{-11}}N{{m}^{2}}k{{g}^{-2}}$
Now, for the other planet the mass is half the mass of earth and the radius is half the radius of earth.
So, we can write the mass the planet as,
${M}'=\dfrac{M}{2}$
Also, the radius of the planet can be given as,
${R}'=\dfrac{R}{2}$
Equation for acceleration due to gravity of the planet will be,
${g}'=\dfrac{G{M}'}{{{{{R}'}}^{2}}}$
By putting the values of mass and radius on the equation for acceleration due to gravity, we get that,
 $\begin{align}
  & {g}'=\dfrac{G\left( \dfrac{M}{2} \right)}{{{\left( \dfrac{R}{2} \right)}^{2}}} \\
 & {g}'=2\dfrac{GM}{{{R}^{2}}} \\
 & {g}'=2g \\
\end{align}$
Now, the acceleration due to gravity at the surface of earth has a value, $g=10m{{s}^{-2}}$ .
Putting this value on the above equation, we get,
${g}'=2g=2\times 9.8=19.6m{{s}^{-2}}$
So, the value of the acceleration due to gravity on the surface of the planet will be $19.6m{{s}^{-2}}$
The correct option is, (D).

Note: Acceleration due to gravity of a planet is directly proportional to the mass of the planet and inversely proportional to the square of the radius of the planet. So, the higher the mass of the planet, the higher is the acceleration due to gravity and lower the radius of the planet, the higher is the acceleration due to gravity.