Video Transcript
A planet has a mass of 8.08 times
10 to the 24th kilograms. A small asteroid with a mass of
1,420 kilograms is in space near the planet. The asteroid is 23,900 kilometers
away from the planet’s center of mass. What is the magnitude of the
gravitational potential energy of the asteroid-and-planet system? Give your answer to three
significant figures.
Drawing a sketch of this scenario,
let’s say that this here is our planet. We’ll say it has a mass 𝑚 sub p,
with that mass value given to us. And let’s say further that out here
is our asteroid, and we’ll label the asteroid’s mass 𝑚 sub a. The centers of these two masses are
separated by a distance we’ll call 𝑟. And that distance is given to us as
23,900 kilometers. Knowing all this, we want to
calculate the magnitude of the gravitational potential energy of the
asteroid-and-planet system.
At this point, we must be very
careful because we may think that the gravitational potential energy between these
two objects is equal to the mass of the asteroid times 𝑔 times its height above the
surface of the planet. But this representation of GPE is
only true when the gravitational field is uniform. In the case of our planet and
asteroid though, if we were to sketch in some of the gravitational field lines
around the planet, we would see that the field is not uniform, but rather it’s
radial. That is, the field lines move out
from the center of the planet, like spokes on a bicycle wheel.
In a situation like this, we can no
longer use this equation to calculate gravitational potential energy. Instead, we need to recall the form
of this equation that applies to this kind of field, a radial field. When a field is structured this
way, then the gravitational potential energy between two masses, big 𝑀 and little
𝑚, where big 𝑀 is the mass creating the field, is equal to negative the product of
those masses times the universal gravitational constant divided by the distance
between the centers of mass of the masses.
To get started using this equation,
we can recall that the universal gravitational constant, to three significant
figures, is equal to this value here. And so applied to our scenario, the
gravitational potential energy of our asteroid-and-planet system equals negative big
𝐺 times the mass of the planet — this is the mass creating the radial field — times
the mass of the asteroid divided by the distance between their centers.
One important point is that our
question wants us to solve for the magnitude of the gravitational potential energy
of this system. To calculate that, we would take
the absolute value of a negative number. And so to simplify things, we could
drop both the absolute value bars and the negative sign.
We’re now ready to plug in for
these values and solve for the magnitude of 𝐺𝑃𝐸. With these values substituted in,
notice that our radial distance has units of kilometers, whereas in our numerator,
we have distance in units of meters. We’ll want these units to agree
before we calculate the magnitude of 𝐺𝑃𝐸. So recalling that 1,000 meters is
equal to one kilometer, we can recognize that 23,900 kilometers is equal to that
number with three zeros added on the end meters, in other words, 23,900,000
meters.
Looking at the units in this
expression, notice now what happens. One factor of meters cancels from
numerator and denominator. And then the kilogram unit in both
of our masses cancels out with one over kilogram squared here. We’re left with newtons times
meters, and a newton times a meter is one joule, the SI base unit of energy. When we compute this fraction, to
three significant figures, we find a result of 3.20 times 10 to the 10th joules. But then, if we recall that 10 to
the ninth or one billion joules is equal to what’s called a gigajoule, then we can
equivalently express our answer as 32.0 gigajoules. This is the magnitude of the
gravitational potential energy of the asteroid-and-planet system.
