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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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

Space Physics

Further Mechanics and Thermal Physics

Engineering Physics

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Study anywhere. Anytime. Across all devices.

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