Potential energy between two objects

[Zorodius]

The following question appeared in my book:

From an inertial frame in space, we watch two identical uniform spheres fall toward one another owing to their mutual gravitational attraction. Approximate their initial speed as zero and take the initial gravitational potential energy of the two-sphere system as U_i. When the separation between the two spheres is half the initial separation, what is the kinetic energy of each sphere?

My solution was this:

Since the mechanical energy of the two-sphere system will be conserved,

K_i + U_i = K_f + U_f

Let K be the kinetic energy of one of the spheres in the final configuration of the system. Since both spheres are identical, K is also the kinetic energy of the other sphere, making the net kinetic energy 2K. Bearing in mind that the kinetic energy of each sphere was given as initially zero, this means:

U_i = 2K + U_f

\frac {U_i - U_f}{2} = K

The gravitational potential energy of such a system is given by:

U = -\frac{GMm}{r}

Let R be the distance between the two spheres at the system's initial configruation, then R/2 is the distance between the two spheres in the system's final configuration, and:

\frac{-\frac{GMm}{R} - -\frac{GMm}{R/2}}{2} = K

\frac{-\frac{GMm}{R} + \frac{2GMm}{R}}{2} = K

\frac{\frac{GMm}{R}}{2} = K

\frac{-U_i}{2} = K

The book's answer is U_i / 4, not -U_i / 2. Where is my mistake?

 

[Zorodius]

Is it just that the book authors made an error? I still don't see anything wrong with my solution. Also, A positive value for Ui means that, since Ui itself contains a negative, the book's value for kinetic energy would be negative, and that doesn't sound right at all.

 

[arildno]

Your answer looks right to me..

 

[JohnDubYa]

Since the potential energy is an inverse-r law, then halving the separation distance between two masses doubles the potential energy of the system. This in turn halves the kinetic energy of the system, which means that the kinetic energy of the individual masses must halve as well. I like your answer.

 

[gnome]

This

This in turn halves the kinetic energy of the system, which means that the kinetic energy of the individual masses must halve as well.

makes no sense at all, since the initial kinetic energy is 0, and 0/2 is still 0.

You're getting confused by the signs of these values. Maybe it'll be easier if you think in terms of |U|, the absolute value of the potential energy, i.e. a positive number. So the initial potential energy is -|U_i|.

The potential energy which is doubling is a negative number, so the potential energy is decreasing from -|U_i| to -2|U_i|. The change in potential energy is -|U_i|. Conservation of energy requires that kinetic energy will increase by the same amount. The initial kinetic energy was 0, so the final kinetic energy is U_i. At the end, total energy = U + K = -2|U_i| + |U_i| = -|U_i|, so all is well. [edited to re-order the terms to be consistent]

Since you are looking at the system from an external inertial frame, and the two spheres are identical in mass and each started with 0 velocity, conservation of momentum requires that the velocities of the two spheres are always equal in magnitude and opposite in direction. Direction is irrelevant as far as kinetic energy is concerned, so each sphere ends up with the same kinetic energy: U_i/2.

So Zorodius is correct as to the absolute value; his book is correct as to the sign.

 

[JohnDubYa]

Sorry, I did get the signs confused. Since the objects are moving to where they naturally want to go, then they must LOSE potential energy. That means they must gain the same amount of kinetic energy.