A horizontal circular platform (m = 66.1 kg, r = 3.01m) rotates about a frictionless vertical axle. A student (m = 56.3kg) walks slowly from the rim of the platform toward the center. The angular velocity, omega, of the system is 3.9 rad/s when the student is at the rim. Find omega when the student is 2.81m from the center.

wf = 3.307 rad/s

Explanation:

Given:-

- The mass of circular platform, mc = 66.1 kg

- The radius of the platform, ro = 3.01 m

- The mass of the student, ms = 56.3 kg

- The combined initial angular velocity of (platform + student), wi = 3.9 rad/s

 At the rim

Find:-

Find omega when the student is 2.81m from the center.

Solution:-

- We will consider the circular platform as a disc whose moment of inertia from a central vertical axis is given by:

                     Idisc = 0.5*mc*ro^2

                     Idisc = 0.5*66.1*3.01^2

                     Idisc = 299.436305 kg.m^2

- The Inertia of the student is function of distance r it is from the central vertical axis. Initially the inertia of student would be r = ro = 3.01 at the rim of the platform is given by:

                    (Istd)_i = ms*ro^2

                    (Istd)_i =56.3*3.01^2

                    (Istd)_i = 510.08363 kg.m^2

- The final inertia of the student after he jumps to a point that is 2.81 radially away from the central vertical axis. The new inertia would be defined by r = 2.81 m.

                    (Istd)_f = ms*r^2

                    (Istd)_f =56.3*2.81^2

                    (Istd)_f = 444.55043 kg.m^2

- We apply the principle of conservation of angular momentum ( L ) before and after the jump.

                    Li = Lf

Li = Initial angular momentum = wi*( Idisc +  (Istd)_i )

Lf = Final angular momentum = wf*(Idisc +  (Istd)_f )

- The final angular velocity wf can be determined:

                 wi*( Idisc +  (Istd)_i ) = wf*(Idisc +  (Istd)_f )

                 wf = [ ( Idisc +  (Istd)_i ) / (Idisc +  (Istd)_f ) ] * wi

                 wf = [ (510.08363 + 299.436305) / ( 510.08363 + 444.55043)]*3.9

                 wf = ( 809.519935 / 954.63406 ) * 3.9

                 wf = 0.84798*3.9

                 wf = 3.307 rad/s