NCERT Solutions for Chapter: Gravitation, Exercise 5: LA

A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let $r$ be the distance of the body from the centre of the star and let its linear velocity be $v$ angular velocity $\omega$ kinetic energy K, gravitational potential energy U, total energy E and angular momentum l. As the radius r of the orbit increases, determine which of the above quantities increase and which ones decrease.
 

Six point masses of mass $m$ each are at the vertices of a regular hexagon of side $l$. Calculate the force on any of the masses.
 

 A satellite is to be placed in equatorial geostationary orbit around earth for communication. Calculate height of such a satellite. 

$M=6\times {10}^{24}\text{kg}, R=6400 \text{km}, T=24\text{h}, G=6.67\times {10}^{-11} \text{SI Units}$

A satellite is to be placed in equatorial geostationary orbit around earth for communication. Find out the minimum number of satellites that are needed to cover entire earth so that at least one satellites is visible from any point on the equator.
$R=6400 km,$$M=6\times {10}^{24}\text{kg}, T=24\text{h}, G=6.67\times {10}^{-11}\text{SI Units}$

 Earth’s orbit is an ellipse with eccentricity $0.0167$. Thus, earth’s distance from the sun and speed as it moves around the sun varies from day to day. This means that the length of the solar day is not constant through the year. Assume that earth’s spin axis is normal to its orbital plane and find out the length of the shortest and the longest day. A day should be taken from noon to noon. Does this explain variation of length of the day during the year?
 

 A satellite is in an elliptic orbit around the earth with aphelion of $6R$ and perihelion of $2R$ where $R=6400 \text{km}$ is the radius of the earth. Find eccentricity of the orbit. Find the velocity of the satellite at apogee and perigee. What should be done if this satellite has to be transferred  to a circular orbit of radius $6R$ ?
$G=6.67 \times {10}^{-11} \text{SI Units}, M=6\times {10}^{24}\text{kg}$.