Calculate the gravitational acceleration at Jupiter’s surface (technically Jupiter is a gas cloud, so it doesn’t have a surface per se), given that Jupiter’s radius is 6.9×10^4 km. Express this acceleration as a fraction of the Earth’s gravitational acceleration at the surface. If your body mass was 60 kg, what would be the force of gravity on you if you happened to be on Jupiter’s surface?
The Correct Answer and Explanation is:
To calculate the gravitational acceleration at Jupiter’s surface, we use the universal law of gravitation:g=G⋅MR2g = \frac{G \cdot M}{R^2}g=R2G⋅M
Where:
- ggg is the gravitational acceleration,
- GGG is the gravitational constant =6.674×10−11 N\cdotpm2/kg2= 6.674 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2=6.674×10−11N\cdotpm2/kg2,
- MMM is the mass of Jupiter =1.898×1027 kg= 1.898 \times 10^{27} \, \text{kg}=1.898×1027kg,
- RRR is the radius of Jupiter =6.9×104 km=6.9×107 m= 6.9 \times 10^4 \, \text{km} = 6.9 \times 10^7 \, \text{m}=6.9×104km=6.9×107m
Step 1: Compute Jupiter’s surface gravity
gJ=6.674×10−11×1.898×1027(6.9×107)2g_J = \frac{6.674 \times 10^{-11} \times 1.898 \times 10^{27}}{(6.9 \times 10^7)^2}gJ=(6.9×107)26.674×10−11×1.898×1027gJ=1.266×10174.761×1015≈26.6 m/s2g_J = \frac{1.266 \times 10^{17}}{4.761 \times 10^{15}} \approx 26.6 \, \text{m/s}^2gJ=4.761×10151.266×1017≈26.6m/s2
Step 2: Express Jupiter’s gravity as a fraction of Earth’s
Earth’s surface gravity: gE=9.8 m/s2g_E = 9.8 \, \text{m/s}^2gE=9.8m/s2gJgE=26.69.8≈2.71\frac{g_J}{g_E} = \frac{26.6}{9.8} \approx 2.71gEgJ=9.826.6≈2.71
So, gravity on Jupiter is about 2.71 times stronger than on Earth.
Step 3: Calculate the gravitational force on a 60 kg person
Using Newton’s second law F=m⋅gF = m \cdot gF=m⋅g:F=60 kg×26.6 m/s2=1596 NF = 60 \, \text{kg} \times 26.6 \, \text{m/s}^2 = 1596 \, \text{N}F=60kg×26.6m/s2=1596N
Explanation
Gravitational acceleration is the rate at which an object speeds up due to gravity. It depends on two factors: the mass of the planet and the square of the distance from the planet’s center. Using Newton’s law of gravitation, we can find the acceleration due to gravity on Jupiter even though it does not have a solid surface. Scientists often define a notional “surface” at the point where atmospheric pressure equals 1 bar (same as Earth’s sea level).
We start by plugging values into the gravitational formula. Jupiter is much more massive than Earth, and while it is also larger in radius, the increase in mass has a stronger effect than the increase in radius. This results in a stronger gravitational pull.
After performing the calculation, we find that gravity on Jupiter is about 2.71 times that on Earth. This means that if you weigh 60 kg on Earth, you would feel significantly heavier on Jupiter due to the increased gravitational force. Specifically, your weight (which is the force of gravity on you) would be about 1596 newtons, compared to 588 newtons on Earth.
Even though you would still have the same mass of 60 kg, the gravitational force acting on that mass would be much higher. This distinction between mass (which stays constant) and weight (which changes depending on gravity) is crucial in physics. Jupiter’s immense gravity makes it an extreme environment, and this is one reason why sending humans or probes near its surface presents such technical challenges.
