Two disks are rotating about the same axis. Disk A has a moment of inertia of 3.8 kg · m2 and an angular velocity of +6.7 rad/s. Disk B is rotating with an angular velocity of -8.9 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 -2.3 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?
Answer in kg*m2
To find the moment of inertia of disk B, 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 torques.
The angular momentum of an object is given by the product of its moment of inertia and angular velocity. Mathematically, it can be expressed as:
L = I * ω
Where:
L = Angular momentum
I = Moment of inertia
ω = Angular velocity
Initially, disk A and disk B are rotating about the same axis, and their individual angular momenta can be expressed as:
LA = IA * ωA
LB = IB * ωB
After the two disks are linked together, they rotate as a single unit. The total angular momentum of the system remains constant. We can express the total angular momentum before and after linking the disks as follows:
Before linking: LA + LB
After linking: (IA + IB) * ωtotal
Since there are no external torques acting on the system, the total angular momentum before and after must be equal. Therefore, we can equate the two expressions:
LA + LB = (IA + IB) * ωtotal
Plugging in the given values:
IA * ωA + IB * ωB = (IA + IB) * ωtotal
Substituting the given values:
IA = 3.8 kg·m^2 (moment of inertia of disk A)
ωA = +6.7 rad/s (angular velocity of disk A)
ωB = -8.9 rad/s (angular velocity of disk B)
ωtotal = -2.3 rad/s (angular velocity of the linked disks)
We can now solve for IB:
(3.8 kg·m^2 * 6.7 rad/s) + (IB * -8.9 rad/s) = (3.8 kg·m^2 + IB) * -2.3 rad/s
Expanding and rearranging the equation:
25.46 kg·m^2 - 8.9IB = -8.74 kg·m^2 - 2.3IB
Combining like terms:
25.46 kg·m^2 + 8.74 kg·m^2 = -2.3IB + 8.9IB
34.2 kg·m^2 = 6.6IB
Dividing both sides by 6.6:
IB = 34.2 kg·m^2 / 6.6
Therefore, the moment of inertia of disk B is approximately 5.18 kg·m^2.