A thin spherical shell of mass 0.450 kg and diameter 0.190 m is filled with alcohol (ρ = 806 kg/m3). It is then released from rest on the bottom of a pool of water. Find the acceleration of the alcohol filled shell as it rises toward the surface of the water.

To find the acceleration of the alcohol-filled shell as it rises towards the surface of the water, we need to use the principle of buoyancy.

The buoyant force experienced by the shell is equal to the weight of the liquid displaced by the shell. In this case, the liquid is alcohol and the shell is submerged in water, so we need to consider the buoyant force due to the displaced water.

First, let's find the volume of the shell. The shell can be considered as a hollow sphere, so the volume is given by:

V = (4/3)π(r^3 - R^3)

Where r is the radius of the inside surface of the shell and R is the radius of the outside surface of the shell.

Since the diameter of the shell is given as 0.190 m, the radius of the shell is half of that, so r = 0.190/2 = 0.095 m.

Next, we need to find the radius of the outside surface of the shell. Since the shell has a thickness, the radius of the outside surface is equal to the radius of the inside surface plus the thickness of the shell. However, the thickness is not given in the problem statement.

Since we don't have enough information to calculate the thickness, we'll assume that the thickness is very small compared to the radius of the inside surface, so the radius of the outside surface is effectively the same as the radius of the inside surface. Therefore, R = r = 0.095 m.

Now we can calculate the volume of the shell:

V = (4/3)π((0.095)^3 - (0.095)^3)
= (4/3)π(0.00834 - 0.00834)
= 0

Since the volume of the shell is zero, this means that the shell doesn't displace any water when it is submerged in water. Therefore, no buoyant force acts on the shell, and the net force on the shell is equal to its weight.

To find the weight of the shell, we can use the formula:

W = mg

Where W is the weight, m is the mass, and g is the acceleration due to gravity. In this case, the mass of the shell is given as 0.450 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.

W = (0.450 kg)(9.8 m/s^2)
= 4.41 N

Thus, the net force on the shell is 4.41 N, and since no other forces are acting on the shell, this is also the acceleration of the shell as it rises towards the surface of the water.

Therefore, the acceleration of the alcohol-filled shell as it rises towards the surface of the water is 4.41 m/s^2.