The centripetal acceleration of a satellite orbiting Saturn at a distance of one Saturn radius above its surface is calculated using the formula Ac = v^2/r. The gravitational force at this location is determined to be 2.79 m/s², which is closely related to the gravitational intensity defined as GM/r², where G is the gravitational constant, M is Saturn's mass (568.3 x 10^24 kg), and r is the distance from the center of Saturn (two Saturn radii). The discussion clarifies that gravitational intensity and centripetal acceleration yield similar results, reinforcing the connection between these two concepts.
PREREQUISITESStudents studying physics, particularly those focusing on gravitational forces and orbital mechanics, as well as educators seeking to clarify concepts related to centripetal acceleration and gravitational intensity.
Coco12 said:Homework Statement
What is the centripetal acceleration of a satellite orbiting Saturn at the location exactly one Saturn radius above the surface of Saturn
Homework Equations
Ac=v^2/r
In the previous question found out that the fg at that location is 1/4 of 36.0 N
The Attempt at a Solution
I read somewhere that they made centripetal acceleration equal to Gm/4pi ^2
rude man said:I would not lend much credulity to what you read. The dimensions of Gm/4pi^2 are not the dimensions of acceleration.
Equate gravitational acceleration to centripetal acceleration.
Coco12 said:The answer is the same as the gravitational intensity which is 2.60N . How does that work??
zahbaz said:Well, you have two masses to consider, the satellite's mass and Saturn's mass. After you make some cancellations in your equations, you may not need both values, though.
Anyway, if this satellite is in orbit, it's experiencing a centripetal force. The real question: what force IS PROVIDING this centripetal force? What's the equation for this force that rude man is getting at? Equate this to your centripetal.
Coco12 said:Mv^2/r=gm1m2/r^2?
Oh no, the m in the first equation is the same as m1 in the equation after the equal sign.rude man said:How come three different masses?
Coco12 said:Oh no, the m in the first equation is the same as m1 in the equation after the equal sign.
So when canceled would be: v^2/r=Gm/ r^2?
Coco12 said:Mass would be the mass of Saturn and r would be the radius of Saturn *2?
The thing is , all of this info is not given in the problem, so how are we supposed to know it?
zahbaz said:What's the book's answer? I was anticipating the reasoning you said you followed.
Below I pulled Saturn's mass and radius from Google, thought if it's a book problem, I'm sure your text has similar values printed inside.
mv^2/r = GmM/r^2
v^2/r = GM/r^2
where M = mass of saturn = 568.3 x 10^24 kg
r = two saturn radii = 2(58,232,000 m)
a = GM/r^2
a = 27.9 m/s^2
zahbaz said:What is this gravitational intensity you keep bringing up?
PS... I edited my post in the time you replied. I had made a calc error. I got a = 2.79 m/s^2, which is pretty close to 2.60m/s^2. I'm wondering if your book has different, perhaps rounded values for Saturn's mass and radius.
zahbaz said:Oh okay... So gravitational intensity is gravitational force per unit mass... otherwise stated as F/m.
We know gravitational force is GmM/r^2, so therefore gravitational intensity is GM/r^2. Which is... ta dah! The same thing we had before!
Yeah, I'm not sure what digits were used in the prior calculation that got you 2.60m/s^2, but the theory is identical in both approaches. If it was a two stage problem, I'd stick with the 2.6.