Br answer explanation.plzz help!

[Temper888]

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Hey guys, I was doing questions from BR physics and I could not understand the answer explanation to a question. Please help me🙁

Q) Suppose two satellites are orbiting the Earth with the same orbital radius and the same period.However, one satellite is 20 times more massive than the other.This is possible, because.
A) the gravitational force is inversely proportional to r^2.
B) the gravitational force is proportional to mass.

The answer is B. The explanation as follows:B says that the mass of a body increases, the force of gravity on that body increases proportionately. However, since F=ma, the net force on the satellite also increases proportionality with mass, and these two effects cancel each other out.

I do not understand where did the F=ma come from and how does it cancel the force of gravity?

 

[chiddler]

Hey guys, I was doing questions from BR physics and I could not understand the answer explanation to a question. Please help me🙁

Q) Suppose two satellites are orbiting the Earth with the same orbital radius and the same period.However, one satellite is 20 times more massive than the other.This is possible, because.
A) the gravitational force is inversely proportional to r^2.
B) the gravitational force is proportional to mass.

The answer is B. The explanation as follows:B says that the mass of a body increases, the force of gravity on that body increases proportionately. However, since F=ma, the net force on the satellite also increases proportionality with mass, and these two effects cancel each other out.

I do not understand where did the F=ma come from and how does it cancel the force of gravity?

The question says that why can two sattelites orbit with the same radius and period of one is a lot more massive than the other? More massive = slower speed, right?

No that's not right. More massive = more force because F = ma = G*m*M/r^2. a = GM/r^2. This tells you that, if radius is fixed, the only thing that determines the acceleration of the object is the mass of the object that it is orbiting. The mass of the satellite has nothing to do with it.

To answer your question specifically, F = ma. So if the period and radius is the same, that means that acceleration is the same for the two satellites, right? BUT one is more massive than the other. So for the same acceleration to be possible, then there must be more force. Therefore, the amount of force is proportional to the mass of the object.

Understand?

 

[davcro]

Would someone be willing to post a free body diagram of the forces acting on the satellite in this problem?

I can think of two forces: the force of gravity from the Earth and the centripetal force.

 

[davcro]

Okay, did more thinking about this one.

The only force acting on the satellite is gravity. Therefore sum of the forces acting on the satellite is:

Sigma F = G*MeMs/r^2

Newton's second law says that sum of the forces equals mass * acceleration. Thus

Ms*a = G*MeMs/r^2

Because the force of gravity is proportional to the mass of the satellite, the acceleration (and the period) of the satellite does not depend on its mass.

 

[MedPR]

Alternatively:

F=GmM/r^2=ma

ma=GmM/r^2 -- therefore, mass cancels out.

a=GM/r^2=w^2r

w=sqrt(GM/r^3)

w=2pi/T

2pi/T = sqrt(GM/r^3)

The last equation proves that the orbital period depends only on G, the mass of the thing it is orbiting (earth, in your problem) and the radius.

 

[chiddler]

Alternatively:

F=GmM/r^2=ma

ma=GmM/r^2 -- therefore, mass cancels out.

a=GM/r^2=w^2r

w=sqrt(GM/r^3)

w=2pi/T

2pi/T = sqrt(GM/r^3)

The last equation proves that the orbital period depends only on G, the mass of the thing it is orbiting (earth, in your problem) and the radius.

very well done. would not have thought of connecting those two equations because they are taught so far apart in class.

 

[MedPR]

very well done. would not have thought of connecting those two equations because they are taught so far apart in class.

What class are you taking? I think those equations are from the same chapter in TBR.

I'm sure you could probably do it faster/simpler with energy conservation, but I just started working on that chapter...

 

[varsityblue]

Yes, an energy conversion would be faster.

 

[MedPR]

Yes, an energy conversion would be faster.

Can you demonstrate?

 

[chiddler]

What class are you taking? I think those equations are from the same chapter in TBR.

I'm sure you could probably do it faster/simpler with energy conservation, but I just started working on that chapter...

i meant in undergrad physics classes