Video: Calculating Jupiter’s Mass

Jupiter has a radius at its equator of 71492 km, and the gravitational acceleration at that point is 23.1 m/s². 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 × 10²⁴ kg?

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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.

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