question_answer 59)

                  A star lie the sun has several bodies moving around it as 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 \[\upsilon \], 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.

Answer:

                  (i) Linear velocity, \[\upsilon =\,\sqrt{\frac{GM}{r}}\]. When increase, then \[\upsilon \] decrease.                 (ii) Angular velocity, \[\omega =\,\frac{2\pi }{T}=\,\frac{2\pi }{K{{r}^{3/2}}}\].                 When r increase, \[\omega \] decreases.                 (iii) K.E. \[=\,\frac{GMm}{2r}\]. When r increases, K.E. decrease.                 (iv) Gravitation P.E., \[U=\,\frac{-GM\,m}{r}\]. When \[r\] increases, U increases.                 (v) Total energy = K.E. + P.E.                 \[=\,\frac{-GMm}{2r}\]. When \[r\] increases, total energy increase.                 (vi) Angular momentum, \[l=\,m\upsilon r=mr\,\sqrt{gr}\].                 When r increases, \[l\] also increases.