   Chapter 15.8, Problem 38E

Chapter
Section
Textbook Problem

Use cylindrical or spherical coordinates, whichever seems more appropriate.38. A solid right circular cone with constant density has base radius a and height h.(a) Find the moment of inertia of the cylinder about its axis.(b) Find the moment of inertia of the cylinder about a diameter of its base.

(a)

To determine

To find: The moment of inertia about its axis for the given region by using spherical or cylindrical coordinates.

Explanation

Given:

The region B is the right circular cone with radius a and height h.

The density function is a constant K.

Formula used:

If f is a cylindrical region E given by h1(θ)rh2(θ),αθβ, u1(rcosθ,rsinθ)zu1(rcosθ,rsinθ) where 0βα2π , then,

Ef(x,y,z)dV=αβh1(θ)h2(θ)u1(rcosθ,rsinθ)u2(rcosθ,rsinθ)f(rcosθ,rsinθ,z)rdzdrdθ (1)

If g(x) is the function of x and h(y) is the function of y and k(z) is the function of z, then, abcdefg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz . (2)

The cylindrical coordinates (r,θ,z) corresponding to the rectangular coordinates (x,y,z) is,

r=x2+y2θ=tan1(yx)z=z

Calculation:

The base of the right circular cone lies in the xy-plane. So, its axis is z-axis. Here, it is more appropriate to use the cylindrical coordinates because the given solid is a cone. Thus, the moment of inertia about z axis is given by,

Iz=K02π0a0h(1ra)(x2+y2)dzdydx=K02π0a0h(1ra)(r2)(r)dzdrdθ=K02π0a0h(1ra)r3dzdrdθ

Integrate with respect to z and apply the limit

(b)

To determine

To find: The moment of inertia about the diameter of its base for the region in part (a).

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