George W. answered • 01/04/21
Physical Science Blogger; Stock Options Trader; AP Physics Tutor
Hello Mallory!
This question regards the gravitational force of attraction that exists between two objects with well-defined masses. Before answering the question, let's think conceptually about the question. If F = ( m )( a ), where F = Force, m = Mass in kilograms, and a = Acceleration in meters per second squared, a particle of mass m1 will accelerate towards earth with an acceleration of g = ( 9.8 m / s^2 ). If the mass of this particle were doubled, would its acceleration towards the earth double? Well, the answer is no, and here's why.
Keep in mind that forces occur IN PAIRS. The falling particle has a LARGE force exerted upon it by the earth; however, the particle exerts an equal and opposite force upon the earth! How is this possible? Well, let's see what mathematics tells us. If F1 = F2, then ( m1 )( a1 ) = ( m2 )( g ), where m1 = mass of earth, m2 = mass of the particle, a = acceleration of the earth towards the falling object, and g = gravitational constant of acceleration under which the falling particle accelerates towards earth. Needless to say, m1 is much larger than m2. Thus, when we solve for a1, we have a1 = ( m2 / m1 )( g ). Thus, the rate at which the earth accelerates towards falling bodies is practically negligible.
Now, let's imagine that two particles are falling towards earth. Each particle has a mass of m1, and they fall towards earth under the influence of a force F = ( m1 )( g ). If these masses combined, we'd have 2F = ( 2m1 )( g ). Therefore, the force exerted by the earth upon our new mass has doubled, but the force that our new mass exerts upon the earth would double as well. Thus, if F1 = F2, then 2F1 = 2F2, and ( 2F2 / 2m1 ) still equals g!
Furthermore, a quantity of work ( W ) is done upon falling objects. If work ( W ) = Fd, where F = Force and d = distance, work has the unit of energy known as Joules ( J ). If an object of mass m1 falls from some height = h, it's initial potential energy ( PE ) = mgh. Doubling the mass would double the initial potential energy ( PE ) of the object. If the resulting force is doubled, then ( 2F / 2m ) = g! Differing masses must accelerate at g = 9.8 m / s^2, because differing masses accelerating at differing rates would violate the laws regarding the Conservation of Energy.
Now, getting to the question :) . The gravitational force of attraction BETWEEN two objects is quantified using F = Gm1m2 / r^2. Another gentleman answered the question in a more direct manner, and this is what I will do at this time. If F = Gm1m2 / r^2, and both masses are doubled, then F = G( 2m1 )( 2m2 ) / r^2. However, we must take into account that the distance between the two objects is tripled. Therefore, F = G ( ( 2m1 )( 2m2 ) ) / ( 3 m )^2 ). Thus, the final force = ( 4 / 9 )( F ).
( 4F / 9 ) is the answer.
Best of luck in your studies!
P.S. I promise not to be this long-winded or detailed with any student that wants more direct answers to questions :) . Getting lost within physics equations is my method of dealing with this global pandemic.
