A disk of radius 0.4 cm and mass 0.7 kg is rotating at 5 rad/s. A second disk of radius 0.5 cm and mass 0.85 kg is rotating at 6 rad/s in the other direction. What is the combined angular velocity of the disks when the second disk is placed on top of the first?
Should I add up the momemta of the two disks separately and then combine them into one "disk" to find the speed? I have this equation set up.
1/2(0.7)(0.04)^2(5)+1/2(0.85)(0.05)^2(-6)=1/2((0.7)(0.04)^2 + (0.85)(0.05)^2)x
If they are on the same rotatonal axis, addthem to get momentum. Then since momentum is conserved, set that equal to added moments of inertia*final angular velocity.
To find the combined angular velocity of the two disks, you need to consider the conservation of angular momentum. The angular momentum of a rotating object is given by the formula:
L = I * ω
Where L is the angular momentum, I is the moment of inertia of the object, and ω is the angular velocity.
In this case, we have two disks rotating in opposite directions. The moment of inertia of a disk is given by the formula:
I = (1/2) * m * r^2
Where m is the mass of the disk and r is its radius.
To find the combined angular velocity when the second disk is placed on top of the first, you can add up the angular momenta of the individual disks and equate it to the angular momentum of the combined system.
Let's calculate it step by step:
1. Find the angular momentum of the first disk:
L1 = (1/2) * (0.7 kg) * (0.4 cm)^2 * (5 rad/s)
2. Find the angular momentum of the second disk (taking into account its opposite direction of rotation):
L2 = (1/2) * (0.85 kg) * (0.5 cm)^2 * (-6 rad/s)
3. Add the two angular momenta together to get the total angular momentum of the combined system:
L_combined = L1 + L2
4. Find the total moment of inertia of the combined system:
I_combined = [(1/2) * (0.7 kg) * (0.4 cm)^2] + [(1/2) * (0.85 kg) * (0.5 cm)^2]
5. Finally, calculate the combined angular velocity:
L_combined = I_combined * ω_combined
ω_combined = L_combined / I_combined
Now, you can substitute the values you have into these equations to find the combined angular velocity.