A 1.6 kg disk with radius 0.63 m is rotating freely at 55 rad/s around an axis perpendicular to its center. A second disk that is not rotating is dropped onto the first disk so that their centers align, and they stick together. The mass of the second disk is 0.45 kg and its radius is 0.38 m. What is the angular velocity of the two disks combined?

To find the angular velocity of the two disks combined, we can use the principle of conservation of angular momentum.

The angular momentum of an object can be calculated by multiplying its moment of inertia by its angular velocity. For a rotating disk, the moment of inertia 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.

Let's calculate the angular momentum of the first disk before the collision:

L1 = I1 * ω1,

where L1 is the angular momentum, I1 is the moment of inertia of the first disk, and ω1 is its angular velocity.

Given:
Mass of first disk (m1) = 1.6 kg,
Radius of first disk (r1) = 0.63 m,
Angular velocity of first disk (ω1) = 55 rad/s.

First, calculate the moment of inertia (I1) of the first disk:

I1 = (1/2) * m1 * r1^2.

Substituting the given values:

I1 = (1/2) * 1.6 kg * (0.63 m)^2 = 0.8004 kg·m^2.

Now, calculate the angular momentum of the first disk:

L1 = I1 * ω1 = 0.8004 kg·m^2 * 55 rad/s = 44.022 kg·m^2/s.

Next, let's calculate the angular momentum of the second disk before the collision, which is not rotating:

L2 = I2 * ω2,

where L2 is the angular momentum, I2 is the moment of inertia of the second disk, and ω2 is its angular velocity.

Given:
Mass of second disk (m2) = 0.45 kg,
Radius of second disk (r2) = 0.38 m,
Angular velocity of second disk (ω2) = 0 rad/s (as it is not rotating).

Calculate the moment of inertia (I2) of the second disk:

I2 = (1/2) * m2 * r2^2.

Substituting the given values:

I2 = (1/2) * 0.45 kg * (0.38 m)^2 = 0.04071 kg·m^2.

Calculate the angular momentum of the second disk:

L2 = I2 * ω2 = 0.04071 kg·m^2 * 0 rad/s = 0 kg·m^2/s.

Since the two disks are now stuck together, the total angular momentum after the collision will be the sum of their individual angular momenta:

Lt = L1 + L2.

Substituting the calculated values:

Lt = 44.022 kg·m^2/s + 0 kg·m^2/s = 44.022 kg·m^2/s.

Finally, to find the combined angular velocity (ωt) of the two disks, rearrange the equation for angular momentum:

Lt = It * ωt,

where It is the moment of inertia of the combined system and ωt is the combined angular velocity.

Since the disks stick together, their combined moment of inertia It can be calculated by adding their individual moments of inertia:

It = I1 + I2.

Substituting the calculated values:

It = 0.8004 kg·m^2 + 0.04071 kg·m^2 = 0.84111 kg·m^2.

Now, solve for ωt:

Lt = It * ωt.

Rearranging the equation:

ωt = Lt / It = 44.022 kg·m^2/s / 0.84111 kg·m^2 ≈ 52.33 rad/s.

Therefore, the angular velocity of the two disks combined is approximately 52.33 rad/s.

Compute the moment of inertia of each disc using the formula

I = (1/2) M R^2

For example, for the initially rotating disc,
I1 = (1/2)(1.6)(0.63)^2 = 0.318 kg m^2
I2 = (1/2)(0.46)(0.38)^2 = ____?

Then apply the law of conservation of angular momentum in this form:

I1*w1 = (I1 + I2) w2

and solve for w2.