Two disks are rotating about the same axis. Disk A has a moment of inertia of 3.7 kg · m2 and an angular velocity of +7.3 rad/s. Disk B is rotating with an angular velocity of -9.5 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 -2.6 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 find the moment of inertia of disk B, we can use the principle of conservation of angular momentum.
The angular momentum of an object is given by the product of its moment of inertia and its angular velocity. Mathematically, it can be expressed as:
L = Iω
Where:
L is the angular momentum,
I is the moment of inertia, and
ω is the angular velocity.
Initially, disk A has a moment of inertia of 3.7 kg · m^2 and an angular velocity of +7.3 rad/s. Disk B has an unknown moment of inertia and an angular velocity of -9.5 rad/s. When the disks are linked together, they rotate as a single unit with an angular velocity of -2.6 rad/s.
According to the conservation of angular momentum, the total angular momentum before and after linking the disks should be the same.
Thus, the sum of the angular momentum of disk A and disk B before linking should be equal to the angular momentum of the combined system after linking.
(LA)i + (LB)i = (LA)f + (LB)f
where:
(LA)i and (LB)i are the initial angular momenta of disks A and B respectively,
(LA)f and (LB)f are the final angular momenta of disks A and B respectively.
For disk A:
(LA)i = I1 * ω1 = 3.7 kg·m^2 * 7.3 rad/s
For disk B:
(LB)i = I2 * ω2 = I2 * (-9.5 rad/s)
After linking the disks:
(LA)f = (LA+fLB)f = (I1 + I2) * (-2.6 rad/s)
Now, equate the initial and final angular momenta:
(LA)i + (LB)i = (LA)f + (LB)f
Solving this equation will give us the moment of inertia, I2, of disk B.