Potential Energy: Accounting for Change in g with Altitude

[JPC]

hey

Ep = g* m * z
z = (z1 - z2)

but, apparently that can only work for a short distance z value , because the value of g changes with altitude

So , is there any equation, where the change of the g value with the change of altitude is taken acount of ?

 

[Hootenanny]

Welcome to the forums,

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html#ufm

 

[JPC]

But you sure that's the equation, it sounds too simple

the thing u found is a bit like :

Ec = mgz = GMmz / z² = GMm / z


but i mean , like during the fall of the object, the value of g changes

like if you have an object 500 miles over the Earth's surface ;
during the fall, the g value won't be the same
g at 500 miles over surface isn't the same as g at 300
see what i mean ?

 

[Hootenanny]

You would have to integrate the force between the limits of the distance thus;

$$\Delta U=\int^{h+R}_{R}\frac{GMm}{r^2}dr$$

Where R is the radius of the Earth and h is the height above the surface. Apologies that I don't have time for a more detailed explanation right now.

 

[cristo]

But you sure that's the equation, it sounds too simple

the thing u found is a bit like :

Ec = mgz = GMmz / z² = GMm / z


but i mean , like during the fall of the object, the value of g changes

like if you have an object 500 miles over the Earth's surface ;
during the fall, the g value won't be the same
g at 500 miles over surface isn't the same as g at 300
see what i mean ?

Note that g=GM/R², where here, R is the radius of the earth. This is the acceleration due to gravity on the Earth's surface. As you move off the surface, the value of g changes, since R is now not simply the Earth's radius.

As Hootenanny says, to obtain the potential, one would integrate the expression, between the limits of the distance (however, I think he missed a "m" from the integrand!)

 

[Hootenanny]

As Hootenanny says, to obtain the potential, one would integrate the expression, between the limits of the distance (however, I think he missed a "m" from the integrand!)

Indeed I did, good catch, you've just saved a hell of a lot of confusion later on...

 

[JPC]

hey but what does the kind of sloppy sign mean ?
im only in grade 11

and what's d in the equation ?

 

[Hootenanny]

hey but what does the kind of sloppy sign mean ?
im only in grade 11

and what's d in the equation ?

The equation I gave you involved an integrand, which involves calculus, which you clearly haven't met yet. In that case could you post the exact problem you are trying to solve then we can all see what's what?

 

[turdferguson]

Youre thinking is very sound having never encountered calculus. The simple equation for PE is only a rough approximation close to the Earth's surface. With calculus, were able to find exact answers to problems where gravity acts over long distances with varying force.

Basically, the squiggle is a symbol for integration, or finding the "area under the curve." For a straight line like y = 3, the area under the curve is the base times height. W=Fd, so if a constant 3 N force acts over 2 m, the work done is 6 J. However, calculus enables us to find the exact (no estimate) work done when force is not constant. The d before the r means that r is the variable changing. You can easily find the area under a certain section of a parabola for example.

 

[cristo]

Youre thinking is very sound having never encountered calculus. The simple equation for PE is only a rough approximation close to the Earth's surface. With calculus, were able to find exact answers to problems where gravity acts over long distances with varying force.

Basically, the squiggle is a symbol for integration, or finding the "area under the curve." For a straight line like y = 3, the area under the curve is the base times height. W=Fd, so if a constant 3 N force acts over 2 m, the work done is 6 J. However, calculus enables us to find the exact (no estimate) work done when force is not constant. The d before the r means that r is the variable changing. You can easily find the area under a certain section of a parabola for example.

IMHO, this is not the correct place to attempt to teach the OP calculus. I think that a sort of hand-waving explanation here will just further confuse the OP.

It is far better to leave the discussion of calculus, ask the OP to state the specific question he was having trouble with, and then help him with the question using the mathematics that he understands.

 

[storygeek]

Refresh on energy principle

The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches.

Why is the potential neg near a planet? It is because the work taken to push an object out into space is positive?

BTW, JPC, are you asking the question for homework or for your curiosity?

 

[JPC]

Ok

i see a bit your general idea for calculus , but yeah , still a lot of confusion
Just wondering , Calculus is learned in what grade , or how many years after you finished school ?

My general idea was to build a program , which would be able to calculate the gravitational potential energy with big distances

But , maybe i can do that with a kind of aprox way :
with a 'Do until loop'
like the program , cuts the distances in many short distances, and does them all one by one with the general Ep = mgh equation
and then add them all up

But , i was thinking, that maybe i could make my program give exact results, so that's why i posted here

 

[Hootenanny]

Hi JPC,

I'm from the UK, so I don't know about US grades, but I first learned calc in Year 12 (16/17 years old). If this isn't homework then we can give you a lot more help than we have been doing. Unfortunately, I don't have much time to help you now (12:30 AM in the UK); but I'm going to recommend to the mentors that this be moved to general physics where it should receive more attention.

