What is the length of a sector on a ring with radius R and angle dθ?

[GreenPrint]

Homework Statement

See Attachment

Homework Equations

The Attempt at a Solution

Ok so I understand for the radius, the distance from m to M would be (d^2+r^2)^(1/2)
but sense the formula for gravitation is a inverse square of the displacement in between the two bodies it would be

1/(R^2+d^2)

I don't understand why the answer is in terms of d/(R^2+d^2)^(3/2)
thanks for any help

 

[omoplata]

The formula F = \frac{G M m}{r^2}, where r is the distance between the bodies is valid only for point masses (We use it when calculating planetary motion because the size of the planets are so small compared to the distances between them that they can be approximated to be point masses).

In this case only one object is a point mass. The other is not.

The solution can be found by finding the force on the mass m due to an infinitesimally small point mass on the ring, and the integrating over the whole ring to find the total force.

 

[GreenPrint]

I would like to grasp a good understanding of these types of problems...

So I have to integrate...
well... any infinity small point on the ring is a distance sqrt(d^2 + r^2)...
um so... an infinity small point mass on the ring dM...
um I'm not exactly sure what to do or what function I'm integrating...
I'm not exactly sure what the limits of the integral would be... hm...

I'm not really sure what to do

 

[omoplata]

well... any infinity small point on the ring is a distance sqrt(d^2 + r^2)...

Correct.

To find the mass of a very small part of the ring, consider the length of that part and the fact that the mass distribution of the ring is uniform.

 

[GreenPrint]

well length of a sector is r^2/2 * angle (I actually had to look that one up)
if the angle becomes infinity small the length of the sector would be r^2/2 dtheta

so the length of the whole ring can be expressed as

1/2 * integral[0,2pi] R^2 dtheta

so a very small piece of the ring has a length of R^2/2 dtheta and is a distance of sqrt(R^2 + d^2)

I notice that both these quantities are in terms of R...

I don't see how the length of a section of the ring is related to the mass...

 

[omoplata]

well length of a sector is r^2/2 * angle

This cannot be correct. Look at the dimensions of (r^2/2*angle).

if the angle becomes infinity small the length of the sector would be r^2/2 dtheta

so the length of the whole ring can be expressed as

1/2 * integral[0,2pi] R^2 dtheta

You have the right idea about using integration to find the total length (circumference) of the ring here, but you need to find the correct expression for the length of the sector.

I don't see how the length of a section of the ring is related to the mass...

Consider the fact that the ring has a uniform mass distribution. If it's M for the total length of the ring, what is it going to be for the length of the section?