Finding # of g's experienced on Jupiter

[Razael]

Homework Statement

Jupiter is about 320 times as massive as the Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter since people can’t survive more than a few g’s. Calculate the number of g’s a person would experience at the equator of such a planet. Use the following data for Jupiter, and take centripetal acceleration into account.

Mass = 1.9 x 10²⁷kg
Equatorial radius = 7.1 x 10²km (7.1 x 10²m)
Rotation period = 9 hours, 55 minutes (35700 seconds/rotation)

Homework Equations

F = GMm/r² = ma (or ma_r)
ma_r = mv² / r

The Attempt at a Solution

F = (6.67 x 10^-11)(1.97 x 10²⁷)m / (7.1 x 10⁷)² = ma_r

m's cancel, I get a centripetal acceleration of 26.06m/s²

I also found velocity; 26.06m = mv² / r
m's cancel once again, V = \sqrt{}26.06(7.1 x 10^7 = 43014.65m/s. Not sure if that's important.

And from there I have no clue. I don't know how to use the rotation period (I don't think they'd give me it if it wasn't important) and I have no idea how to use the centripetal acceleration to find g.

 

[ehild]

You have to find the normal force Fn on the person at the surface of Jupiter. The gravity of Jupiter acts towards the centre, the normal force away from the centre, and as a result, the person moves on a circle with radius of Jupiter and period equal to the rotation period of Jupiter, so with speed equal to that of the equator. (The speed you calculated would be the speed of a space probe orbiting just above the surface of Jupiter. )
So you have to calculate the speed of the equator first, and then Fcp belonging to that speed. This is equal to the difference between the force of gravity and Fn. From that, you get Fn, and Fn/m is the "G" in question.

By the way, the Equatorial radius = 7.1 x 10⁴km.

 

[Razael]

Okay that was very confusing, but I thought of something else.

Given the period time, I figured I could use the V = 2(pi)(r) / T equation.

V= 2(pi)(7.1 x 10^7m) / 35700s = 12495.97m/s

Fcp = mv^2 / r

Fcp = (1.97 x 10^27kg)(12495.97m/s) / (7.1 x 10^7) = 4.33 x 10^27N

I don't know where to go from here. This is force in the horizontal direction, is it not?

 

[Razael]

Anyone?

 

[ehild]

Well, again. The question was: "Calculate the number of g’s a person would experience". What does it mean? How many g-s you feel here on the Earth?

ehild

 

[Razael]

Ah. For some reasoning I was thinking that centripetal acceleration was horizontal.

26.06 / 9.8 = 2.66 g's felt, correct?

But then where does the rotational period come into play?

 

[ehild]

The 2.66 g would be the case if the Jupiter had not revolved. But it does and together with its equator, the person moves on a circle. For that, a certain centripetal force is needed, and the magnitude of this force can be calculated from the speed of the man (here comes in the period) and the radius of Jupiter. You have calculated that speed, V= 2(pi)(7.1 x 10^7m) / 35700s = 12495.97m/s. Determine the centripetal acceleration which corresponds to this speed.

The person moves with this centripetal acceleration round, and this motion is the result of two forces: one is the gravitational pull of Jupiter, F_G = GMm/r²,
the other force is the normal force F_n from the ground, it points upward. If m is the mass of the person
ma_cp =mv²/r= GMm/r²-Fn.

Determine Fn/m and divide it with g=9.8.

ehild