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?
To find the combined angular velocity of the two disks when the second disk is placed on top of the first, we need to 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 written as L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia of a solid disk rotating about its axis can be calculated using the formula I = (1/2) * m * r^2, where I is the moment of inertia, m is the mass of the disk, and r is the radius of the disk.
Given the data for the first disk: radius (r1 = 0.4 cm) and mass (m1 = 0.7 kg), we can find its moment of inertia (I1) using the formula. Similarly, for the second disk: radius (r2 = 0.5 cm) and mass (m2 = 0.85 kg), we can calculate its moment of inertia (I2).
For the first disk:
I1 = (1/2) * m1 * r1^2
= (1/2) * 0.7 kg * (0.4 cm)^2 (Note: Convert cm to meters as the SI unit is used)
= (1/2) * 0.7 kg * (0.004 m)^2
= 5.6 * 10^(-6) kg m^2
For the second disk:
I2 = (1/2) * m2 * r2^2
= (1/2) * 0.85 kg * (0.5 cm)^2
= (1/2) * 0.85 kg * (0.005 m)^2
= 5.3125 * 10^(-6) kg m^2
Now that we have the individual moment of inertia values for each disk, let's calculate the angular momentum of each disk. Since they are rotating in opposite directions, the angular velocity of the second disk will be negative. So, angular velocity for the first disk (ω1) is 5 rad/s, and for the second disk (ω2) is -6 rad/s.
The angular momentum for each disk is given by:
L1 = I1 * ω1
= 5.6 * 10^(-6) kg m^2 * 5 rad/s
= 2.8 * 10^(-5) kg m^2/s
L2 = I2 * ω2
= 5.3125 * 10^(-6) kg m^2 * (-6) rad/s
= -3.1875 * 10^(-5) kg m^2/s
Now, when the second disk is placed on top of the first, their angular momenta add up. So, the combined angular momentum (L_combined) is given by:
L_combined = L1 + L2
= 2.8 * 10^(-5) kg m^2/s + (-3.1875 * 10^(-5) kg m^2/s)
= -3.875 * 10^(-6) kg m^2/s
Finally, we can find the combined angular velocity (ω_combined) by dividing the combined angular momentum by the combined moment of inertia.
ω_combined = L_combined / (I1 + I2)
= -3.875 * 10^(-6) kg m^2/s / (5.6 * 10^(-6) kg m^2 + 5.3125 * 10^(-6) kg m^2)
= -3.875 * 10^(-6) kg m^2/s / 1.09125 * 10^(-5) kg m^2
= -3.55 rad/s
Therefore, the combined angular velocity of the disks when the second disk is placed on top of the first is -3.55 rad/s. Note that the negative sign indicates the opposite direction of rotation compared to the initial rotation of the first disk.