Question

what is the gain in potential energy of object of mass m raised from the surface of earth to height equal to radius R of the earth?

Answer 1

should i consider the variation in g with height???...because in some problems we consider and in some we don't...for ex:(here) at height=R,g'=g/4...so the answer will change....

Answer 2

Yes the value of gravitational acceleration g will b change

Answer 3

Can u calculate the work done for taking the object fron the surface of earth to the height R

Answer 4

U know
Gain of potential energy=work done

Answer 5

U know
g=GM/r^2

Answer 6

If r increase then the value of g will decrease

Answer 7

yes i know these..but i am getting the correct answer(as in the book) only if i consider that g wont change...

Answer 8

here is what i did..
g'=g/(1+h/R)^2=>g'=g/4
work done=-(final potential energy - initial potential energy)
=-(-mg'R-0)
=mg'R
=mgR/4
=GMm/4R {since GM/R^2}
but,when i do it in the below method i am getting the right answer
work done=-(GMm/2R - GMm/R) {final P.E - initial P.E}
=-(-GMm/2R)
=GMm/2R
what is wrong in the first method...??

Answer 9

U used
g'=g/4

Answer 10

But when the height of ur object is h=R/2
Then g will b more than g/4

Answer 11

And the value of g will change when the object will go frm the surface of earth to the height h=R

Answer 12

So the value of g is a variable

Answer 13

U used g=g/4 is a constant

Answer 14

Do u know hw to calculate work done by a variable force

Answer 15

why height of the object is R/2?? @shamim

Answer 16

oh now i get it..g changes at each and every point...

Answer 17

yes i do know...W=integral(F.dr)
W=integral 0toR (mgx/(1+x/R)^2.dx)...is this correct?

Answer 18

\[\Delta U = \dfrac{GM_Em}{R_E} - \dfrac{GM_Em}{2R_E}\]

Answer 19

Limit is correct

Answer 20

Limit will b
0 to R

Answer 21

ok what about the rest of the expression??

Answer 22

is the integral correct?

Answer 23

i tried this integration...but the answer does not math

Answer 24

So i will stick to the 2nd method....

Answer 25

@Farcher how to do this question with 'g'?

Answer 26

No u r not correct !!

Answer 27

U know
Work done=gain in potential energy

Answer 28

Work done
dw=F.dr

Answer 29

W=integration of Fdr

Answer 30

that is what i have done...right?

Answer 31

Limit is 0 to R

Answer 32

F=mg

Answer 33

i have written it from 0 to R only..

Answer 34

But g=GM/r^2

Answer 35

yes... i have substituted g as g/(1+x/R)^2 as it is constantly varying...

Answer 36

So
W=integration of (GMm/r^2)dr

Answer 37

This formula comes in the chapter gravitation

Answer 38

Here u hv to use
g=GM/r^2

Answer 39

U know G,M is constant

Answer 40

r is a variable

Answer 41

r is the distance between the centre of earth to the object

Answer 42

yes u are also correct...but i have done it using a formula which gives the variation of g with height..

Answer 43

So
W=GMm* integration (r^-2)dr

Answer 44

W=GMm*(-1/r), limit 0 to R

Answer 45

but GMm*(1/0) is infinity??

Answer 46

To find the potential energy of a mass m at a distance r from a mass M you have to define a zero of potential .
The zero of potential energy is usually taken as infinity.
So the limits of integration from infinity to r.

For this problem since you want a difference in potential energy you need to integrate from R_E to 2R_E.

So there is no division by zero.

Answer 47

Sorry

Answer 48

Limit of integration will b R to 2R

Answer 49

Because
When the object is on the surface of earth then r=R
Because the distance frm the centre of earth to the object is =R

Answer 50

When the object is at height R
Then r=2R
Because the distance frm the centre of earth to the object is=2R

Answer 51

thanks @Farcher and @shamim ! but haven't u heard of this formula:
g'=g/(1+h/R)^2 for finding gravitational acceleration at ht. h ?

Answer 52

I know ur formula

Answer 53

In ur formula h is variable

Answer 54

Nd h is the distance frm the surface of earth to the object

Answer 55

We need the distance frm the centre of earth to the object

Answer 56

So we need r

Answer 57

not h

Answer 58

ok..