To solve this problem, we can apply the principle of conservation of angular momentum. The angular momentum of an object is given by the product of its moment of inertia and angular velocity.
The angular momentum of the system before the disk is placed on the turntable is zero since the disk is not rotating.
After the disk is placed on the turntable, the angular momentum of the system is given by the sum of the angular momenta of the disk and the turntable.
The moment of inertia of a circular disk is given by the formula:
I = (1/2) * m * r^2
where m is the mass of the disk and r is the radius of the disk.
Given that the mass of the disk is 0.2 kg and the radius is 24 cm (0.24 m), the moment of inertia of the disk is:
I_disk = (1/2) * 0.2 kg * (0.24 m)^2
The moment of inertia of the turntable is given by the same formula, with the mass and radius of the turntable:
I_turntable = (1/2) * 1.7 kg * (0.24 m)^2
The initial angular momentum of the system is zero, so we can set the sum of the angular momenta of the disk and the turntable equal to zero:
I_disk * w_disk + I_turntable * w_turntable = 0
where w_disk is the angular velocity of the disk and w_turntable is the angular velocity of the turntable.
Substituting the expressions for the moment of inertia of the disk and turntable, we get:
(1/2) * 0.2 kg * (0.24 m)^2 * w_disk + (1/2) * 1.7 kg * (0.24 m)^2 * w_turntable = 0
Simplifying the equation, we have:
(0.864 * 0.2 kg * w_disk) + (0.864 * 1.7 kg * w_turntable) = 0
Now we can solve for w_turntable, the new angular velocity of the combination:
w_turntable = -((0.864 * 0.2 kg * w_disk) / (0.864 * 1.7 kg))
Given that the initial angular velocity of the turntable is 3.4 rad/s, we can substitute the value into the equation:
w_turntable = -((0.864 * 0.2 kg * 3.4 rad/s) / (0.864 * 1.7 kg))
Solving for w_turntable, we get:
w_turntable = -1.7 rad/s
Therefore, the new angular velocity of the combination is -1.7 rad/s.