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A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Letr 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 I. As the radius r of the orbit increases, determine which of the above quantities increase and which ones decrease. As shown in figure, where a body of mass m is revolving around a star of mass M. Linear velocity of the body, |
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Answer» SOLUTION :`implies` As shown in figure, where a body of mass m is revolving around a star of mass M. Linear velocity of the body,  `v = sqrt(GM)/R` ` :. V prop 1/sqrtr` Therefore , when r increases , v DECREASES. Angular velcity of the body `omega=(2i)/T` According to Kepler.s `3^(rd)` law, `T^(2) prop r^3` ` :. T = kr ^(3/2)` ` :. omega = (2pi)/(kr^(3/2)) "" ( :. omega= (2pi)/T)` `:. omega prop 1/(r^(3/2))` Kinetic energy of the body , `K = 1/2 mv^2 =1/2 m xx (GM)/r = (GMM)/(2r)` `""( :. v = sqrt((GM)/r))` `:. K prop 1/r` Gravitational potential energy of the body , `U = - (GMm)/(r) implies U prop -1/r` Total energy of the body `E = K +U = (GMm)/(2r) + (-(GMm)/r)= - (GMm)/(2r)` ` :. E prop -1/r` Angular momentum of the body, `L = mvr =mrsqrt((GM)/r) =msqrt(GMr) ` `:. L prop sqrtr` From the above equations it is clear that v , W and K decreases with INCREASE in r And U,E and L increases with increase in r. |
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