Question

If a planet has the same mass as the earth, but has twice the radius, how does the surface gravity, g, compare to g on the surface of the earth? If a planet has the same mass as the earth, but has twice the radius, how does the surface gravity, g, compare to g on the surface of the earth? It is twice as large. It is half as large. It is the same. It is four times smaller.

Answer 1

Answer:

$\large \boxed{\text{Divided by four}}$

Explanation:

The formula for gravitational force (F) is

$F = \dfrac{GmM}{d^{2} }$

where m and M are the masses of the two objects, d is the distance between their centres, and G is the gravitational constant.

If we hold m, and M constant, we can write

$F = \dfrac{k}{d^{2}}$

where k = Gmm

Thus, gravitational force is inversely proportional to the square of the distance between the objects.

 Let d₂ = the radius of the planet

And  d₁ = the radius of the Earth. Then

d₂ = 2d₁

$\dfrac{F_{2}}{F_{1}} = \dfrac{k}{d_{2}^{2}} \div \dfrac{k}{d_{1}^{2}}=\dfrac{d_{1}^{2}}{d_{2}^{2}} =\dfrac{d_{1}^{2}}{(2d_{1})^{2}} = \dfrac{1}{4}\\\\\text{If the radius of the planet is doubled, its surface gravity is \large \boxed{\textbf{divided by four}} }$

Answer 2

On a planet which has the same mass as the earth, but has twice the radius of the earth, the surface gravity, g on the planet is four times smaller than that of earth. Choice D.

According to Newton's law of universal gravitation.

The gravitational force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of their distance apart.

Mathematically, Newton's law of gravitational attraction can be written as follows;

• F = (GmM) / d².

According to the question, the planet in question has the same mass as earth; As such the gravitational force experienced by any object can only be different from that of earth as a result of the difference in the planet's radius compared to earth.

Since the gravitational force, F can be expressed as the weight of the object;

• mg = GmM/d²

• g = GM/d².

Since, the object has the same mass as the earth, but has a radius twice that of the earth;

• d(planet) = 2d(earth)

• d(planet) = 2d(earth)g(earth)/g(planet) = Gm/d² × (2d²)/Gm

d(planet) = 2d(earth)g(earth)/g(planet) = Gm/d² × (2d²)/GmBy cross product;

• d(planet) = 2d(earth)g(earth)/g(planet) = Gm/d² × (2d²)/GmBy cross product;g(earth) = 4g(planet).

d(planet) = 2d(earth)g(earth)/g(planet) = Gm/d² × (2d²)/GmBy cross product;g(earth) = 4g(planet).g(planet) = g(earth)/4

Ultimately, the surface gravity, g on the planet is four times smaller than that of earth.

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