With what angular velocity the Earth should spin in order that a body lying at 45° latitude may become weightless?

Question

With what angular velocity the Earth should spin in order that a body lying at 45° latitude may become weightless? (A) √(g/R) (B) √(2 g/R) (C) 2 √(g/R) (D) None of these.

Tardigrade Admin · Accepted Answer

g'=g-ω2 ⋅ R cos 2 λ if g'=0 at λ=45° ⇒ ω2 R cos 2 45°=g ω2 R((1/2))=g ⇒ ω=√(2g/R)

Q. With what angular velocity the Earth should spin in order that a body lying at $4 5^{\circ}$ latitude may become weightless?

$\frac{g}{R}$

$\frac{2 g}{R}$

$2 \frac{g}{R}$

None of these.

$g^{'} = g - ω^{2} \cdot R cos^{2} λ$
if $g^{'} = 0$ at $λ = 4 5^{\circ}$
$\Rightarrow ω^{2} R cos^{2} 4 5^{\circ} = g$
$ω^{2} R (\frac{1}{2}) = g$
$\Rightarrow ω = \frac{2 g}{R}$

Q. With what angular velocity the Earth should spin in order that a body lying at 45∘ latitude may become weightless?

Rg​​

R2g​​

2Rg​​