if the earth rotates x times faster than its present speed about its axis in rder that the bodies lying on its equator just fly off into space ,then x nearly equals??

Question

jamesj · Answer

So what is the centripetal force?  mv^2/r.  You need that to be greater than the gravitational force, mg.

jamesj · Answer

Alternatively, write centripetal force as
$$ F_c = m\omega^2 r $$

Now use that to find the angular velocity $ \omega $ required for things to spin off the earth and compare that with the earth's current angular velocity.

aravindg · Answer

well i tried like this 
g'=g(1-rw^2/g cos^2 theta)

jamesj · Answer

at the equator theta = 0.

aravindg · Answer

ys

aravindg · Answer

g'=g(1-rw^2/g)

jamesj · Answer

The equation you've written down is the acceleration of a body experiencing both forces, as the centripetal force acts against gravity here
$$ F_g - F_c = mg - m\omega^2 r = m(g - \omega^2r) = m.g.(1 - \omega^2r/g) $$

jamesj · Answer

So whatever you like.  You want to find the $ \omega $ such that that acceleration, what you call g', is equal to zero.

aravindg · Answer

wel according to my equation rw^2/g should become equal to 1 is it?

jamesj · Answer

Yes.  The force is zero and g' = 0 precisely when
$$ frac{r \omega^2}{g} = 1 $$
Equivalently, when
$$ r\omega^2 = g $$
or 
$$ \omega = \sqrt{g/r} $$

jamesj · Answer

* $$ \frac{r\omega^2}{g} = 1 $$

aravindg · Answer

and james u made a typo its centrifugal force not centripetal

jamesj · Answer

yes, fine.

aravindg · Answer

so where does this x come?

aravindg · Answer

i mean we have to compare with present omega and xomega

jamesj · Answer

Find the $ \omega $ for which things would fly off at the equator.  And compare it with the current angular velocity of the earth, call that $ \omega_0 $.  Then by definition of x, "how many times faster than the earth spins now",
$$ x = \frac{\omega}{\omega_0} $$

aravindg · Answer

for that we need value of w.do u knw tht?

jamesj · Answer

C'mon.  You can figure out omega from the formula you wrote down above.

And omega_0, the current angular velocity of the earth you can figure out!! How long does it take the earth to complete one rotation, $ 2\pi $ ?

aravindg · Answer

hmm.7.27*10^-5

aravindg · Answer

:)

jamesj · Answer

right.

aravindg · Answer

wat abt radius isnt it 6400000m

aravindg · Answer

wow!!i got the final answer as 17!

jamesj · Answer

Yes, I got that too.

aravindg · Answer

yey thanks for help

aravindg · Answer

one last qn and i am ovr with gravitation

aravindg · Answer

ready?