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## Homework Statement

Now , I Will Begin by stating the calculation and i will post the query at the end.

so here we begin,

we know that If W is the work done by a conservative force,

potential energy change is given by:

ΔW= - ΔU .....(1)

ie when work done by conservative force is positive , Potential energy decreases ..right?

now im going to derive the work done in lifting a body up from say earth's surface (r1) to some height(r2) let small body be of mass 'm' and the Huge body (earth ,say) be 'M".

now F(conservative force) ,points downwards ,and the displacement is upwards

so, dw = -Fdr

integrating over the limit , r1 to r2,(Using Newtons Law F = GMm/r^2)

............r2.....................................r2

∫dw = -∫fdr → w = - GmMv ∫(1/r^2)dr → w= GmM(1/r2 - 1/r1)

...........r1..................................... r1

now since r2>r1 (Its Going up) , W is Negative , and Hence by (1) Potential Energy Change is Positive (Which is true),

but Suppose we consider a case where a body 'Falls' from r1 to r2

now here F(gravitational force) and dr(displacement) both Point along the same direction.

so , dw = fdr

..........r2 ............................. r2

∫dw = ∫fdr → w= GMm(-1/r) → w= GMm(1/r1-1/r2)

..........r1................................r1

Now Since r1>r2,

w is coming negative

and then by ....(1) ΔU is coming Positive .... I mean How is THis ? THe Change in Potential energy levels should be negative.

Now i do know that potential energy at any point is by convention considered negative , but this here is the 'change' and it should come negative too , but it doesnt ... i dont know why this is happening ? can anyone explain ,where i am going wrong ?

if i just solve it using change in energy levels,im getting it right , but why not this way ?

NOTE : Here i am thinking of the work done by the gravitational force , which is pointing downwards