What is the gravitational potential
energy between two spheres, each of mass 9.5 kilograms, separated by a
center-to-center distance of 20.0 centimeters? We can call this gravitational
potential energy between the two spheres capital 𝑈.
And to start on our solution, we’ll
recall the mathematical forms of gravitational potential energy. When it comes to gravitational
potential energy, there are two relationships we can recall. The first relationship is the
general equation, that gravitational potential energy between two masses 𝑚 one and
𝑚 two equals the product of their masses times big 𝐺 all divided by the distance
between their centers. And this potential energy is the
negative of this fraction. There’s also a special formulation
of GPE, which is when a relatively smaller mass lowercase 𝑚 is near a planetary
body whose acceleration due to gravity, we’ve called it lowercase 𝑔, is known.
In that case, the GPE of the
smaller mass is equal to 𝑚 times 𝑔 times the height it moves through. In our case, because our two masses
have the same value, we’ll use the more general expression for gravitational
potential energy. If each mass has a value we
symbolize with lowercase 𝑚 and the center-to-center distance between the masses we
represent with 𝑟, and if we let the universal gravitational constant equal exactly
6.67 times 10 to the negative 11 to cubic meters per kilogram second squared, then
given the values of 9.5 kilograms for 𝑚 and 20.0 centimeters for 𝑟, we can plug
all these values into our equation being careful to convert our distance from units
of centimeters to units of meters.
Entering this expression on our
calculator, to two significant figures, we find a result of negative 3.0 times 10 to
the negative eighth joules. That’s the gravitational potential
energy between these two spheres.