1.71 kg of glass (ρ = 2.66 x 10^3 kg/m3) is shaped into a hollow spherical shell that just barely floats in water. What are the (a) outer and (b) inner radii of the shell? Do not assume the shell is thin.
Volume = 2.66 * 10^3/1.71
V=1555.56 m^3
V=pi(r^2)h
Volume of Glass = (4/3) pi (Ro^3 - Ri^3)
mass of glass = rho V
so
1.71 = 2660 (4/3) pi (Ro^3-Ri^3)
so
(Ro^3 - Ri^3) = 4.8214*10^-4
Now Archimedes assume rho water = 1000
1000 (4/3)pi Ro^3 = 1.71
solve for Ro^3
go back and get Ri^3
take cube roots :)
By the way, your formula is for the volume of a cylinder, not a sphere. Use (4/3)pi r^3
To find the outer and inner radii of the shell, we need to understand a few concepts and formulas.
First, let's calculate the volume of the glass. The density (ρ) of the glass is given as 2.66 x 10^3 kg/m^3, and the mass is given as 1.71 kg. We can use the formula:
Volume (V) = Mass (m) / Density (ρ)
V = 1.71 kg / (2.66 x 10^3 kg/m^3)
V = 0.000643 m^3
Next, let's consider the volume of the hollow spherical shell. The volume of a spherical shell is given by the formula:
V = (4/3)π(R^3 - r^3)
Where R is the outer radius and r is the inner radius of the shell. We need to find these radii.
The volume of the hollow spherical shell is given as 0.000643 m^3. Substituting this into the formula, we get:
0.000643 m^3 = (4/3)π(R^3 - r^3)
Now, let's solve this equation to find the radii. Since we know the value of V and π (pi), we can rearrange the equation to solve for R^3 - r^3:
R^3 - r^3 = (3/4)((0.000643) / π)
R^3 - r^3 = 0.000163 m^3 / π
Now, let's calculate R^3 - r^3:
R^3 - r^3 = 0.000163 m^3 / π
R^3 - r^3 = 0.0000518 m^3
To find the outer and inner radii, we can assume that the thickness of the shell is small compared to the radii. In this case, we can approximate R^3 - r^3 as (R^3). Therefore,
R^3 - r^3 ≈ R^3
0.0000518 m^3 ≈ R^3
Now, we can solve for R:
R ≈ (0.0000518 m^3)^(1/3)
R ≈ 0.0378 m
Finally, to find the inner radius, we can use the relationship between the outer radius and the volume of the glass to get:
0.000643 m^3 = (4/3)π(R^3 - r^3)
Substituting the known values:
0.000643 m^3 = (4/3)π((0.0378 m)^3 - r^3)
Simplifying the equation:
r^3 ≈ (0.0378 m)^3 - (0.000643 m^3 / (4/3)π)
r^3 ≈ 0.00000148 m^3 / (4/3)π
r ≈ (0.00000148 m^3 / (4/3)π )^(1/3)
r ≈ 0.00416 m
Therefore, the (a) outer radius of the shell is approximately 0.0378 m, and the (b) inner radius is approximately 0.00416 m.