Question

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The region bounded by the cone \(z=16-r\) and the plane \(z=0\)

Short answer

Answer: The center of mass is located along the z-axis, at the height \(Z_c = \frac{16384}{15}\). The coordinates of the center of mass are \((0, 0, \frac{16384}{15})\) in the cylindrical coordinates system.

Step-by-step solution

Sketch the region

First, we will sketch the given region. The equation \(z = 16 - r\) defines a cone with its vertex at \((0, 0, 16)\) and a base on the plane \(z = 0\). In the \(xz\)-plane, we can rewrite the equation as \(z = 16 - x\), which is the equation for the cross-section of the cone. So, the cone has a circular base of radius 16 on the plane z=0, and its vertex is located directly above the centroid of the base. Since the cone is symmetric around the z-axis, the centroid of the cone will also lie on the z-axis.

Set up a coordinate system

We will choose a convenient coordinate system for our problem, which is cylindrical coordinates \((r, \theta, z)\). In this coordinate system, the cone can be described by the equation \(z = 16 - r\). The volume of the cone can be obtained by integrating the area of each cross section (\(\pi r^2\)) over the height of the cone (0 to 16).

Calculate the volume of the cone

To find the volume V of the cone, we need to integrate the area of cross sections over the height of the cone. So first, let's calculate the volume: \(V = \displaystyle\int_{0}^{16} A(z) \, dz\), where \(A(z)=\pi r^2\) We know that \(r = 16 - z\), so we can rewrite the area as: \(A(z) = \pi (16 - z)^2\) Now we can integrate to find the volume: \(V = \displaystyle\int_{0}^{16} \pi (16 - z)^2 \, dz\)

Calculate the location of the centroid along the z-axis

The centroid along the z-axis can be found using the following integral: \(Z_c = \frac{1}{V} \displaystyle\int_{0}^{16} zA(z)\,dz = \frac{1}{V} \displaystyle\int_{0}^{16} z\pi (16 - z)^2 \, dz\) After calculating the above integral for \(Z_c\), we will have the centroid location along the z-axis.

Evaluate the integrals

Now we need to evaluate both integrals to find the volume and the centroid location. For the volume, we have: \(V = \displaystyle\int_{0}^{16} \pi (16 - z)^2 \, dz = \frac{1}{3} \pi (16)^3 = \frac{4096}{3}\pi\) For the centroid along the z-axis, we have: \(Z_c = \frac{1}{V} \displaystyle\int_{0}^{16} z\pi (16 - z)^2 \, dz = \frac{1}{\frac{4096}{3}\pi} \cdot \frac{1}{5}\pi (16)^4 = \frac{4}{5}\cdot \frac{4096}{3} = \frac{16384}{15}\)

Report the result

The center of mass of the solid cone is located along the z-axis, at the height \(Z_c = \frac{16384}{15}\). So, the coordinates of the center of mass are \((0, 0, \frac{16384}{15})\) in the cylindrical coordinates system.

Key concepts

Cylindrical Coordinates

Cylindrical coordinates offer a streamlined approach to tackle problems involving rotational symmetry, such as finding the center of mass in conical regions. This system is defined by three parameters: the radial distance \( r \), the angular coordinate \( \theta \), and the height \( z \). The radial distance \( r \) measures how far point lies from the designated z-axis.
Here’s why it’s particularly useful for solving problems related to a cone:

• Cones are rotationally symmetric around the z-axis, making it natural to describe them using radial distance.
• The height (or depth) is naturally accommodated in the \( z \)-coordinate, aligning well with vertical symmetry.

By using cylindrical coordinates, calculations such as integrations for volume or centroid location become far more manageable, translating complex 3D tasks into simpler, often one-dimensional problems.

Volume of a Cone

The volume of a cone can be calculated through integration over its height in the direction of the z-axis. A cone's volume is typically a third of what it would be if the base were extended into a cylinder occupying the entire height. In mathematical terms:

• Base area \( A(z) = \pi r^2 \)
• Height \( h = 16 \) (in this particular setup)
• Use integration to express changing areas: \( V = \int_{0}^{16} A(z) \, dz \)

This calculation results in:\[V = \frac{1}{3}\pi (16)^3 = \frac{4096}{3}\pi\]Understanding this can help simplify further centroid calculations or adjustments if the cone's specifications are modified.

Centroid Calculation

Determining the centroid of a solid object, like a cone, involves integrating its mass distribution across all dimensions. For symmetrical bodies, this often simplifies significantly:

• The symmetry implies the center lies on the primary axis—here, the z-axis.
• The centroid calculation can be done using the following integral: \( Z_c = \frac{1}{V} \int_{0}^{16} zA(z)\,dz \)

Carrying out this integration yields:\[Z_c = \frac{1}{\frac{4096}{3}\pi} \cdot \frac{1}{5}\pi (16)^4 = \frac{16384}{15}\]This position \( Z_c \) on the z-axis marks the perfect balance point of the cone, guiding where the weight is effectively "centered."

Symmetry in Solids

Solids with symmetrical features, such as cones, benefit significantly from symmetry, simplifying both calculations of center of mass and physical predictions in equilibrium. Here are key points about symmetry:

• In a symmetric body, the centroid is often located along the axis of symmetry, reducing multi-dimensional calculations to just one axis.
• Symmetry helps disregard complex geometric features—like non-linear edges—focusing on core axial calculations.
• This reduces computational complexity in engineering and science, where balanced weight is critical.

By leveraging symmetry, many engineering problems become far more tractable, allowing for straightforward solutions to otherwise complex designs.