Chapter 5: Q10P (page 268)

a) Find the volume inside the cone $3 z^{2} = x^{2} + y$ , above the plane $z = 2$ and inside the sphere $x^{2} + y^{2} + z^{2} = 36$ . Hint: Use spherical coordinates.
b) Find the centroid of the volume in (a)

Short Answer

Expert verified

(a). The required volume is $64 τ τ$ .

(b). The centroid is $(0, 0, \frac{231}{64})$ .

Step by step solution

Given information

The solid is bounded by the cone $3 z^{2} = x^{2} + y$ , the plane $z = 2$ and the sphere $x^{2} + y^{2} + z^{2} = 36$ .

Concept of spherical coordinates

The spherical coordinates $(r, θ, ϕ)$ is related to Cartesian coordinates (x,y,z)by:

$r = \sqrt{x^{2} + y^{2} + z^{2} θ} = \tan^{- 1} (\frac{y}{x}) ϕ = \cos^{- 1} (\frac{z}{r})$

The cylindrical coordinates $(r, θ, ϕ)$ is related to Cartesian coordinates(x,y,z) by:

$r = \sqrt{x^{2} + y^{2} θ} = t a n^{- 1} (\frac{y}{x}) z = z$

Substitute 0 for in the equation of cone.

(a)

Calculate the volume of cone as follows:

$z^{2} = \frac{x^{2}}{3} z = \frac{x^{2}}{\sqrt{3}} α = \arctan (\frac{x}{z}) = \frac{τ τ}{3}$

The required volume is as follows:

$V = \int_{\frac{2}{\cos θ}}^{6} r^{2} d r \int_{0}^{τ τ / 3} \sin θ d θ \int_{0}^{2 τ τ} d ϕ = \frac{2 τ τ}{3} {[r^{3}]}_{\frac{2}{\cos θ}}^{0} \int_{0}^{τ τ / 3} \sin θ {[r^{3}]}_{\frac{2}{\cos θ}}^{6} d θ = \frac{2 τ τ}{3} {(108 - \frac{4}{\cos^{2} θ})}_{0}^{τ τ / 3} = 64 τ τ$

Thus, the required volume is $64 τ τ$ .

Use rcosθ for z in the formula for centroid

(b)

Here, $x = y = 0 z = \frac{1}{V} \int_{v} z d V = \frac{1}{64 τ τ} \int_{\frac{2}{\cos θ}} r^{3} d r \int_{0}^{τ τ / 3} \sin θ \cos θ d θ \int_{0}^{2 τ τ} d ϕ = \frac{1}{128} {[- 324 \cos (2 θ) - \frac{8}{\cos^{2} θ}]}_{0}^{τ τ / 3} = \frac{231}{64}$ .

Thus, the centroid is $(0, 0, \frac{231}{64})$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

a) Find the volume inside the cone $3 z^{2} = x^{2} + y$ , above the plane $z = 2$ and inside the sphere $x^{2} + y^{2} + z^{2} = 36$ . Hint: Use spherical coordinates.
b) Find the centroid of the volume in (a)

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

Substitute 0 for in the equation of cone.

Use rcosθ for z in the formula for centroid

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Torque and Rotational Motion

Electric Charge Field and Potential

Wave Optics

Geometrical and Physical Optics

Space Physics

Study anywhere. Anytime. Across all devices.

a) Find the volume inside the cone3z2=x2+y, above the planez=2 and inside the sphere x2+y2+z2=36. Hint: Use spherical coordinates.b) Find the centroid of the volume in (a)

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

Substitute 0 for in the equation of cone.

Use rcosθ for z in the formula for centroid

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Torque and Rotational Motion

Electric Charge Field and Potential

Wave Optics

Geometrical and Physical Optics

Space Physics

Study anywhere. Anytime. Across all devices.

a) Find the volume inside the cone $3 z^{2} = x^{2} + y$ , above the plane $z = 2$ and inside the sphere $x^{2} + y^{2} + z^{2} = 36$ . Hint: Use spherical coordinates.
b) Find the centroid of the volume in (a)