Suppose there existed a planet that went around the sun twice as fast as the earth. What would be its orbital size as compared to that of the earth ?
Suppose there existed a planet that went around the sun twice as fast as the earth. What would be its orbital size as compared to that of the earth ?
Time taken by the Earth to complete one revolution around the Sun,
$T_{ e }=1$ year
Orbital radius of the Earth in its orbit, $R_{e}=1$ $AU$
Time taken by the planet to complete one revolution around the Sun, $T_{p}=\frac{1}{2} T_{e}=\frac{1}{2}$ year
Orbital radius of the planet $=R_{ p }$
From Kepler's third law of planetary motion, we can write:
$\left(\frac{R_{p}}{R_{e}}\right)^{3}=\left(\frac{T_{r}}{T_{e}}\right)^{2}$
$\frac{R_{p}}{R_{e}}=\left(\frac{T_{p}}{T_{e}}\right)^{\frac{2}{3}}$
$=\left(\frac{\frac{1}{2}}{1}\right)^{\frac{2}{3}}=(0.5)^{\frac{2}{3}}=0.63$
Similar Questions
A spherical asteroid having the same density as that of earth is floating in free space. A small pebble is revolving around the asteroid under the influence of gravity near the surface of the asteroid. What is the approximate time period of the pebble?
A spherical asteroid having the same density as that of earth is floating in free space. A small pebble is revolving around the asteroid under the influence of gravity near the surface of the asteroid. What is the approximate time period of the pebble?
Match List$-I$ With List$-II$
$(a)$ Gravitational constant $(G)$
$(i)$ $\left[ L ^{2} T ^{-2}\right]$
$(b)$ Gravitational potential energy
$(ii)$ $\left[ M ^{-1} L ^{3} T ^{-2}\right]$
$(c)$ Gravitational potential
$(iii)$ $\left[ LT ^{-2}\right]$
$(d)$ Gravitational intensity
$(iv)$ $\left[ ML ^{2} T ^{-2}\right]$
Choose the correct answer from the options given below:
Match List$-I$ With List$-II$
| $(a)$ Gravitational constant $(G)$ | $(i)$ $\left[ L ^{2} T ^{-2}\right]$ |
| $(b)$ Gravitational potential energy | $(ii)$ $\left[ M ^{-1} L ^{3} T ^{-2}\right]$ |
| $(c)$ Gravitational potential | $(iii)$ $\left[ LT ^{-2}\right]$ |
| $(d)$ Gravitational intensity | $(iv)$ $\left[ ML ^{2} T ^{-2}\right]$ |
Choose the correct answer from the options given below:
- [NEET 2022]
Kepler discovered
Kepler discovered
The planet Mars has two moons, if one of them has a period $7\, hours,\, 30\, minutes$ and an orbital radius of $9.0 \times 10^{3}\, {km} .$ Find the mass of Mars.
$\left\{\operatorname{Given} \frac{4 \pi^{2}}{G}=6 \times 10^{11} {N}^{-1} {m}^{-2} {kg}^{2}\right\}$
The planet Mars has two moons, if one of them has a period $7\, hours,\, 30\, minutes$ and an orbital radius of $9.0 \times 10^{3}\, {km} .$ Find the mass of Mars.
$\left\{\operatorname{Given} \frac{4 \pi^{2}}{G}=6 \times 10^{11} {N}^{-1} {m}^{-2} {kg}^{2}\right\}$
- [JEE MAIN 2021]
A geostationary satellite is orbiting around an arbitary planet $^{\prime} P ^{\prime}$ at a height of $11 R$ above the surface of $^{\prime} P ^{\prime} ,$ $R$ being the radius of $^{\prime} P .^{\prime}$ The time period of another satellite in hours at a height of $2R$ from the surface of $^{\prime} P ^{\prime}$ is $........$.$^{\prime} P ^{\prime}$ has the time period of $24\, hours.$
A geostationary satellite is orbiting around an arbitary planet $^{\prime} P ^{\prime}$ at a height of $11 R$ above the surface of $^{\prime} P ^{\prime} ,$ $R$ being the radius of $^{\prime} P .^{\prime}$ The time period of another satellite in hours at a height of $2R$ from the surface of $^{\prime} P ^{\prime}$ is $........$.$^{\prime} P ^{\prime}$ has the time period of $24\, hours.$
- [JEE MAIN 2021]