Question

Two satellites ${{\text{S}}_{\text{1}}}\text{ and }{{\text{S}}_{2}}$ revolve around a planet in a coplanar orbits in the same sense. The periods of revolution are $1\text{h and 8h}$ respectively. The radius of orbit of ${{\text{S}}_{\text{1}}}$ is ${{10}^{4}}\text{km}$. When ${{\text{S}}_{\text{2}}}\text{ is closest to }{{\text{S}}_{1}}$, find (a) the speed of ${{\text{S}}_{\text{2}}}$ relative of ${{\text{S}}_{\text{1}}}$ and (b) the angular speed of ${{\text{S}}_{\text{2}}}$ as observed by an astronaut in ${{\text{S}}_{\text{1}}}$?

Answer 1

Hint: We can find the angular velocity of an object if we know the time period of the satellite. Also, the linear velocity of both the satellites can be calculated if we know the radius of the orbit.

Complete step by step answer:
We have two satellites ${{\text{S}}_{\text{1}}}\text{ and }{{\text{S}}_{2}}$ whose time period are $1\text{h and 8h}$ respectively. So the angular velocity associated with a satellite whose time period is T, is given by
$\text{Angular Velocity (}\omega \text{)}=\dfrac{2\pi }{T}$
So the angular velocity of the satellite ${{\text{S}}_{\text{1}}}$ is given by,
${{\omega }_{1}}=\dfrac{2\pi }{{{T}_{1}}}$
${{\omega }_{1}}=\dfrac{2\pi }{1}$
${{\omega }_{1}}=2\pi \text{ rad(hr}{{\text{)}}^{-1}}$
So the angular velocity of the satellite ${{\text{S}}_{2}}$ is given by,
${{\omega }_{2}}=\dfrac{2\pi }{{{T}_{2}}}$
${{\omega }_{2}}=\dfrac{2\pi }{8}$
${{\omega }_{2}}=0.25\pi \text{ rad(hr}{{\text{)}}^{-1}}$
So the angular speed of ${{\text{S}}_{2}}$ as seen from the astronaut in ${{\text{S}}_{1}}$ is,
${{\omega }_{21}}={{\omega }_{1}}-{{\omega }_{2}}$
${{\omega }_{21}}=\left( 2-0.25 \right)\pi$
${{\omega }_{21}}=1.75\pi \text{ rad(hr}{{\text{)}}^{-1}}$
So the angular speed of ${{\text{S}}_{\text{2}}}$ with respect to an astronaut in ${{\text{S}}_{\text{1}}}$ is ${{\omega }_{21}}=1.75\pi \text{ rad(hr}{{\text{)}}^{-1}}$
According to Kepler's third law of planetary motion, we know that the square of time period is directly proportional to cube of the semi major axis.
${{T}^{2}}\propto {{a}^{3}}$
Here the path of rotation is considered circular. So we can equate, the ratio of square of time period of ${{\text{S}}_{\text{2}}}$ and ${{\text{S}}_{\text{1}}}$ to the ratio of cube of radius of the orbit of ${{\text{S}}_{\text{2}}}$ and ${{\text{S}}_{\text{1}}}$.
$\dfrac{{{T}_{2}}^{2}}{{{T}_{1}}^{2}}=\dfrac{{{r}_{2}}^{3}}{{{r}_{1}}^{3}}$
We can substitute the values of the time periods and the radius of ${{\text{S}}_{\text{1}}}$ in order to calculate the radius of orbit of ${{\text{S}}_{\text{2}}}$.
$\dfrac{{{8}^{2}}}{{{1}^{2}}}=\dfrac{{{r}_{2}}^{3}}{{{({{10}^{4}}\text{km})}^{3}}}$
${{r}_{2}}={{r}_{1}}{{\left( 8 \right)}^{\dfrac{2}{3}}}$
$\therefore {{r}_{2}}=4\times {{10}^{4}}\text{ km}$
So the linear velocity of a body can be written as the product of radius and angular velocity,
$\text{v}=r\omega$
So the linear velocity of ${{\text{S}}_{\text{1}}}$ is given by,
${{\text{v}}_{\text{1}}}={{r}_{1}}{{\omega }_{1}}$
${{v}_{1}}=({{10}^{4}})(2\pi )$
${{\text{v}}_{\text{1}}}=2\pi \times {{10}^{4}}\text{ km(hr}{{\text{)}}^{-1}}$
The linear velocity of ${{\text{S}}_{2}}$ is given by,
${{\text{v}}_{2}}={{r}_{2}}{{\omega }_{2}}$
${{v}_{2}}=(4\times {{10}^{4}})(0.25\pi )$
${{\text{v}}_{2}}=\pi \times {{10}^{4}}\text{ km(hr}{{\text{)}}^{-1}}$
The speed of ${{\text{S}}_{2}}$ relative to ${{\text{S}}_{1}}$ is given by the difference between ${{\text{v}}_{\text{2}}}\text{ and }{{\text{v}}_{1}}$.
$\text{Relative Velocity (}{{\text{v}}_{21}})={{\text{v}}_{1}}-{{\text{v}}_{2}}$
${{\text{v}}_{21}}=(2-1)\pi \times {{10}^{4}}$
$\therefore {{\text{v}}_{21}}=\pi \times {{10}^{4}}\text{ km(hr}{{\text{)}}^{-1}}$
So the relative speed of ${{\text{S}}_{2}}$ to ${{\text{S}}_{1}}$ is ${{\text{v}}_{21}}=\pi \times {{10}^{4}}\text{ km(hr}{{\text{)}}^{-1}}$

Note:
Angular velocity is the measure of how fast an object rotates about a point.
Two types of angular velocities are there,
1. Orbital Angular Velocity: Velocity with which an object rotates around an object.
2. Spin Angular Velocity: Velocity with which an object rotates about its center of mass.