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Q.

The imaginary angular velocity of the earth for which the effective acceleration due to gravity at the equator shall be zero is equal to [AMU 2012J (Take, g = 10 $m s^{- 2}$ for the acceleration due to gravity, if the earth were at rest, radius of earth equal to 6400 km and $λ$ = 60)

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a

5 X $10^{- 3}$ rad $s^{- 1}$

b

1.25 x $10^{- 3}$ rad $s^{- 1}$

c

2.50 x $10^{- 3}$ rad $s^{- 1}$

d

3.75 X $10^{- 3}$ rad $s^{- 1}$

answer is B.

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Detailed Solution

Acceleration due to gravity,

$g = g - ω^{2} R \cos^{2} λ$

$\Rightarrow 0 = g - ω^{2} R \cos^{2} 60 °$

$0 = g - \frac{ω^{2} R}{4}$

$\Rightarrow ω = 2 \sqrt{\frac{g}{R}} = 2 \sqrt{\frac{10}{6400 \times 1000}}$

Angular velocity, $ω = \frac{1}{400} = 2.50 \times 10^{- 3} {rads}^{- 1}$

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The imaginary angular velocity of the earth for which the effective acceleration due to gravity at the equator shall be zero is equal to [AMU 2012J (Take, g = 10 ms-2 for the acceleration due to gravity, if the earth were at rest, radius of earth equal to 6400 km and λ = 60)