Questions: A solid of constant density is bounded below by the plane z=0, above by the cone z=4r, r ≥ 0, and on the sides by the cylinder r=2. Find the center of mass. The center of mass is (x̄, ȳ, z̄).

Solution Steps

To find the center of mass of the given solid, we need to calculate the coordinates \((\bar{x}, \bar{y}, \bar{z})\). Since the solid is symmetric about the z-axis, \(\bar{x}\) and \(\bar{y}\) will be zero. To find \(\bar{z}\), we need to compute the volume of the solid and the moment about the xy-plane. The volume can be found using cylindrical coordinates, integrating over the region defined by the cone and the cylinder. The moment about the xy-plane is found by integrating \(z\) over the same region. Finally, \(\bar{z}\) is the moment divided by the volume.

The solid is bounded below by the plane \( z = 0 \), above by the cone \( z = 4r \) for \( r \geq 0 \), and on the sides by the cylinder \( r = 2 \).

The volume \( V \) of the solid can be calculated using cylindrical coordinates. The limits of integration are:

• \( z \) from \( 0 \) to \( 4r \)
• \( r \) from \( 0 \) to \( 2 \)

The volume is given by: \[ V = \int_0^2 \int_0^{4r} r \, dz \, dr = \frac{64\pi}{3} \]

The moment \( M_z \) about the xy-plane is calculated as follows: \[ M_z = \int_0^2 \int_0^{4r} z \cdot r \, dz \, dr = 64\pi \]

The \( z \)-coordinate of the center of mass \( \bar{z} \) is given by: \[ \bar{z} = \frac{M_z}{V} = \frac{64\pi}{\frac{64\pi}{3}} = 3 \]

Since the solid is symmetric about the z-axis, we have \( \bar{x} = 0 \) and \( \bar{y} = 0 \).

Final Answer