A 22.8 kg turntable with a radius of 47 cm

is covered with a uniform layer of dry ice
that has a mass of 4.08 kg. The angular
speed of the turntable and dry ice is initially
0.52 rad/s, but it increases as the dry ice
evaporates.
What is the angular speed of the turntable
once all the dry ice has evaporated? Answer
in units of rad/s. 1.0 m

You are probably suppoased to assume that the turntable is frictionless and "coasting" (unpowered) while the dry ice evaporates. This is a rather unrealistic assumption

Anyway, in that case, you should assume that angular momentum is conserved.
I1 w1 = I2 w2.

If the dry ice is evenly distributed, the moment of inertia I is proportional to total mass, M. Therefore
M1 w1 = M2 w2.
w2 = (22.8)/(22.8-4.7) x 0.52

You are probably supposed to assume that the turntable is frictionless and "coasting" (unpowered) while the dry ice evaporates. This is a rather unrealistic assumption

Anyway, in that case, you should assume that angular momentum is conserved.
I1 w1 = I2 w2.

If the dry ice is evenly distributed, the moment of inertia I is proportional to total mass, M. Therefore
M1 w1 = M2 w2.
w2 = [(22.8 + 4.7)/(22.8)] x 0.52

If you model the turntable-ice mixture, then the moment of inertia will be proportional to mass.

wf=Iinitial/I final* wi

=k(4.08+22.8)/k(22.8) * .52 rad/sec

To find the final angular speed of the turntable once all the dry ice has evaporated, we need to apply the law of conservation of angular momentum. The law states that the total angular momentum of a system remains constant unless acted upon by an external torque.

First, we need to calculate the initial angular momentum of the system. The angular momentum (L) of an object can be calculated by multiplying its moment of inertia (I) and its angular velocity (ω):

L = I * ω

The moment of inertia (I) of the turntable can be calculated using the formula:

I = 1/2 * m * r^2

where m is the mass of the turntable and r is its radius.

Substituting the given values:

m = 22.8 kg
r = 47 cm = 0.47 m

I = 1/2 * 22.8 kg * (0.47 m)^2 = 2.685 kg * m^2

Now, we can calculate the initial angular momentum (L_initial) using the initial angular velocity (ω_initial):

L_initial = I * ω_initial

Substituting the given values:

ω_initial = 0.52 rad/s
L_initial = 2.685 kg * m^2 * 0.52 rad/s = 1.3972 kg * m^2/s

Next, we need to find the moment of inertia (I_final) and the final angular momentum (L_final) once all the dry ice has evaporated. Since the dry ice is on the turntable, its moment of inertia is included in the total moment of inertia.

The moment of inertia of the turntable with the dry ice is:

I_total = I_turntable + I_dry ice

The moment of inertia of the dry ice (I_dry ice) can be calculated using the formula:

I_dry ice = m_dry ice * r^2

where m_dry ice is the mass of the dry ice.

Substituting the given value:

m_dry ice = 4.08 kg
I_dry ice = 4.08 kg * (0.47 m)^2 = 0.900168 kg * m^2

The total moment of inertia (I_total) is the sum of the moment of inertia of the turntable (I_turntable) and the moment of inertia of the dry ice (I_dry ice):

I_total = I_turntable + I_dry ice

Since the moment of inertia of the turntable (I_turntable) remains the same, we get:

I_total = I + I_dry ice = 2.685 kg * m^2 + 0.900168 kg * m^2 = 3.585168 kg * m^2

Finally, we can find the final angular velocity (ω_final) using the final angular momentum (L_final) and the total moment of inertia (I_total):

L_final = I_total * ω_final

We can rearrange this equation to solve for ω_final:

ω_final = L_final / I_total

Since the angular momentum is conserved, the final angular momentum (L_final) will be the same as the initial angular momentum (L_initial). Therefore:

L_final = L_initial = 1.3972 kg * m^2/s

Substituting the values:

ω_final = 1.3972 kg * m^2/s / 3.585168 kg * m^2 ≈ 0.3895 rad/s

Therefore, the angular speed of the turntable once all the dry ice has evaporated is approximately 0.3895 rad/s.