This problem uses cylindrical coordinates 2,r, 0 with r? = r² + y?.Let H, m > 0. Consider a solid cone G defined by z < H and z > mr,where m = cot(4.) and the angle between the lateral surface of thecone and the central axis of the cone is o.. See Figure 1.%3D2= H2=mrFIGURE 1. A cone of height H and central angle o..Find the centroid (7, ỹ, z) of G.Note that the cone becomes very narrow as o, → 0*, or equivalently,as m → +o. Take your answer for the centroid above and computethe limitlim (7, ỹ, 2).Finally, does your answer surprise you?

We use the term centroid, center of gravity, or center of mass to find where the whole mass of a…

This problem uses cylindrical coordinates z, r, 0 with r² = x² + y°. Let H, m > 0. Consider a solid cone G defined by z < H and z > mr, where m = cot(4.) and the angle between the lateral surface of the cone and the central axis of the cone is ,. See Figure 1. 2mr FIGURE 1. A cone of height H and central angle o,. Find the centroid (2, 9, 2) of G. Note that the cone becomes very narrow as o, → 0*, or equivalently, as m + +oo. Take your answer for the centroid above and compute the limit lim (7, §, 2). Finally does vour answer surprise vou?

This problem uses cylindrical coordinates z, r, 0 with r² = x² + y°. Let H, m > 0. Consider a solid cone G defined by z < H and z > mr, where m = cot(4.) and the angle between the lateral surface of the cone and the central axis of the cone is ,. See Figure 1. 2mr FIGURE 1. A cone of height H and central angle o,. Find the centroid (2, 9, 2) of G. Note that the cone becomes very narrow as o, → 0*, or equivalently, as m + +oo. Take your answer for the centroid above and compute the limit lim (7, §, 2). Finally does vour answer surprise vou?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

This problem uses cylindrical coordinates 2,r, 0 with r? = r² + y?.
Let H, m > 0. Consider a solid cone G defined by z < H and z > mr,
where m = cot(4.) and the angle between the lateral surface of the
cone and the central axis of the cone is o.. See Figure 1.
%3D
2= H
2=mr
FIGURE 1. A cone of height H and central angle o..
Find the centroid (7, ỹ, z) of G.
Note that the cone becomes very narrow as o, → 0*, or equivalently,
as m → +o. Take your answer for the centroid above and compute
the limit
lim (7, ỹ, 2).
Finally, does your answer surprise you?

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