Let S be the part of the sphere
x^2 + y^2 + z^2 = 25
that lies above the plane z = 4. Let S have constant density k.
(a) Find the center of mass.
(b) Find the moment of inertia about the z-axis.
To find the center of mass and the moment of inertia about the z-axis, we need to set up and evaluate the appropriate triple integrals.
(a) To find the center of mass, we need to find the average values of x, y, and z over the region S. The center of mass (x̅, ȳ, ẑ) can be calculated using the following formulas:
x̅ = (1/M) ∫∫∫ xρ dV
ȳ = (1/M) ∫∫∫ yρ dV
ẑ = (1/M) ∫∫∫ zρ dV
where M is the total mass of S and ρ is the density. Since the density is constant, ρ = k.
The first step is to find the limits of integration. From the equation of the sphere, we have x^2 + y^2 + z^2 = 25, which implies that z = ±√(25 - x^2 - y^2). Since we are only interested in the part of the sphere above the plane z = 4, we have z = √(25 - x^2 - y^2).
Now we can write the integral for the x-coordinate:
x̅ = (1/M) ∫∫∫ xρ dV
= k/M ∫∫∫ x dV
We can evaluate this integral by using spherical coordinates. The limits of integration for ρ, θ, and φ are:
0 ≤ ρ ≤ 5
0 ≤ θ ≤ 2π
arcsin(4/5) ≤ φ ≤ π/2
The integral becomes:
x̅ = k/M ∫∫∫ x dV
= k/M ∫₀⁵ ∫₀²π ∫ₐʳᶜˢⁱⁿ(⁴/⁵) π/₂ xρ² sinφ dφ dθ dρ
We can now evaluate this triple integral to find the x-coordinate of the center of mass.
(b) To find the moment of inertia about the z-axis, we need to calculate the following integral:
Iz = ∫∫∫ (x^2 + y^2)ρ dV
Again, since the density is constant, ρ = k. Using spherical coordinates, the limits of integration for ρ, θ, and φ are the same as before.
Iz = ∫∫∫ (x^2 + y^2)ρ dV
= k ∫₀⁵ ∫₀²π ∫ₐʳᶜˢⁱⁿ(⁴/⁵) π/₂ (ρ^2 sinφ)(ρ^2 sinφ cos^2θ + ρ^2 sinφ sin^2θ) dφ dθ dρ
We can now evaluate this triple integral to find the moment of inertia about the z-axis.
To find the center of mass of the given part of the sphere, we need to set up a triple integral to calculate the mass and then find the coordinates of the center of mass.
(a) Center of Mass:
Step 1: Calculate the mass of S.
Since S has constant density k, the mass can be calculated by integrating the density over the volume of S.
The volume of S can be obtained by integrating over the region where z lies between 4 and the upper hemisphere of the sphere with radius 5.
To set up the integral, we can use spherical coordinates. The equation of the sphere in spherical coordinates is:
ρ^2 = x^2 + y^2 + z^2
We need to convert the equation of the sphere to spherical coordinates:
ρ^2 = r^2
ρ = 5
Since z = 4 is the plane above which S lies, we know that z will range from 4 to √(25 - ρ^2), where ρ varies between 0 and 5.
The integral for the mass becomes:
m = ∫∫∫ kρ^2 sin(φ) dρ dθ dφ
where the limits of integration are:
ρ: 0 to 5
θ: 0 to 2π
φ: 0 to arccos(4/5)
Step 2: Calculate the first moments with respect to each coordinate axis using the definition:
M_x = ∫∫∫ xρ^2 sin(φ) dρ dθ dφ
M_y = ∫∫∫ yρ^2 sin(φ) dρ dθ dφ
M_z = ∫∫∫ zρ^2 sin(φ) dρ dθ dφ
Step 3: Find the center of mass using the formulas:
X_cm = M_x / m
Y_cm = M_y / m
Z_cm = M_z / m
(b) Moment of Inertia about the z-axis:
The moment of inertia about the z-axis is given by the integral:
I_z = ∫∫∫ (x^2 + y^2)ρ^2 sin(φ) dρ dθ dφ
Using the same coordinate system and limits of integration as in part (a), we can evaluate this integral to find the moment of inertia about the z-axis.