6. Two disks are rotating about the same axis. Disk A has a moment of inertia of 4.00 kg.m² and an angular velocity of +6.80 rad/s. Disk B is rotating with an angular velocity of -11.8 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 -2.50 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? kg.m2 36 skp
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![### Physics Problem: Rotational Motion of Disks
**Problem Statement:**
Two disks are rotating about the same axis. Disk A has a moment of inertia of \(4.00 \, \text{kg.m}^2\) and an angular velocity of \(+6.80 \, \text{rad/s}\). Disk B is rotating with an angular velocity of \(-11.8 \, \text{rad/s}\). 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 \(-2.50 \, \text{rad/s}\). 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?
**Calculation:**
To find the moment of inertia of disk B, we can use the law of conservation of angular momentum, which states that the total angular momentum before the disks are combined is equal to the total angular momentum after they are combined, because no external torques are acting on the system.
The total angular momentum \( L \) is given by:
\[ L = I \cdot \omega \]
Where:
- \( I \) is the moment of inertia
- \( \omega \) is the angular velocity
Initially:
\[ L_{\text{initial}} = I_A \cdot \omega_A + I_B \cdot \omega_B \]
After combining:
\[ L_{\text{final}} = (I_A + I_B) \cdot \omega_f \]
Equating the initial and final angular momentum:
\[ I_A \cdot \omega_A + I_B \cdot \omega_B = (I_A + I_B) \cdot \omega_f \]
Given:
- \( I_A = 4.00 \, \text{kg.m}^2 \)
- \( \omega_A = +6.80 \, \text{rad/s} \)
- \( \omega_B = -11.8 \, \text{rad/s} \)
- \( \omega_f = -2.50 \, \text{rad/s} \)
Substitute these values into the equation:
\[ (4.00 \, \text{kg.m}^2 \cdot 6.80 \, \text{rad/s}) + (I_B \cdot -11.8 \,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04c06aee-69db-4b48-aaad-c9b35ad590a4%2Fefbe1f4e-e523-4ac7-8a4d-179b12e0e023%2Fys6nna_processed.png&w=3840&q=75)
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