18-09-09 Center of Gravity.

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Where is the center of gravity of a very long pole of length L placed vertically on the earths' surface....

(If you think that the answer is obvious.. Think again :P)

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Asish Mahapatra ·2009-09-21 20:19:27

centre of gravity:
consider an element of length dx situated at height x abv the earths surface.

grav force on it due to earth = GdmM/(R+x)² (M=mass of earth)
= GÎ»dxM/(R+x)²

Centre of grav is where the point where the earth's grav force can be thought to act on the object.

Total grav force = âˆ«(0 to l)GÎ»dxM/(R+x)²
= -GÎ»M[1/R+x]₀^l
= GÎ»M[1/R - 1/(R+L)]
= GÎ»LMR(R+L)
= GmMR(R+L) (m=mass of rod)
Let the centre of grav be situated at h height frm earths surface

So, GmMR(R+L) = GmM/(R+h)²

=> R²+h²+2Rh = R²+RL
=> h²+2Rh - RL = 0

=> h = -R + √R²+RL

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eureka123 ·2009-09-21 20:27:17

Qualitatively we can say that centre of gravity will lie at a relatively lower position

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18-09-09 Center of Gravity.

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