Q37 of 38 Page 63

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?

Given


The eccentricity of earth’s orbit e = 0.0167


Let the earth - sun distance at the aphelion and perihelion be ra and rp respectively, and the angular velocity at those points be ωa and ωp. Now since the angular momentum of the earth is conserved we have




Now, from the ellipse we have



Where a is the semi - major axis of the ellipse




Since e = 0.0167



Now let the mean angular velocity of earth corresponding to the mean solar day be ω, then



And since the mean angular velocity is the geometric mean of the two, we have



Now we know that the mean solar day corresponds to 1° on the orbit of earth. A mean solar day has 24 h for which the earth rotates 361°. Thus for aphelion the earth rotates 361.034° and hence the time for the day is 24 hours and 8.1 seconds, and the for the perihelion the earth rotates 359.96° and the time is 23 hours and 52 seconds. Thus the longest day is 8.1 seconds long and the shortest day is 8 seconds shorter. However, the variation of length of day in the year is not explained by this phenomenon as the total time of the day still remains around 24 hours, while the amount of time the sun is visible changes.


More from this chapter

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34

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 ω, 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.

 35

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.

 36

A satellite is to be placed in equatorial geostationary orbit around earth for communication.

(a) Calculate height of such a satellite.


(b) 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.


[M = 6 × 1024 kg, R = 6400 km, T = 24h, G = 6.67 × 10 - 11 SI units]


 38

A satellite is in an elliptic orbit around the earth with aphelion of 6R and perihelion of 2 R where R = 6400 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 × 10 - 11 SI units and M = 6 × 1024 kg]