Ratio of two objects and its gravitational force and mass

Question

The gravitational force between two objects is directly proportional to the product of their masses of the two objects and inversely proportional to the square of the distance between the two objects. Consider two balls of clay placed on a table, initially equal in mass. Half of one ball of clay is pulled off and rolled into the other ball, and the distance between two balls is then triple. The gravitational force between the two balls after they have been resized and moved is what fraction of the gravitational force between the two balls before they were resized and moved.

Answer 1

Let F = gravitational force 

    m₁ = mass of first ball

    m₂ = mass of second ball

    d = distance between the balls

 

F = k(m₁)(m₂)/d²

 

Let F₁ be the force when m₁ = m₂     F₁ = k(m₁)²/d²

 

Let F₂ be the force when mass of first ball is (1/2)m₁ and mass of second ball is (3/2)m₁, and the original distance is tripled.

 

   F₂ = k(1/2)(m₁)(3/2)(m₁)/(3d)² = k(3/4)(m₁)²/(9d²)

 

                                              =[ k(m₁)²/d² ](3/4)/9 = (1/12)F₁

 

F₂ is 1/12 of F₁

 

                                              

Answer 2

The first sentence states Newton's law of universal gravitation, so write that down.  Perhaps in your book the masses are called m₁ and m₂, and the distance is called r.

 

The second and third sentences give an initial and final situation.  You could summarize them in a table like this:

 

              m₁        m₂        r

initial       m         m         d

final     (1/2)m  (3/2)m   3d

 

Then you can add a column headed by F.  Plug in the initial masses and distance, and you'll get an expression for initial force involving G, m, and d.  Plug in the final masses and distance, and you'll get another expression, for final force.

 

Divide final force by initial force, cancel the G, m, and d, and you'll get the fraction.