A student holds a spinning bicycle wheel while sitting motionless on a stool tha

Question

A student holds a spinning bicycle wheel while sitting motionless on a stool that is free to rotate about a vertical axis through its center (see the figure below). The wheel spins with an angular speed of 17.6 rad/s and its initial angular momentum is directed up. The wheel's moment of inertia is 0.140 kg·m2 and the moment of inertia for the student plus stool is 2.60 kg·m2. HINT Lwheel Lwheel -a (a) Find the student's final angular speed (in rad/s) after he turns the wheel over so that it spins at the same speed but with its angular momentum directed down 2.464 X rad/s (b) Will the student's final angular momentum be directed up or down? up down

Explanation / Answer

Consider (wheel + student + stool) as a single system :

since to change the direction of spinning of wheel, student has to exert a torque on the wheel and then there will be similar amount of torque exerted by wheel on the man(according to Newton's third law of motion). Since we are considering the (wheel + student + stool) as a single system, so net torque acting on the system will be zero.Hence, we can say that angular momentum will remain constant and we can apply law of conservation of angular momentum.

Given :    (considering upward direction of angular momentum as positive )

Initial velocity of wheel, Ww = 17.6 rad / s

Let angular velocity of stool be Ws

Momement of inertia of wheel, Iw = 0.140 kg.m2

moment of inertia of the student + stool , Is = 2.60 kg.m2

Initial angular momentum of system(when angular momentum of wheel acting upward)

Li = + Lw = Iw * Ww (since only wheel is rotating)

= 0.140 * 17.6 = 2.464

Final angular momentum of the system :

Lf = Ls - Lw   (since angular momentum of wheel pointed downward )

apply law of conservation of angular momentum :

Li = Lf  

Lw = Ls - Lw

Ls = 2 * Lw   

Is * Ws = 2 * 2.464

2.60 * Ws = 2 * 2.464

Ws = 1.9 rad / s   

Since, Ls = 2 Lw is positive só, angular momentum of student Ls will act in the initial direction of angular momentum of wheel , so in upward direction

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