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Gravitational potential difference

An object at a height "h"

Δ U = - G M m R + h G M m R

Δ U = G M m 1 R 1 R + h

We can eliminate reference to gravitational constant and mass of Earth by using relation of gravitational acceleration at Earth’s surface ( g = g 0 ),

g = G M R 2

G M = g R 2

Substituting in the equation of change in potential energy, we have :

Δ U = m g R 2 1 R 1 R + h

Δ U = m g R 2 X h R R + h

Δ U = m g h 1 + h R

It is expected that this general formulation for the change in potential energy should be reduced to approximate form. For h<<R, we can neglect “h/R” term and,

Δ U = m g h

Maximum height

For velocity less than escape velocity (the velocity at which projectile escapes the gravitation field of Earth), the projected particle reaches a maximum height and then returns to the surface of Earth.

When we consider that acceleration due to gravity is constant near Earth’s surface, then applying conservation of mechanical energy yields :

1 2 m v 2 + 0 = 0 + m g h

h = v 2 2 g

However, we have seen that “mgh” is not true measure of change in potential energy. Like in the case of change in potential energy, we come around the problem of variable acceleration by applying conservation of mechanical energy with reference to infinity.

Maximum height

The velocity is zero at maximum height, "h".

K i + U i = K f + U f

G M m R + 1 2 m v 2 = 0 + G M m R + h

G M R + h = G M R v 2 2

R + h = G M G M R v 2 2

h = G M G M R v 2 2 R

h = G M G M + v 2 R 2 G M R v 2 2

h = v 2 R 2 G M R v 2

We can also write the expression of maximum height in terms of acceleration at Earth’s surface using the relation :

G M = g R 2

Substituting in the equation and rearranging,

h = v 2 2 g v 2 R

This is the maximum height attained by a projection, which is thrown up from the surface of Earth.

Example

Problem 1: A particle is projected vertically at 5 km/s from the surface Earth. Find the maximum height attained by the particle. Given, radius of Earth = 6400 km and g = 10 m / s 2 .

Solution : We note here that velocity of projectile is less than escape velocity 11.2 km/s. The maximum height attained by the particle is given by:

h = v 2 2 g v 2 R

Putting values,

h = 5 X 10 3 2 2 X 10 5 X 10 3 2 6.4 X 10 6

h = 25 X 10 6 2 X 10 25 X 10 6 6.4 X 10 6

h = 1.55 x 10 6 = 1550000 m = 1550 k m

It would be interesting to compare the result, if we consider acceleration to be constant. The height attained is :

h = v 2 2 g = 25 X 10 6 20 = 1.25 X 10 6 = 1250000 m = 1250 k m

As we can see, approximation of constant acceleration due to gravity, results in huge discrepancy in the result.

Escape velocity

In general, when a body is projected up, it returns to Earth after achieving a certain height. The height of the vertical flight depends on the speed of projection. Greater the initial velocity greater is the height attained.

Here, we seek to know the velocity of projection for which body does not return to Earth. In other words, the body escapes the gravitational influence of Earth and moves into interstellar space. We can know this velocity in verities of ways. The methods are equivalent, but intuitive in approach. Hence, we shall present here all such considerations :

1: binding energy :

Gravitational binding energy represents the energy required to eject a body out of the influence of a gravitational field. It is equal to the energy of the system, but opposite in sign. In the absence of friction, this energy is the mechanical energy (sum of potential and kinetic energy) in gravitational field.

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
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or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Gravitation fundamentals. OpenStax CNX. Sep 26, 2007 Download for free at http://cnx.org/content/col10459/1.2
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