From the dimensional consideration, which of the following equations is correct? (Here, T = Time period, R = Radius, M = Mass and G = Gravitational constant)

Question

From the dimensional consideration, which of the following equations is correct? (Here, T = Time period, R = Radius, M = Mass and G = Gravitational constant) (A) T=2π√(R3/GM) (B) =π√(GM/ ) (C) T=2π√(GM/R2) (D) T=2π√(R2/GM)

Tardigrade Admin · Accepted Answer

textT=2π√(R3/GM)=2π√(R3/gR2)=2π√(R/g) [As, GM=gR2] Now, by substituting the dimensions of each quantity on both sides, we get [T]=[(L/LT-2)]1/2=[T]

Q. From the dimensional consideration, which of the following equations is correct?
(Here, $T =$ Time period, $R =$ Radius, $M =$ Mass and $G =$ Gravitational constant)

$T = 2 π \frac{R ^{3}}{GM}$

$= π GM$

$T = 2 π \frac{GM}{R ^{2}}$

$T = 2 π \frac{R ^{2}}{GM}$

$T = 2 π \frac{R ^{3}}{GM} = 2 π \frac{R ^{3}}{g R ^{2}} = 2 π \frac{R}{g}$ [As, $GM = g R^{2}]$
Now, by substituting the dimensions of each quantity on both sides, we get
$[T] = [\frac{L}{L T ^{- 2}}]^{1/2} = [T]$

Q. From the dimensional consideration, which of the following equations is correct? (Here, T= Time period, R= Radius, M= Mass and G= Gravitational constant)

T=2πGMR3​​

=πGM​​

T=2πR2GM​​

T=2πGMR2​​