Final answer:
The acceleration due to gravity on another planet with four times Earth's mass and twice Earth's radius is equal to Earth's gravity, so the value of n is 1.
Step-by-step explanation:
To calculate the acceleration due to gravity on another planet, we can apply Newton's law of universal gravitation, which states that the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers.
For a planet with a mass M and radius R, and an object with mass m on its surface, the force of gravity is given by:
F = G (m * M) / R^2, where G is the gravitational constant.
The weight of the object, which is the force due to gravity, is also w = m * g, where g is the acceleration due to gravity. Equating the two expressions for the force gives us:
m * g = G (m * M) / R^2
Now, cancelling the mass m from both sides (since it's not zero) gives us an expression for the acceleration due to gravity:
g = (G * M) / R^2
On Earth, this acceleration is approximately 9.80 m/s².
To find the acceleration due to gravity on the other planet, let's assume its mass is 4M₀ (where M₀ is Earth's mass) and its radius is 2R₀ (where R₀ is Earth's radius).
Plugging these values into the above formula gives us:
g' = (G * 4M₀) / (2R₀)^2
= G * 4M₀ / 4R₀^2
= G * M₀ / R₀^2, which is the gravity on Earth, g.
Therefore, the acceleration due to gravity on the other planet is the same as on Earth, and n = 1.