### Video Transcript

A speck of dust with a mass of 0.22
grams is at a distance of 1250 meters from a tiny shard of ice that has a mass of
1.5 grams. Both of the objects are located in
deep space, extremely far away from any stars or other objects of any kind. The gravitational force that the
objects exert on each other is negligible, as is the gravitational force on either
the speck of dust or the shard of ice from any other objects. Which of these statements about the
gravitational potential energy of the speck of dust and the shard of ice is
correct? (A) The gravitational potential
energies of both objects are zero. (B) The gravitational potential
energies of both objects are equal and nonzero. Or (C) the gravitational potential
energies of both objects are nonzero, but they are not equal.

To answer this question, let’s
start by briefly recalling that we can use gravitational potential energy, or GPE,
to calculate changes in the energy of an object in a gravitational field. We can also recall that the
gravitational potential energy transferred to increase the height of the position of
an object by ℎ is given by GPE equals 𝑚𝑔ℎ, where 𝑚 is the mass of the object and
𝑔 is the gravitational field strength at the position of the object.

Notice that both of these
qualitative and quantitative descriptions of GPE refer to a gravitational field. This is key to answering this
question. To help us think about this
further, let’s clear some room on screen.

Here, we’re considering two tiny
objects out in deep space. We were told that their
gravitational influence on one another is negligible, or essentially zero. So we know that these objects are
too tiny and far apart to influence each other gravitationally. And it would be incorrect to think
that either object has potential energy due to its own gravitational field. So while, yes, a lone speck of dust
would produce a very weak, but still nonzero, gravitational field around it, we have
to remember that an object’s GPE is always measured with respect to some other
position within the field. And an object can’t move relative
to its own gravitational field.

Now, we were also told that there
is negligible, or basically zero, gravitational influence on the two objects from
any other object, since they’re isolated in deep space. Thus, we can reason that there is
effectively no external gravitational field around either object or, more
accurately, that the strength of the field is zero. With a strength of zero, the field
can do no work. And no gravitational potential
energy can be transferred to an object.

Note that if we try to use a field
strength of zero in the formula for GPE, the entire expression simply equals
zero. Therefore, the best answer is
(A). The gravitational potential
energies of both objects are zero.