To solve this problem, we need to find the acceleration due to gravity on a planet with given mass and diameter relative to Earth.$\\$Given:$\\$• Mass of the planet $M_p = \frac{1}{10} M_e$ (where $M_e$ is the mass of Earth)$. \\$• Diameter of the planet $D_p = \frac{1}{2} D_e$ (where $D_e$ is the diameter of Earth)$. \\$The acceleration due to gravity $g$ on a planet is given by:$\\ g = \frac{G M}{R^2} \\$Where:$\\$• $G$ is the gravitational constant.$\\$• $M$ is the mass of the planet.$\\$• $R$ is the radius of the planet.$\\$Since the diameter is half, the radius of the planet $R_p = \frac{1}{2} R_e$ (where $R_e$ is the radius of Earth)$. \\$Substitute the given values into the formula for gravity:$\\ g_p = \frac{G \left(\frac{1}{10} M_e\right)}{\left(\frac{1}{2} R_e\right)^2} \\$Simplify the expression:$\\ g_p = \frac{G \cdot \frac{1}{10} M_e}{\frac{1}{4} R_e^2} \\ g_p = \frac{4G \cdot \frac{1}{10} M_e}{R_e^2} \\ g_p = \frac{4}{10} \cdot \frac{G M_e}{R_e^2} \\ g_p = \frac{2}{5} \cdot g_e \\$Where $g_e$ is the acceleration due to gravity on Earth (approximately $9.8 \,$m/s$^2$).$\\$Calculate $g_p: \\ g_p = \frac{2}{5} \cdot 9.8 \\ g_p = 3.92 \,$m/s$^2 \\$Therefore, the acceleration due to gravity on the planet is $3.92 \,$m/s$^2$. This corresponds to Option 2.