If a disk rotating at 900 rpm couples to a stationary disk with five times the moment of inertia, what is the angular speed of the combination?
To calculate the angular velocity of the combined system, you can use the principle of conservation of angular momentum. The angular momentum of an object is the product of its moment of inertia and angular velocity.
Let's assume the initial angular velocity of the rotating disk is ω1 and the moment of inertia of the stationary disk is I2, which is five times the moment of inertia of the rotating disk (I1).
According to the conservation of angular momentum, the total angular momentum before the coupling is equal to the total angular momentum after the coupling.
Angular momentum before coupling = Angular momentum after coupling
(I1 * ω1) = (I1 + I2) * ω2
Here, ω2 is the angular velocity of the system after the coupling.
Since you have the initial angular velocity (ω1 = 900 rpm), you need to find the final angular velocity (ω2).
First, you need to determine the relationship between the moments of inertia (I1 and I2).
Given that I2 = 5 * I1,
Substituting this into the equation:
(I1 * ω1) = (I1 + 5 * I1) * ω2
I1 * ω1 = 6 * I1 * ω2
Dividing both sides by I1 gives:
ω1 = 6 * ω2
Now, you can solve for ω2:
ω2 = ω1 / 6
Substituting the value of ω1 (900 rpm) into the equation:
ω2 = 900 rpm / 6
ω2 = 150 rpm
Therefore, the angular speed of the combined system is 150 rpm.