Chapter 5: Q9P (page 268)

Let the solid in Problem 7 have density $= c o s θ$ .
Show that then $I_{z} = \frac{3}{10} M a^{2} s i n^{2} α$ .

Short Answer

Expert verified

The required value of $\frac{I_{z}}{M} i s \frac{3}{10} a^{2} \sin^{2} α$ .

Step by step solution

Given information

The density of the solid is $\cos θ .$

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:

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

Evaluate the mass of the solid

Calculate the mass of the solid as follows:

$M = \int ρ d V = \int_{0}^{a} r^{2} d r \int_{0}^{a} \cos θ d θ \int_{0}^{2 τ τ} d ϕ = \frac{2 τ τ}{3} a^{3} \int_{0}^{a} \sin θ d θ (\cos θ) = \frac{τ τ a^{3}}{3} \sin α$

The moment of inertia is as follows:

$I_{z} = \int ρ (x^{2} + y^{2}) d V = ρ (r^{2} \sin^{2} θ) d V = \int_{0}^{2 τ τ} d ϕ \int_{0}^{α} \cos θ \sin^{3} θ d θ \int_{0}^{a} r^{4} d r = - \frac{2 τ τ a^{5}}{5} \int_{0}^{α} \sin^{3} θ d θ (\sin θ)$

Use the above values to prove .

Write integral as follows:

$I_{Z} = - \frac{τ τ a^{5}}{10} {(\sin^{4} θ)}_{0}^{α} = \frac{τ τ a^{5}}{10} \sin^{4} α$

Substitute $\frac{τ τ a^{5}}{10} \sin^{4} α$ for $I_{z}$ and $\frac{τ τ a^{5}}{3} \sin α$ for M in $\frac{I_{z}}{M}$ .

$\frac{I_{z}}{M} = \frac{3}{10} a^{2} \sin^{2} α$

Therefore, the required value of $I_{z}$ is $\frac{3}{10} M a^{2} \sin^{2} α$ .

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Let the solid in Problem 7 have density $= c o s θ$ .
Show that then $I_{z} = \frac{3}{10} M a^{2} s i n^{2} α$ .

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

Evaluate the mass of the solid

Use the above values to prove .

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Let the solid in Problem 7 have density =cosθ.Show that then Iz=310Ma2sin2α.

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

Evaluate the mass of the solid

Use the above values to prove .

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Let the solid in Problem 7 have density $= c o s θ$ .
Show that then $I_{z} = \frac{3}{10} M a^{2} s i n^{2} α$ .