Two disks are rotating about the same axis. Disk A has a moment of inertia of `3.4 kg-m^(2)` and an angular velocity of `+72 "rad"//s`. Disk B is rotating with an angular velocity of `-9.8 "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 `-2.4 "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?

Options

A. `4.4 kg. m^(2)`
B. `5.2 kg m^(2)`
C. ` 2.4 kg. m^(2)`
D. `3.6 kg m^(2)`

Answer

To solve the problem, we will use the principle of conservation of angular momentum. The total angular momentum before the disks are linked together must equal the total angular momentum after they are linked, since no external torques are acting on the system.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Moment of inertia of Disk A, \( I_A = 3.4 \, \text{kg-m}^2 \)
- Angular velocity of Disk A, \( \omega_A = +72 \, \text{rad/s} \)
- Angular velocity of Disk B, \( \omega_B = -9.8 \, \text{rad/s} \)
- Final angular velocity after linking, \( \omega_f = -2.4 \, \text{rad/s} \)
- Moment of inertia of Disk B, \( I_B \) (unknown).

2. **Write the Angular Momentum Conservation Equation**:
The total initial angular momentum \( L_i \) is the sum of the angular momentum of both disks:
\[ L_i = I_A \omega_A + I_B \omega_B \]
The total final angular momentum \( L_f \) when both disks are linked together is:
\[ L_f = (I_A + I_B) \omega_f \]

3. **Set Up the Equation**:
Since angular momentum is conserved:
\[ I_A \omega_A + I_B \omega_B = (I_A + I_B) \omega_f \]

4. **Substitute Known Values**:
Plugging in the known values:
\[ 3.4 \times 72 + I_B \times (-9.8) = (3.4 + I_B)(-2.4) \]

5. **Calculate the Left Side**:
Calculate \( 3.4 \times 72 \):
\[ 3.4 \times 72 = 244.8 \]
So, the equation becomes:
\[ 244.8 - 9.8 I_B = -2.4(3.4 + I_B) \]

6. **Expand the Right Side**:
Expanding the right side:
\[ 244.8 - 9.8 I_B = -2.4 \times 3.4 - 2.4 I_B \]
Calculate \( -2.4 \times 3.4 \):
\[ -2.4 \times 3.4 = -8.16 \]
Thus, the equation is:
\[ 244.8 - 9.8 I_B = -8.16 - 2.4 I_B \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ 244.8 + 8.16 = 9.8 I_B - 2.4 I_B \]
Simplifying:
\[ 252.96 = 7.4 I_B \]

8. **Solve for \( I_B \)**:
Divide both sides by \( 7.4 \):
\[ I_B = \frac{252.96}{7.4} \approx 34.18 \, \text{kg-m}^2 \]

### Final Answer:
The moment of inertia of Disk B is approximately \( 34.18 \, \text{kg-m}^2 \).

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