Video Transcript
Jupiter has a radius at its equator
of 71492 kilometers and the gravitational acceleration at that point is 23.1 meters
per second squared. Calculate Jupiter’s mass from its
radius and the gravitational acceleration at its equator. What is the ratio of the calculated
mass of Jupiter to NASA’s Jupiter fact sheet value of 1898 times 10 to the 24th
kilograms?
We can label the mass of the planet
Jupiter capital 𝑀 and the ratio of this mass to the fact sheet value mass 𝑀 over
𝑀 sub 𝑐. To solve first for the overall mass
of the planet Jupiter based on the information about its radius and its acceleration
due to gravity, we can recall that in general the universal gravitational constant
times an object’s mass—typically a large object such as a planet—divided by the
radius squared of that object is equal to the gravitational acceleration it causes
at its surface.
In our example, we’re told the
acceleration due to gravity at Jupiter’s equator as well as the radius of the planet
at that point. For capital 𝐺, the universal
gravitational constant, we’ll use a value of 6.67 times 10 to the negative 11th
cubic metres per kilogram second squared. We can now rearrange this
expression to solve for capital 𝑀, the mass of Jupiter. We see it’s equal to 𝑟 squared
times little 𝑔 over big 𝐺.
And when we plug in for these
values, we’re careful to write our radius of the planet in units of metres so that
it agrees with the units in the rest of our expression. To three significant figures, the
mass of Jupiter is 1.77 times 10 to the 27th kilograms. That’s its mass calculated based on
its equatorial radius, the universal gravitational constant, and its acceleration at
its radius.
Next, we want to solve for the
ratio of this mass we’ve just solved for to the fact sheet value mass of Jupiter
which is 1898 times 10 to the 24th kilograms. Entering both these values in in
scientific notation, when we calculate this fraction to three significant figures,
it’s 0.933. So our calculated value for
Jupiter’s mass is within 10 percent of the fact sheet mass value.
