Name: Inertial Frames of Reference
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: Imagine if you are on a bus or a train, can you say you are in an inertial frame of reference? If yes, how can you predict it? Let us explore more in this vi...

Question

The Jupiter's period of revolution around the Sun is $12$ times that of the Earth. Assuming the planetary orbits to be circular, find,

$(a)$ how many times the distance between the Jupiter and the Sun exceeds that between the Earth and the Sun,

$(b)$ the velocity and the acceleration of Jupiter in the heliocentric reference frame.

Take ( $G = 6.67 \times 10^{- 11} N m^{2} {kg}^{- 2}$ , Radius of Jupiter's orbit = $R_{J} = 777.8 \times 10^{9} m$ )

Accepted Answer

For any planet of mass $M$ revolving in an orbit of radius $(a)$ , the centripetal force is given as,

$M R ω^{2} = \frac{G M m_{s}}{R^{2}}$ .

Here, $G$ is the universal gravitational constant and $m_{s}$ is the mass of the Sun.

or $ω = \sqrt{\frac{G m_{s}}{R^{3}}} . . . (1)$

So, time period

$T = \frac{2 π}{ω} = \frac{2 π R^{\frac{3}{2}}}{\sqrt{G m_{s}}} . . . (2)$

$∴ T \propto R^{\frac{3}{2}}$

$(a)$ Thus, $\frac{T_{J}}{T_{E}} = {(\frac{R_{J}}{R_{E}})}^{\frac{3}{2}}$

Here, $T_{J}$ and $T_{E}$ are the time period of Jupiter and Earth, respectively. As $T_{J} = 12 years$ and $T_{E} = 1 year$ (as per given in the question), so

$\frac{R_{J}}{R_{E}} = {(\frac{T_{J}}{T_{E}})}^{\frac{2}{3}} = (12)^{\frac{2}{3}} = 5.2 . . . (3)$

$(b)$ As gravitational force provides necessary centripetal acceleration, so for Jupiter

$\frac{M {v_{J}}^{2}}{R_{J}} = \frac{G M m_{s}}{{R_{J}}^{2}}$

where, $v_{J}$ is the speed of Jupiter.

$\Rightarrow v_{J} = \sqrt{\frac{G m_{s}}{R_{J}}} \Rightarrow v_{J} = \sqrt{\frac{(6.67 \times 10^{- 11}) \times (1.97 \times 10^{30})}{777.8 \times 10^{9}}} \Rightarrow v_{J} = 13 km s^{- 1}$

Acceleration of Jupiter in heliocentric reference frame,

$a = \frac{v_{J}^{2}}{R_{J}} = \frac{{(13 \times 10^{3})}^{2}}{777.8 \times 10^{9}}$

$\Rightarrow a = 2.2 \times 10^{- 4} km s^{- 2}$

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