Q41 of 44 Page 9

An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s Third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that

T =


where k is a dimensionless constant and g is acceleration due to gravity.


According to Kepler’s Third law, the square of the time period of a satellite revolving around a planet, is proportional to the cube of the radius of the orbit of the satellite. Let T be the time period of revolution and r is the radius of the orbit, then




Since, the revolution of the satellite follows the laws of gravitation the time period of revolution depends on acceleration due to gravity g and the radius of the planet R,




Where k, is a dimensionless constant of proportionality.


Now the dimensional formula for time period [T] = [M0L0T1], for radius of planet [R] = [M0L1T0], for radius of orbit of satellite [r] = [M0L1T0] and for acceleration due to gravity [g] = [M0L1T-2]. Therefore,




Equating the exponents of like terms on both sides, we have





Substituting these values in the expression for T we obtain,




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If velocity of light c, Planck’s constant h and gravitational constant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

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In an experiment to estimate the size of a molecule of oleic acid 1 mL of oleic acid is dissolved in 19 mL of alcohol. Then 1 mL of this solution is diluted to 20 mL by adding alcohol. Now 1 drop of this diluted solution is placed on water in a shallow trough. The solution spreads over the surface of water forming one molecule thick layer. Now, lycopodium powder is sprinkled evenly over the film and its diameter is measured. Knowing the volume of the drop and area of the film we can calculate the thickness of the film which will give us the size of oleic acid molecule.

Read the passage carefully and answer the following questions:


(a) Why do we dissolve oleic acid in alcohol?


(b) What is the role of lycopodium powder?


(c) What would be the volume of oleic acid in each mL of solution prepared?


(d) How will you calculate the volume of n drops of this solution of oleic acid?


(e) What will be the volume of oleic acid in one drop of this solution?


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(a) How many astronomical units (A.U.) make 1 parsec?

(b) Consider a sunlike star at a distance of 2 parsecs. When it is seen through a telescope with 100 magnification, what should be the angular size of the star? Sun appears to be (1/2)° fromthe earth. Due to atmospheric fluctuations, eye can’t resolve objects smaller than 1 arc minute.


(c) Mars has approximately half of the earth’s diameter. When it is closest to the earth it is at about 1/2 A.U. from the earth. Calculate what size it will appear when seen through the same telescope.


(Comment : This is to illustrate why a telescope can magnify planets but not stars.)