Two disks are rotating about the same axis. Disk A has a moment of inertia of 3.6 kg · m2 and an angular velocity of +7.9 rad/s. Disk B is rotating with an angular velocity of -10.6 rad/s. The two disks are then linked together without the aid of any external torques, so that they rotate as a single unit with an angular velocity of -2.5 rad/s. The axis of rotation for this unit is the same as that for the separate disks. What is the moment of inertia of disk B?
To solve this problem, we can use the principle of conservation of angular momentum. According to this principle, the total angular momentum of a system remains constant unless acted upon by an external torque.
The angular momentum of an object is given by the product of its moment of inertia (I) and its angular velocity (ω). Mathematically, angular momentum (L) is defined as:
L = I * ω
In the initial state, before the disks are linked together, the total angular momentum of the system is the sum of the angular momentum of each individual disk:
L_initial = L_A + L_B
where L_A is the angular momentum of Disk A and L_B is the angular momentum of Disk B.
After the disks are linked together, they rotate as a single unit with a new angular velocity. Let's call this angular velocity ω_f. The total angular momentum after the linking is:
L_final = I_total * ω_f
where I_total is the total moment of inertia of the combined system.
Since we know the initial angular velocities and the final angular velocity, we can set up the equation:
L_initial = L_final
L_A + L_B = I_total * ω_f
Substituting the expressions for angular momentum and simplifying, we get:
I_A * ω_A + I_B * ω_B = (I_A + I_B) * ω_f
Now, let's plug in the given values:
I_A = 3.6 kg·m^2 (moment of inertia of Disk A)
ω_A = +7.9 rad/s (angular velocity of Disk A)
I_B = ? (moment of inertia of Disk B)
ω_B = -10.6 rad/s (angular velocity of Disk B)
ω_f = -2.5 rad/s (final angular velocity of the combined system)
The equation becomes:
3.6 kg·m^2 * 7.9 rad/s + I_B * (-10.6 rad/s) = (3.6 kg·m^2 + I_B) * (-2.5 rad/s)
Now, let's solve for I_B:
28.44 kg·m^2 - 10.6 I_B = -9 I_B - 2.5 I_B
Grouping the terms with I_B on one side, we get:
10.6 I_B - 9 I_B - 2.5 I_B = 28.44 kg·m^2
-2.9 I_B = 28.44 kg·m^2
Dividing both sides by -2.9, we find:
I_B = -28.44 kg·m^2 / -2.9
I_B = 9.8 kg·m^2
Therefore, the moment of inertia of Disk B is 9.8 kg·m^2.