We are tasked with finding the center of mass of a region defined by:

1. $y^2 + z^2 \leq \frac{1}{4}$
2. $(x-1)^2 + y^2 + z^2 \leq 1$
3. $x \geq 1$

Let's break this down and then outline a strategy to use cylindrical coordinates.

Step 1: Understanding the Region

• The inequality $y^2 + z^2 \leq \frac{1}{4}$ defines a cylinder along the $x$-axis with radius $\frac{1}{2}$ (since $y^2 + z^2 = r^2$, the radial part in cylindrical coordinates).

• The inequality $(x-1)^2 + y^2 + z^2 \leq 1$ describes a sphere with radius $1$, centered at $(1, 0, 0)$.

• The condition $x \geq 1$ limits the region to only half of the space, cutting off the region along the plane $x = 1$.

Step 2: Convert to Cylindrical Coordinates

In cylindrical coordinates $(r, \theta, z)$, we have the relationships:

$x = x, \quad y = r \cos \theta, \quad z = r \sin \theta$

For the two shapes:

• The cylinder becomes: $r \leq \frac{1}{2}$ (since $y^2 + z^2 = r^2$).
• The sphere becomes: $(x - 1)^2 + r^2 \leq 1$, or equivalently:

$x \in [1, 2] \quad \text{and} \quad r^2 \leq 1 - (x-1)^2$

The region is essentially a half-cylinder intersecting with the sphere.

Step 3: Finding the Center of Mass

The center of mass $\bar{x}, \bar{y}, \bar{z}$ is given by:

$\bar{x} = \frac{1}{M} \int_V x \, dV, \quad \bar{y} = \frac{1}{M} \int_V y \, dV, \quad \bar{z} = \frac{1}{M} \int_V z \, dV$

Where $M$ is the total mass, $dV$ is the volume element in cylindrical coordinates, given by:

$dV = r \, dr \, d\theta \, dx$

We will perform the integrals over the appropriate bounds, which we now describe.

Step 4: Bounds for the Integrals

1. $x$-coordinate bounds: The $x$-coordinate runs from $x = 1$ to $x = 2$, since the region is constrained to $x \geq 1$ and the sphere is centered at $x = 1$ with radius 1.

2. $r$-coordinate bounds: For each $x$, the radial coordinate $r$ is constrained by both the cylinder and the sphere. So $r \leq \min\left(\frac{1}{2}, \sqrt{1 - (x-1)^2}\right)$.

3. $\theta$-coordinate bounds: The angle $\theta$ runs from $0$ to $2\pi$, since the region is symmetric around the $x$-axis.

Step 5: Center of Mass Calculations

We can now set up the integrals for the center of mass in cylindrical coordinates. Let's calculate $\bar{x}, \bar{y}, \bar{z}$, using the volume element $dV = r \, dr \, d\theta \, dx$, for which we need the mass $M$.

Do you want to go into the full integration details for finding the center of mass?

Related Questions:

1. What is the physical interpretation of the center of mass in this region?
2. How does the symmetry of the region influence the expected values of $\bar{y}$ and $\bar{z}$?
3. Could we change the coordinate system to spherical coordinates and simplify the problem further?
4. How would the center of mass shift if the constraints on $x$ were altered (e.g., $x \geq 0$)?
5. How do the bounds for the integrals change if we extend the region beyond the defined limits?

Tip:

When using cylindrical coordinates for integration, always ensure that the limits for $r$ depend on the radial constraints, particularly when dealing with combined shapes like cylinders and spheres.