Q35 of 38 Page 63

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.

Given


Side of hexagon = l


The force on the masses will be the resultant of the forces of all the other masses. Let the masses be m each. Now the distance AC is given by the parallelogram law i.e.





Due to symmetry we have AC = AE =



Now, for AD we have



Now the force on mass at A due to b is



Also



The force on mass A due to C is



Also due to symmetry, we have



The force on mass at A due to mass at D is



Now, the forces Fae and Fac make equal angles with the direction AD viz. 30°, and therefore by parallelogram law of vector addition their resultant is along AD and therefore the magnitude of the resultant,




Similarly, Fab and Faf also make equal angles with AD viz. 60° and therefore they are also along AD and




Therefore, the net force on mass at A will be





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33

A mass m is placed at P a distance h along the normal through the centre O of a thin circular ring of mass M and radius r (Fig. 8.3).


If the mass is removed further away such that OP becomes 2h, by what factor the force of gravitation will decrease, if h = r?


 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.

 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]


 37

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?