 

[Gokul43201]

My general idea was to build a program , which would be able to calculate the gravitational potential energy with big distances

U=-GMm/r is all you need. It is very accurate and does not need any knowledge of calculus.

 

[jtbell]

U=-GMm/r is all you need. It is very accurate and does not need any knowledge of calculus.

To make it absolutely clear (hopefully): you do not need to know calculus in order to use this formula. You do need to know calculus in order to derive it and understand where it comes from, which is what the discussion above, involving an integral, was about.

Note that r is the distance from the center of the earth, not from the surface of the earth!

 

[Hootenanny]

Just for completeness, from my integral, we would obtain;

$$\Delta U = \int^{R+h}_{R} \frac{GMm}{r^2}\; dr$$

$$=\left[ -\frac{GMm}{r} \right]^{R+h}_{R} = \frac{GMm}{R}-\frac{GMm}{R+h}$$

$$\therefore\boxed{\Delta U = \frac{GMmh}{R\left( R + h\right)}}$$

Where R is the radius of the earth, h is the height above the surface, M is the mass of Earth and m is the mass of the object.

 

[disregardthat]

What does the delta sign stay for?

 

[Hootenanny]

What does the delta sign stay for?

"Change", e.g. the difference in potential energy between two positions.

 

[planish]

... My general idea was to build a program , which would be able to calculate the gravitational potential energy with big distances

But , maybe i can do that with a kind of aprox way :
with a 'Do until loop'
like the program , cuts the distances in many short distances, and does them all one by one with the general Ep = mgh equation
and then add them all up

But , i was thinking, that maybe i could make my program give exact results, so that's why i posted here

I recall a listing for a BASIC program that I was able to adapt for my Sinclair ZX80 lo, these many years ago. It was the "Moon Lander" game that involved the player trying to determine how much fuel to burn in successive burns so as to land with less than some maximum vertical velocity. If you burned too much too often, you go up or hover too long, run out of fuel, and then crash and die. If you did not burn enough, you would land too hard and crash and die. Every time you executed a burn, the combined mass of the lander reduced, so the next burn might have to be shortened.

I think that version was a simple text display of the velocity and altitude, so it should be easy to extract the routine than did the calculation. The ZX80 only had something like 4K or RAM, and you had to type this listing in, so it should be very short, very simple, and easy to deconstruct. If you peruse a couple of '80s vintage tutorials on BASIC (maybe for Apple II, Commodore 64, TRS80, Timex/Sinclair, early IBM PC, etc.) you would probably find some version of it.

Waiddaminnit - Here's some stuff I scooped from the source of a page with a Java applet that emulates the TRS-80's BASIC version of it, and they call it "Lunar Lander 2.0". Clicking on the "open program window" button gives you this:

1  PRINT"WELCOME TO LUNAR LANDER 2.0"
6  H=45
7  S=0
8  B=0
9  F=100
10 X=1
20  PAUSE 2750
30  CLS
35  A=1
40  SET(A,46)
45  A=A+1
50  IF A<127 GOTO 40
60  SET(116,X)
100  PRINT"YOU HAVE";F;"GALLONS OF FUEL"
110  PRINT"YOU ARE";H;"MILES HIGH"
120  PRINT"YOU ARE GOING";S;"MPH"
130  INPUT"HOW MUCH FUEL DO YOU WANT TO BURN?";B
131  RESET(116,H)
132  F=F-B
134  S=S+16
136  S=S-B
138  H=H-S
140  B=0
145  X=45-H
150  IF F<1 GOTO 1000
160  IF H<1 GOTO 1150
165  CLS
170  GOTO 35
1000  PRINT"OUT OF FUEL!"
1005  PAUSE 1900
1010  GOTO 10000
1150  IF S>3 GOTO 1500
1160  PRINT"LANDED SAFELY!"
1165  PAUSE 1900
1170  GOTO 10000
1500  PRINT"CRASH!"
1505  PAUSE 1900
1510  GOTO 10000
10000  CLS
10001  P=2
11000  INPUT"WOULD YOU LIKE TO PLAY AGAIN(1=YES/2=NO)";P
12000  IF P=1 THEN GOTO 1
13000  PRINT"GOODBYE"

"1150 S=3" appears to set the maximum landing velocity above which you die.
I think "SET" turns on a white pixel, indication the position of the lander.

I got that from http://www.vavasour.ca/jeff/level1/entry15.html , which is a frame within http://www.vavasour.ca/jeff/level1/library.html .

 

[JPC]

hey thanks Hootenanny for simplifying the equation

ill be able to do the program during the weekend

PS : planish , try not using numbers as label names , it gets confusing