I already found the common angular velocity, but then the question asks me to fi

Question

I already found the common angular velocity, but then the question asks me to find the thermal energy created in dropping disk 1 onto disk 2 in order to obtain a common angular velocity for the two disks and I have Idea how to do this.

Two disks are initially spinning above one another, as shown in the figure below. Both disks have the same radius, R = 2.59 m. Disk 1 has a moment of inertia I1 = 10.7 kgm2. Disk 2 has a moment of inertiaI2 = 6.4 kgm2. Disk 1 is initially spinning with angular velocity 1 = 25.8 rad/s, and disk 2 is initially spinning with angular velocity 2 = -16.1 rad/s. Disk 1 is then dropped on disk 2, and eventually the two discs reach a common, final angular velocity. Find their final angular velocity. Let counteclockwise rotation as viewed from above be positive. I already found the common angular velocity, but then the question asks me to find the thermal energy created in dropping disk 1 onto disk 2 in order to obtain a common angular velocity for the two disks and I have Idea how to do this.

Explanation / Answer

a)

conservation of angular momentum:

I1 w1 + I2 w2 = (I1 + I2) w

==> w = (I1 w1 + I2 w2)/(I1 + I2)

==> w = (10.7*25.8 - 6.4 * 16.1)/(10.7+6.4)

==> w = 10.118

==> w = 10.1 rad/s

b)

Q = thermal energy =K_initial - K_final

==> Q = (0.5 I1 w1^2 + 0.5 I2 w2^2) - 0.5 (I1+I2) w^2

==> Q = (0.5*10.7*25.8*25.8 + 0.5*6.4*16.1*16.1) - 0.5 * (10.7+6.4) * 10.118*10.118

==> Q = 3515 J

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