Chapter 5: Q24MP (page 275)

Repeat Problem 23a for the paraboloid $\frac{z}{c} = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} 0 < z < c .$

Short Answer

Expert verified

The centroid of the solid paraboloid is $(x, y, z) = (0, 0, \frac{2}{3} c)$

Step by step solution

Step 1: Given Condition

The paraboloid is $\frac{z}{c} = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2 .}} 0 < z < c$

Concept of centroid

The centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure.

Step 3: Calculate the volume

The volume is given by the following formula,

$V = \int_{x} \int_{y} \int_{z} d x d y d z = \int_{x} \int_{y} \int_{z} d x' d y' d z' a b c = a b c \int_{v} d V = 2 πabc \int_{0}^{1} dz' \frac{1}{2} z' = \frac{π}{2} abc$

Thus,

x = y = 0

Calculate the z coordinate

Now, to calculate the value of $z$ ,

$V = \int_{x} \int_{y} \int_{z} z d x d y d z = a b c^{2} \int_{x} \int_{y} \int_{z} d x' d y' d z' z' = a b c^{2} \int_{0}^{2 π} d θ \int_{0}^{1} d z' z \int_{0}^{\sqrt{z}} r d r = 2 πabc \int_{0}^{1} z' dz' \frac{1}{2} z' = {πabc}^{2} \int_{0}^{1} z'^{2} d z' = \frac{π}{3} c z = \frac{2}{3} c$

Hence, the centroid is . $(x, y, z) = (0, 0, \frac{2}{3} c)$ .

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Repeat Problem 23a for the paraboloid $\frac{z}{c} = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} 0 < z < c .$

Short Answer

Step by step solution

Step 1: Given Condition

Concept of centroid

Step 3: Calculate the volume

Calculate the z coordinate

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Most popular questions from this chapter

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Repeat Problem 23a for the paraboloidzc=x2a2+y2b20<z<c.

Short Answer

Step by step solution

Step 1: Given Condition

Concept of centroid

Step 3: Calculate the volume

Calculate the z coordinate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Physics of Motion

Geometrical and Physical Optics

Fields in Physics

Fundamentals of Physics

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

Repeat Problem 23a for the paraboloid $\frac{z}{c} = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} 0 < z < c .$