# Balancing Centrifugal and Gravitational Forces: A Homework Problem

• dirk_mec1
In summary, the conversation discusses the equations for gravitational and centrifugal forces, and how to approach a problem related to tension and balancing forces. The equations for F_gravitational and F_centrifugal are given, and the conversation suggests using the definition of linear density to solve for F_g and F_c as functions of x.
dirk_mec1

## Homework Statement

[PLAIN]http://img194.imageshack.us/img194/2062/57916122.png

## Homework Equations

$$F_{gravitational}= \frac{MmG}{r^2}$$

$$F_{centrifugal}= \frac{mv^2}{r}$$

## The Attempt at a Solution

I got this:

$$dF_{centr} = dF_{grav} \longrightarrow \frac{dm \cdot v^2}{R+x} = \frac{MdmG}{(R+x)^2}$$

Is this correct?

Last edited by a moderator:
The dF's look okay to me. I don't think they would be equal, though. There would be tension in the wire, except perhaps at one value of x.

Tension? How can I calculate this tension and how does this equation then changes?

I wouldn't worry about tension or balancing the forces if the question does not ask for it. I think your equations look correct. Here's how I think you should approach the problem.

From the definition of linear density:
rho = dm/dx.

Therefore you could substitute dm in both equations for rho*dx, so that you could actually solve for Fg and Fc as function of x. Then it's just simple calculus to get to a solution.

dFg = M*rho*G*dx/(R + x)^2
Fg = -M*rho*G/(R + x)

dFc = rho*v^2*dx/(R+x)
Fc = rho*v^2*log(R + x)

Hope this helps.

Thanks a lot that helped!

## 1. How do you calculate centrifugal and gravitational forces?

Centrifugal and gravitational forces can be calculated using the equations Fc = mv^2/r and Fg = mg, respectively. Fc represents the centrifugal force, m is the mass of the object, v is the velocity, r is the radius of the circular motion, Fg represents the gravitational force, and g is the acceleration due to gravity.

## 2. What is the relationship between centrifugal and gravitational forces?

The relationship between centrifugal and gravitational forces is that they are equal in magnitude but opposite in direction. This means that the centrifugal force is always pushing an object away from the center of rotation, while the gravitational force is pulling the object towards the center of mass.

## 3. Why is balancing centrifugal and gravitational forces important?

Balancing centrifugal and gravitational forces is important because it ensures that an object stays in a stable orbit or circular motion. If these forces are not balanced, the object will either fly away from the center of rotation or crash towards it.

## 4. What factors can affect the balance between centrifugal and gravitational forces?

The balance between centrifugal and gravitational forces can be affected by the mass of the object, the velocity of the object, and the radius of the circular motion. These factors can all impact the strength of the forces and their ability to keep the object in its orbit.

## 5. How can you use the concept of balancing forces to solve real-world problems?

The concept of balancing forces, specifically centrifugal and gravitational forces, can be applied to various real-world problems such as designing satellites, understanding the motion of planets and moons, and analyzing the stability of structures like bridges and roller coasters. It is also important in fields such as aviation and space exploration.

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