Calculate weight and mass on Planet X

Question

Planet X has the same mass as the Earth, but 1/2 the radius. (Planet X is more dense than the Earth). Suppose your mass is some 55 kg, so your weight at the sea level on Earth is about 120 lb.

 

1. What would be your mass on a surface of Planet X?
2. What would be your weight on a surface of Planet X?
3. Suppose you stand on a top of a very tall ladder, and so your distance from the center of the Planet X is now equal to the Earth radius. What would be your weight there?

Answer 1

Gravitational force is given by

 

F= GMm/r².

 

G is the gravitational constant, M and m are the masses of the two bodies exerting gravitational force on each other, and r is the distance between their centers of mass. On Earth, M becomes the mass of earth and r becomes the radius of earth. We normally lump GM/r² together as g=9.8 m/s² (i.e. weight is just gravitational force at Earth's surface):

 

F_weight=G M_earthm_person/r²_earth = (GM_earth/r²_earth)m_person = gm.

 

This question is asking what happens to F as you scale the constituent parts of g.

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1. Your mass is constant, always. No change here.

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2. On planet X, there would be a new acceleration due to gravity (call it g_X), given by

 

g_x=GM_X/r_x²

 

We know M_x=M_earth and r_x=r_earth/2 (so r_x²=r²_earth/2² = r²_earth/4). Plugging these in:

 

g_x=[GM_earth]/[r²_earth/4] = [4GM]/[r²] = 4*g.

 

So, F_x=mg_x=(m)(4g_earth) =4(mg_earth)=4 F_earth

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3. The same idea follows, except the ladder causes the r on planet X to increase back to r_earth. Your weight would be the same at this height since the mass of X is the same as Earth's.