If the radius of the earth be increased by a factor of 5 , by what factor its density be changed to keep the value of g the same?

Question

If the radius of the earth be increased by a factor of 5 , by what factor its density be changed to keep the value of g the same? (A) 1 / 25 (B) 1 / 5 (C) 1 / √5 (D) 5

Tardigrade Admin · Accepted Answer

g =( GM / R 2) ⇒ g =( G ρ × (4/3) π R 2/ R 2) ⇒ g =(4/3) π GR ρ ...(i) g '=(4/3) π G (5 R ) ρ' ...(ii) given that g=g', so ρ'=(ρ/5)

Q. If the radius of the earth be increased by a factor of $5$ , by what factor its density be changed to keep the value of $g$ the same?

$1/25$

$1/5$

$1/ 5$

5

$g = \frac{GM}{R ^{2}}$
$\Rightarrow g = \frac{Gρ \times \frac{4}{3} π R ^{2}}{R ^{2}}$
$\Rightarrow g = \frac{4}{3} π GRρ ...$ (i)
$g^{'} = \frac{4}{3} π G (5 R) ρ^{'} ...$ (ii)
given that $g = g^{'}$ , so $ρ^{'} = \frac{ρ}{5}$

Q. If the radius of the earth be increased by a factor of 5 , by what factor its density be changed to keep the value of g the same?

1/25

1/5

1/5​

5

g=R2GM​ ⇒g=R2Gρ×34​πR2​ ⇒g=34​πGRρ...(i) g′=34​πG(5R)ρ′...(ii) given that g=g′, so ρ′=5ρ​

Q. If the radius of the earth be increased by a factor of $5$ , by what factor its density be changed to keep the value of $g$ the same?

$1/25$

$1/5$

$1/ 5$

$g = \frac{GM}{R ^{2}}$
$\Rightarrow g = \frac{Gρ \times \frac{4}{3} π R ^{2}}{R ^{2}}$
$\Rightarrow g = \frac{4}{3} π GRρ ...$ (i)
$g^{'} = \frac{4}{3} π G (5 R) ρ^{'} ...$ (ii)
given that $g = g^{'}$ , so $ρ^{'} = \frac{ρ}{5}$