The time period of revolution of a satellite | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the time period $T$ be proportional to $R^a M^b G^c$,so $T = K R^a M^b G^c$.Writing the dimensional formulas for each quantity:$[T] = [T]^1$$[

Vedclass · Accepted Answer

$K \sqrt{\frac{R^3}{GM}}$

Vedclass · Answer

(C) Let the time period $T$ be proportional to $R^a M^b G^c$,so $T = K R^a M^b G^c$.Writing the dimensional formulas for each quantity:$[T] = [T]^1$$[R] = [L]^1$$[M] = [M]^1$$[G] = [M^{-1} L^3 T^{-2}]$Substituting these into the equation: $[T]^1 = [L]^a [M]^b [M^{-1} L^3 T^{-2}]^c = [M]^{b-c} [L]^{a+3c} [T]^{-2c}$.Comparing the powers of $M$,$L$,and $T$ on both sides:For $T$: $-2c = 1 \Rightarrow c = -1/2$.For $M$: $b - c = 0 \Rightarrow b = c = -1/2$.For $L$: $a + 3c = 0 \Rightarrow a = -3c = -3(-1/2) = 3/2$.Substituting these values back into the expression: $T = K R^{3/2} M^{-1/2} G^{-1/2} = K \sqrt{\frac{R^3}{GM}}$.

The time period of revolution of a satellite $(T)$ around the earth depends on the radius of the circular orbit $(R)$,mass of the earth $(M)$,and universal gravitational constant $(G)$. The expression for $T$,using dimensional analysis is ($K$ is a constant of proportionality):

Similar Questions

What could be a correct expression for the speed of ocean waves in terms of its wavelength $\lambda$,the depth $h$ of the ocean,the density $\rho$ of sea water,and the acceleration of free fall $g$?

The velocity of water waves $v$ may depend upon their wavelength $\lambda$,the density of water $\rho$,and the acceleration due to gravity $g$. The method of dimensions gives the relation between these quantities as:

Consider two physical quantities $A$ and $B$ related to each other as $E = \frac{B - x^2}{At}$,where $E, x,$ and $t$ have dimensions of energy,length,and time respectively. The dimension of $AB$ is

Why does the concept of dimension have basic importance?

If velocity of light $c$,Planck's constant $h$,and gravitational constant $G$ are taken as fundamental quantities,then express time in terms of dimensions of these quantities.

Vedclass Test Series

Exam Paper Generator

Online Exam Module