Use equation (2.190) to show that the razor-thin disk for which the circular speed is given by (2.262a) has surface density (toomre 1963) 1 σ(r ) (2.262b) v1 a2/r2 2tgr show that these formulae correspond to the mestel disk in an appropriate limit. show that the surface density (2.68b) of the kuzmin disk is obtained when £(r) is differentiated with respect to a2, and hence recover the circular speed of this model

An athlete swings a ball, connected to the end of a chain, in a horizontal circle. the athlete is able to rotate the ball at the rate of 8.13 rev/s when the length of the chain is 0.600 m. when he increases the length to 0.900 m, he is able to rotate the ball only 6.04 rev/s. (a) which rate of rotation gives the greater speed for the ball? 6.04 rev/s 8.13 rev/s (b) what is the centripetal acceleration of the ball at 8.13 rev/s? m/s2 (c) what is the centripetal acceleration at 6.04 rev/s? m/s2