Center of Mass and Symmetry in Cylindrical Coordinates

The video tutorial explains how to find the center of mass of a solid bounded by a paraboloid and a plane, assuming constant density. It highlights the importance of symmetry about the z-axis, which simplifies the calculation by setting the x and y coordinates of the center of mass to zero. The tutorial then details the process of calculating the mass and the moment about the XY plane using triple integrals in cylindrical coordinates, ultimately determining the z-coordinate of the center of mass.

What is the significance of the symmetry about the z-axis in this problem?

• It indicates that the density varies with height.

• It means the solid is a perfect sphere.

• It simplifies the calculation by making the x and y coordinates of the center of mass zero.

• It allows us to ignore the density of the solid.

What is the equation of the circle formed by the trace of the plane z=7?

• 4x^2 + y^2 = 7

• x^2 + y^2 = 7

• 4x^2 + 4y^2 = 7

• x^2 + y^2 = 4

In cylindrical coordinates, what is the expression for the differential volume element dV?

• dV = dr dθ dz

• dV = r dz dr dθ

• dV = r dr dθ dz

• dV = dx dy dz

What is the upper limit of integration for z in the mass calculation?

• z = 0

• z = 4r^2

• z = r^2

• z = 7

What is the result of the mass calculation in terms of k?

• 343πk/12

• 4k

• 49πk/8

• 7k

What is the integrand function for the moment about the XY plane?

• kz^2

• kz

• kzR

• kR

What is the final expression for the moment about the XY plane?

• 49πk/8

• 343πk/12

• 7k

• 4k

How is the z-coordinate of the center of mass calculated?

• By dividing the mass by the moment about the XY plane

• By dividing the moment about the XY plane by the mass

• By multiplying the mass by the moment about the XY plane

• By adding the mass and the moment about the XY plane

What is the z-coordinate of the center of mass?

• 7/3

• 14/3

• 49/8

• 343/12

Why was it unnecessary to calculate the moments about the YZ and XZ planes?

• Because the x and y coordinates of the center of mass are zero.

• Because the density was not constant.

• Because the z-coordinate was already known.

• Because the solid was not symmetrical.