Gravity due to a uniform ring of mass

[i_hate_math]

Homework Statement

Several planets (Jupiter, Saturn, Uranus) are encircled by rings, perhaps composed of material that failed to form a satellite. In addition, many galaxies contain ring-like structures. Consider a homogeneous thin ring of mass 2.1 x 1022 kg and outer radius 4.3 x 108 m (the figure). (a) What gravitational attraction does it exert on a particle of mass 76 kg located on the ring's central axis a distance 4.5 x 108 m from the ring center? (b) Suppose that, starting at that point, the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which it passes through the center of the ring?

Homework Equations

Kepler's 3rd Law: T^2=(4*π^2/GM)*R^3
F=GmM/R^2

The Attempt at a Solution

I haven't really been able to attempt this question, I have absolutely no idea how to establish a relation between the mass on the axis and the ring. Please enlighten me with your genius ideas!

 

[gneill]

The problem should remind you of the similar one involving electric force: a ring of charge acting on a point charge along the axis of the ring.

 

[i_hate_math]

The problem should remind you of the similar one involving electric force: a ring of charge acting on a point charge along the axis of the ring.

I haven't previously studied electric force, I'm not sure what you meant.

 

[haruspex]

1. In what direction will the force from the whole ring be?
2. Consider a small segment of the ring, length rdθ, say. What force does it exert at the given point on the axis? What component of that acts in the direction in (1)?

 

[gneill]

Mentor's note:
Please note that the the thread title has been changed to: Gravity due to a uniform ring of mass.

The original title, "Challenging planetary problem", was too vague and non-descriptive of the actual problem.

 

[i_hate_math]

1. In what direction will the force from the whole ring be?
2. Consider a small segment of the ring, length rdθ, say. What force does it exert at the given point on the axis? What component of that acts in the direction in (1)?

I'm guessing for 1) the force would be towards the centre, and 2) I don't see how r*dθ would exert a force, pls instruct me on how to analyse this system, thanks

 

[i_hate_math]

Mentor's note:
Please note that the the thread title has been changed to: Gravity due to a uniform ring of mass.

The original title, "Challenging planetary problem", was too vague and non-descriptive of the actual problem.

Okay I'll be more specific next time

 

[haruspex]

I don't see how r*dθ would exert a force

The ring as a whole has a given mass M, so it has a mass per unit length density. If the ring has radius r, and a small piece of it subtends angle dθ at the centre of the ring then it has a mass (rdθ)M/(2πr). That will exert a gravitational force on the particle.

 

[i_hate_math]

The ring as a whole has a given mass M, so it has a mass per unit length density. If the ring has radius r, and a small piece of it subtends angle dθ at the centre of the ring then it has a mass (rdθ)M/(πr²). That will exert a gravitational force on the particle.

I am starting to understand. Is this related to Kepler's Second Law? I will need to go thru the textbook again if it is

 

[haruspex]

I am starting to understand. Is this related to Kepler's Second Law? I will need to go thru the textbook again if it is

No, it's Newton's law of gravitation, which you quoted up front. What is the gravitational attraction between the particle and the small element of the ring?

 

[i_hate_math]

No, it's Newton's law of gravitation, which you quoted up front. What is the gravitational attraction between the particle and the small element of the ring?

Right, F=GMm/R^2
where m=(rdθ)M/(πr2)

 

[haruspex]

Right, F=GMm/R^2
where m=(rdθ)M/(πr2)

I made an error in my post #8, which you have blindly copied. Trust no-one!
Also, be careful with the M's - you have them mixed up. I suggest fixing on M for the mass of the ring and m for the mass of the particle.

 

[i_hate_math]

I made an error in my post #8, which you have blindly copied. Trust no-one!
Also, be careful with the M's - you have them mixed up. I suggest fixing on M for the mass of the ring and m for the mass of the particle.

I have managed to got the correct answer for this question, thank you very much for the help!