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Assume that there is a smooth tunnel of ...

Assume that there is a smooth tunnel of depth `(R )/(2)` along a diameter of earth. A particle is projected from the bottom of tunnel with speed u. Find the minimum value of u so that the particle is able to escape the gravitational field of earth. M and R represents mass and radius of earth

A

`sqrt((2GM)/(R ))`

B

`2 sqrt((GM)/(R ))`

C

`sqrt((11)/(4)(GM)/(R ))`

D

`sqrt((11)/(8)(GM)/(R ))`

Text Solution

Verified by Experts

The correct Answer is:
C
Gravitation potential at a distance of `(R )/(2)` from the centre of earth is
`V= -(GM)/(2R^(3)) (3R^(2)- (R^(2))/(4))= -(11)/(8) (GM)/(R )`
Applying energy conservation
`(1)/(2) m u^(2)- (11)/(8) (GMm)/(R )= 0 rArr u= sqrt((11)/(4) (GM)/(R ))`
`E_(x) = 2E cos theta = 2 ((kQ)/(a^(2) + y^(2))) ((a)/(sqrt(a^(2) + y^(2))))= (Qa)/(2pi epsi_(0)(a^(2)+ y^(2))^(3//2))`
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