A rod, a ball, garvitational Potential Energy (U), and the power series expns.

In summary, the Homework Statement asks for the GPE of the rod/ball system, using the Power Series Expansion for ln(1+x). The Attempt at a Solution states that I'm not quite sure where to start - the Power Series expansion has confused me slightly, as otherwise I would have just put the variables in the above equation. I'm still rather confused about what I should be doing, so any help would be much appreciated.
  • #1
TFM
1,026
0

Homework Statement



Mass of rod: M
mass of ball: m
Length of Rod: L
distance between rod and ball: x
GPE is zero at infinty

The questiopn asks to take the GPE of the rod/ball system, using the Power Series Expansion for ln(1+x) .

Homework Equations



U = -GMm/r

The Attempt at a Solution



I'm not quite sure where to start - the Power Series expansion has confused me slightly, as otherwise I wouold hqave just put the variables in the above equation?

TFM :confused:
 
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  • #2
It just means you can approximate ln(1+x) with x, for x << 1.
 
  • #3
Where does the Ln(x+1)come from?

TFM
 
  • #4
It'll probably appear as you solve the problem. They say to use the power series to make it easier to solve.
 
  • #5
Do you still use the U=-GMm/r, with U = 0, r = infinty, giving:

0=-GMm/infinity?

TFM
 
  • #6
I guess you want to integrate over the length of the rod and end up with the integral of
(1/l+1) dl from 0 to L... that should give you ln(x+1)

U= -GmM/L [integral 0 to L (dl /sqrt. of x^2+l^2)] and I guess you can say that the square root of x^2+l^2 is x+C where C is some constant, though this makes no sense I can't think of any other way

btw, the way I got that is by saying dU= -Gm(dM)/r.. then setting dM=dl(M/L) and r=sqrt. (x^2+l^2)
 
Last edited:
  • #7
take the intergral of 1/r, which is ln(r), setting your limits from infinity to x. Do you know the expansion to ln(r)?
 
  • #8
from infinity to x? how in the world do you integrate from infinity to x?
 
  • #9
t-money said:
take the intergral of 1/r, which is ln(r), setting your limits from infinity to x. Do you know the expansion to ln(r)?

Why do I need to take the integral of 1/r?

TFM
 
  • #10
I'm Still rather cionfused about what I should be doing:frown:

Any Help?

TFM
 

Related to A rod, a ball, garvitational Potential Energy (U), and the power series expns.

1. What is a rod?

A rod is a long, thin, straight object that is usually made of metal or wood. It is often used to support or connect other objects.

2. What is a ball?

A ball is a round object that is usually made of rubber, plastic, or metal. It can be used for various activities such as sports, games, or as a toy.

3. What is gravitational potential energy (U)?

Gravitational potential energy is the energy that an object possesses due to its position in a gravitational field. It is the potential for an object to do work as a result of its position in a gravitational field.

4. How is gravitational potential energy (U) calculated?

Gravitational potential energy (U) is calculated by multiplying the mass of the object (m) by the acceleration due to gravity (g) and the height (h) of the object above a reference point. The formula is U = mgh.

5. What is a power series expansion?

A power series expansion is a mathematical method for representing a function as an infinite sum of terms. It is a useful tool for approximating functions and solving equations in various fields of mathematics and science.

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