f you weigh 100 pounds on Earth and go to a planet that is 1/2 the mass of earth and 1/2 the size of Earth, what is your new weight on that planet?

Question

Its related to newton law of gravity, but I just don't understand the math.

Answer 1

Newton's universal law of gravity states that the force of gravity between two masses, M and m, is this:

F = GMm / r²

Where F = force (in this case the person's weight)
G = universal gravitational constant (value is tiny, but not relevant)
M = one mass (say, of the Earth)
m = 2nd mass (say, of the person)
r = distance between the two centers of masses


We have 100 lbs = GMm/r² on Earth    (eq 1)

For the new planet, M₂ = M/2  (given)
Assuming "half the size" means "half the diameter" then the new planet has  r₂ = r/2
And the person's weight on the new planet will be: x = GM₂m/r₂² = G(M/2)m / (r/2)²   (eq 2)

The person's mass, m, stays the same, regardless of planet:
Solving equation (1) for m: m = 100*r₂ / (GM)
Solving equation (2) for m: m = (x*r₂/4) / (G*M/2)

When you set the right hand sides equal and solve for x (noticing that r, M, and G all cancel), you will get x=200 lbs

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As an alternate approach: notice that the weight varies in direct proportion to the planet's mass.  Also note that their weight varies inversely with the square of the distance to the planet's center, therefore the person's weight on the new planet is 100 lbs*(1/2)*(1/(1/2)²) = 100*2 = 200 lbs.

Newton's universal law of gravity states that the force of gravity between two masses, M and m,  is this:   

 

F = GMm / r²

 

Where F = force (in this case the person's weight)

          G = universal gravitational constant (value is tiny, but not relevant)

          M = one mass (say, of the Earth)

          m = 2nd mass (say, of the person)

           r = distance between the two centers of masses

 

 

We have 100 lbs =  GMm/r²  on Earth   (eq 1)

 

For the new planet,  M₂ = M/2  is given.

Assuming "half the size" means "half the diameter" then  the new planet has  r₂ = r/2

And the person's weight on the new planet will be:  x = GM₂m/r₂² = G(M/2)m / (r/2)²    (eq 2)

 

The person's mass m stays the same, regardless of planet:

Solving equation (1) for m:   m = 100*r² / (GM) 

Solving equation (2) for m:    m = (x*r²/4) / (G*M/2)

 

When you set the right hand sides equal and solve for x (noticing that r, M, and G all cancel), you will get  x=200 lbs 

 

-----------

As an alternate approach:  notice that the weight varies in direct proportion to the new planet's mass.  Also note that their weight varies inversely with the square of the distance to the new planet's center, therefore the person's weight on the new planet is 100 lbs*(1/2)*(1/(1/2)²) = 100*2 = 200 lbs.