A hypothetical planet has a radius 1.1 times that of Earth, but has the same mass. What is the acceleration due to gravity near its surface?
Express your answer using two significant figures.
To find the acceleration due to gravity near the surface of the hypothetical planet, we can use the formula for gravitational acceleration:
\[g = \frac{{GM}}{{r^2}}\]
Where:
g = acceleration due to gravity
G = gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^−1 s^−2)
M = mass of the planet
r = radius of the planet
In this case, the radius of the hypothetical planet is given as 1.1 times that of Earth, but it has the same mass. Let's assume the mass of Earth as M_e and use this information to find the mass of the hypothetical planet.
Since the mass of the hypothetical planet is the same as Earth, we have:
\[M = M_e\]
Now, let's calculate the radius of the hypothetical planet.
The radius of Earth is typically given as 6371 km. We can calculate the radius of the hypothetical planet by multiplying Earth's radius by the scale factor of 1.1:
\[r = 1.1 \times 6371 \text{ km}\]
Now, we need to convert the radius to meters by multiplying it by 1000:
\[r = 1.1 \times 6371 \times 1000 \text{ m}\]
Substituting the values of M and r into the formula for gravitational acceleration, we can calculate g:
\[g = \frac{{(6.67430 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}) \times M_e}}{{(1.1 \times 6371 \times 1000)^2 \text{ m}^2}}\]
Calculating g will give us the result. Rounding the answer to two significant figures, we get the final result.