A uniform disk of mass 10 m and radius 3.0 r can rotate freely about its fixed c

Question

A uniform disk of mass 10mand radius 3.0rcan rotate freely about its fixed center like a merry-go-round. A smaller uniform disk of massmand radiusrlies on top of the larger disk, concentric with it. Initially the two disks rotate together with an angular velocity of 20rad/s. Then a slight disturbance causes the smaller disk to slide outward across the larger disk, until the outer edge of the smaller disk catches on the outer edge of the larger disk. Afterward, the two disks again rotate together (without further sliding). (a) What then is their angular velocity about the center of the larger disk? (b) What is the ratioK/K0of the new kinetic energy of the two-disk system to the system's initial kinetic energy? A uniform disk of mass 10mand radius 3.0rcan rotate freely about its fixed center like a merry-go-round. A smaller uniform disk of massmand radiusrlies on top of the larger disk, concentric with it. Initially the two disks rotate together with an angular velocity of 20rad/s. Then a slight disturbance causes the smaller disk to slide outward across the larger disk, until the outer edge of the smaller disk catches on the outer edge of the larger disk. Afterward, the two disks again rotate together (without further sliding). (a) What then is their angular velocity about the center of the larger disk? (b) What is the ratioK/K0of the new kinetic energy of the two-disk system to the system's initial kinetic energy? A uniform disk of mass 10mand radius 3.0rcan rotate freely about its fixed center like a merry-go-round. A smaller uniform disk of massmand radiusrlies on top of the larger disk, concentric with it. Initially the two disks rotate together with an angular velocity of 20rad/s. Then a slight disturbance causes the smaller disk to slide outward across the larger disk, until the outer edge of the smaller disk catches on the outer edge of the larger disk. Afterward, the two disks again rotate together (without further sliding). (a) What then is their angular velocity about the center of the larger disk? (b) What is the ratioK/K0of the new kinetic energy of the two-disk system to the system's initial kinetic energy?

Explanation / Answer

a) Calculation of w

I1w1 = I2w2 -----1)

where w1 and w2 are the angular momentum

Calculating moments of inertia
=>I1 = 1/2mr2( 10*9 + 1)= 91/2mr2

=> I1 = 1/2mr2( 10*9 + 1 + 2*4)= 99/2mr2

Putting the values back in the equation,

91w1 = 99w2

w2 = (91/99)*20 = 18.38 rad/s

b) Calculation of ratio

K/K0 = I1(w1)^2/I2(w2)^2

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