# Why do you integrate to find the gravitational field?

• Vitani11
In summary, the student is trying to solve a problem involving the gravitational force between a sphere and a thin rod, but is having difficulty understanding the concept. They either need to break the problem down into smaller pieces to be able to sum the forces or approach the problem from a different angle.
Vitani11

## Homework Statement

I know that in general it is when the gravitational field is not constant, so that part is fine. To find the gravitational force between a sphere with mass M and a radius R and a thin rod of length L and mass m which has one end a distance x away from the center of the sphere, you need to integrate.

Unnecessary

## The Attempt at a Solution

My issue is not the integration, it's just the concept. Is it because each piece of mass (dm) at a length dl on the rod experiences a difference magnitude of gravitational force since each piece of mass is at different locations (and a certain distance away from the spheres center) along the rod? Also, why can you treat the sphere as a point charge in this case? I know it is because it is a sphere, but I don't understand how being a sphere allows it to be treated as a point charge.

Vitani11 said:
Is it because each piece of mass (dm) at a length dl on the rod experiences a difference magnitude of gravitational force since each piece of mass is at different locations (and a certain distance away from the spheres center) along the rod?
Yes.
Vitani11 said:
why can you treat the sphere as a point charge
It's a bit more general, applying to a uniform spherical shell. So the sphere as a whole does not need to be uniform - you can have the density depending on radius.
if you assume validity of the lines of flux model for forces obeying an inverse square law, you can put a Gaussian surface around the sphere and argue i) that the total flux through the surface is proportional to the mass (or charge) enclosed, and ii) by symmetry, the force must be radial and spherically symmetric. It follows that it is independent of the radius of the sphere, so that can be reduced to a point.
Alternatively, you can just do the integration and discover the result.

Of course, this is only for test points outside the spherical shell. There is no field inside.

Vitani11
From the way you worded that, it sounds like the rod is oriented so that its long axis is radially outwards from the spherical mass. So yes, your reasoning is correct. Break down the rod into pieces, and you'll see that each one experiences a different force due to the equation ##\vec F = G \frac {m_1 m_2} {r^2}##. So to find the overall force, you need to break it down into infinite infinitesimal pieces and sum those forces.

Vitani11
TJGilb said:
From the way you worded that, it sounds like the rod is oriented so that its long axis is radially outwards from the spherical mass. So yes, your reasoning is correct. Break down the rod into pieces, and you'll see that each one experiences a different force due to the equation ##\vec F = G \frac {m_1 m_2} {r^2}##. So to find the overall force, you need to break it down into infinite infinitesimal pieces and sum those forces.
I was, thank you.

## 1. Why do you integrate to find the gravitational field?

Integrating is a mathematical process that allows us to find the cumulative effect of small changes over a given interval. In the case of finding the gravitational field, we need to integrate the gravitational force equation over a certain distance in order to determine the total gravitational field at that point.

## 2. What is the difference between integrating to find the gravitational field and simply using the gravitational force equation?

The gravitational force equation only gives us the magnitude of the gravitational force between two objects. Integrating this equation takes into account the distance between the objects and allows us to determine the gravitational field, which is a vector quantity that describes the strength and direction of the gravitational force at a specific point.

## 3. Can you explain the concept of electric potential and how it relates to integrating for the gravitational field?

Electric potential is a measure of the potential energy per unit charge at a given point in an electric field. Similarly, gravitational potential is a measure of the potential energy per unit mass at a given point in a gravitational field. By integrating for the gravitational field, we are essentially finding the change in gravitational potential between two points.

## 4. Why is integration necessary in calculating the gravitational field of a non-uniformly distributed mass?

In the case of a non-uniformly distributed mass, the gravitational force exerted by each individual mass element needs to be accounted for. By integrating over the entire mass distribution, we can take into account the cumulative effect of each individual force and determine the total gravitational field at a given point.

## 5. Can you give an example of a real-world application where integrating to find the gravitational field is important?

An example of where integrating for the gravitational field is important is in the field of astrophysics. In order to accurately model the motion of celestial bodies in a system, such as planets orbiting a star, we need to know the gravitational field at different points. Integrating for the gravitational field allows us to make more precise predictions and understand the dynamics of these systems.

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