A body revolves around the sun 27 times faste | vedclass

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

Question

(B) According to Kepler's third law,the square of the orbital period $T$ is proportional to the cube of the semi-major axis $r$: $T^2 \propto r^3$.Sin

Vedclass · Accepted Answer

$1/9$

Vedclass · Answer

(B) According to Kepler's third law,the square of the orbital period $T$ is proportional to the cube of the semi-major axis $r$: $T^2 \propto r^3$.Since angular velocity $\omega = 2\pi / T$,we have $T = 2\pi / \omega$,which implies $T^2 \propto 1/\omega^2$.Substituting this into Kepler's law: $1/\omega^2 \propto r^3$,which gives $\omega^2 \propto 1/r^3$ or $\omega \propto r^{-3/2}$.Rearranging for $r$,we get $r \propto \omega^{-2/3}$.Given that the body revolves $27$ times faster than the earth,$\omega_{	ext{body}} = 27 \omega_{	ext{earth}}$.The ratio of the radii is $\frac{r_{	ext{body}}}{r_{	ext{earth}}} = \left( \frac{\omega_{	ext{earth}}}{\omega_{	ext{body}}} ight)^{2/3}$.Substituting the values: $\frac{r_{	ext{body}}}{r_{	ext{earth}}} = \left( \frac{1}{27} ight)^{2/3} = \left( (3^3)^{-1} ight)^{2/3} = (3^{-3})^{2/3} = 3^{-2} = \frac{1}{9}$.

$A$ body revolves around the sun $27$ times faster than the earth. What is the ratio of their orbital radii?

Similar Questions

For proposing the universal law of gravitation,which laws were taken for help?

$A$ planet $(P_1)$ is moving around a star of mass $2M$ in an orbit of radius $R$. Another planet $(P_2)$ is moving around another star of mass $4M$ in an orbit of radius $2R$. The ratio of the time periods of revolution of $P_2$ and $P_1$ is . . . . . . .

The average distance of the Earth from the Sun is $L_{1}$. If one year of the Earth is $D$ days,then one year of another planet whose average distance from the Sun is $L_{2}$ will be:

$A$ planet of mass $m$ moves around the sun of mass $M$ in an elliptical orbit. The maximum and minimum distances of the planet from the sun are $r_1$ and $r_2$ respectively. The time period of the planet is proportional to ..........

The figure shows the orbit of a planet $P$ around the sun $S.$ $AB$ and $CD$ are the minor and major axes of the ellipse,respectively.
If $t_1$ is the time taken by the planet to travel along the path $ACB$ and $t_2$ is the time taken to travel along the path $BDA,$ then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module