IIT JEE , AIEEE Forum - Physics , Mechanics

lal

A force F is given by

F= yi + zj + xk

Calculate the work done by the force along the two paths shown in the figure joining two corners O & P of a cube of side a.

ans are 0 and a²

tka

what are y, z and x? are they just variables or are the displacements along y, z and x resp. or are they distances travelled along y, z and x. resp??

NOTE: in this solution, i have assumed y, z and x to be displacements along y axis, z axis and x axis resp.

it is clear that this force is conservative (use curl of a force to find out).

so the work done does not depend on the path taken.

supposing displacement along x axis is 'x' units and displacement along z axis is 'z' units, (disp. along y = 0),

F = yi + zj + xk

s = xi + 0j + zk

work done = f.s = xz

is this right? i mean i just know what to do, just that i don't know exactly how to implement the concept of 'curl of a force'. if i'm right, its good for both you and me

feynmann

The Curl is non zero here !!!!!

So the answer is going to be different for two paths .

For the 2nd case ( shown by green arrow ) W = 0 dx + a dz = a^2

similarly calculate for the other path.

tka

oye the curl is zero!!

when d stands for partial differentiation,

curl = d/dx y + d/dy z + d/dz x = 0 + 0 + 0 = 0

surely the force is conservative

feynmann

My dear Tka , what you have written is the formula for divergence .

Please note curl is a vector qty given by ( in cartesian coordinates) in determinant form as follows

i j k

d/dx d/dy d/dz

y z x

d stands for partial diff . ( the left & rt bar of determinant is omitted )

which is clearly non zero . Got it ?

tka

no yaar. please explain to me the whole theory behind curl of a force, how is it different from what i used (u called it divergence) and explain its use with the help of a simple problem

feynmann

Plz refer to the standard vector calculus book (like that of Spigel's ) . Anyway , where did u get that (wrong) idea that curl is given by the expn . u used ?

tka

i dont have any such book. can u tell me some website that has material on this topic?

anyway i got that thing from some book called vector analysis. its some theorem called stokes theorem

feynmann

You dont have to go as far as Stoke's th. Just search on Google under the heading '' Divergence & Curl of vector fields '' . I think this may clear ur doubt . It is a basic thing . So, don't worry .

spandana

@feynmann

plz xplain the procedure

a dz = a^2?

it should be (a^2)/ 2

feynmann

see, a is a const. qty & z runs from 0 to a

krishna.gopal

We must not get into curl issue here. We will just find out work done on two paths part by part.
For second path feynmann has given right answer. For the first one if we divide it in 4 parts and write work = _{[ ]}

^{[ ]} F.ds where ds = dxi+dyj+dzk then
W = _{[0 ]}

^{[a ]}0dx +_{[0 ]}

^{[a ]}0dy+_{[0 ]}

^{[a ]}a*dz+_{[a ]}

^{[0 ]}adx
=0

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