Chapter 13: Q 65. (page 1067)

Find the specified quantities for the solids described below:
The center of mass of the region from Exercise $55$ , assuming that the density of the region is constant.

Short Answer

Expert verified

The center of mass is given by $(\bar{x}, \bar{y}, \bar{z}) = (0, 0, (\frac{3 R}{8}) (1 + \cos α))$ .

Step by step solution

Given Information

The density of region is constant.

Region above sphere is given by equation $ρ = R$ and below the cone is given by equation $ϕ = α$ .

Evaluation of center of mass of x&y coordinate

For the given information, Center of ,mass if given by

$\bar{x} = \frac{M_{y z}}{M} = \frac{∭_{E} x ρ d x d y d z}{∭ ρ d x d y d z}$

$\vec{y} = \frac{M_{x}}{M} = \frac{∭_{E} y ρ d x d y d z}{∭_{E} ρ d x d y d z}$

As density of region is constant $ρ = k$ , the axis of center lies above $+ z$ axis and base lies in $x y$ plane.

Hence, $\bar{x} = 0, \bar{y} = 0$

Evaluation of center of mass of z coordinate

It can be given by $\bar{z} = \frac{M_{x y}}{M} \frac{∭_{E} z ρ d x d y d z}{∭_{E} ρ d x d y d z}$

$\bar{z} = \frac{M_{x y}}{M} = \frac{∭_{E} z ρ d V}{∭_{E} ρ d V}$

$= \frac{\int_{ϕ = 0}^{α} \int_{θ = 0}^{2 π} \int_{ρ = 0}^{R} (ρ \cos ϕ) k ρ^{2} \sin ϕ d ρ d θ d ϕ}{\int^{a} \int^{2 π} \int^{R} k ρ^{2} \sin ϕ d ρ d θ d ϕ}$

$= \frac{{{{(\frac{\sin^{2} ϕ}{2})|}_{ϕ = 0}^{ϕ = α} k (\frac{ρ^{4}}{4})|}_{ρ = 0}^{R} (θ)|}_{θ = 0}^{2 π}}{{{{k (- \cos ϕ)|}_{ϕ = 0}^{α} (θ)|}_{θ = 0}^{2 π} (\frac{ρ^{3}}{3})|}_{ρ = 0}^{R}}$

Putting limits yields

$\bar{z} = \frac{(\frac{\sin^{2} α}{2}) k (\frac{R^{4}}{4}) (2 π)}{k (1 - \cos α) (2 π) (\frac{R^{3}}{3})}$

$= \frac{\sin^{2} α \times R^{4} (2 k π)}{8} \times \frac{3}{(2 k π) R^{3} (1 - \cos α)}$

$= (\frac{3 R}{8}) (1 + \cos α)$

Hence, required Center of Mass is given by $(\bar{x}, \bar{y}, \bar{z}) = (0, 0, (\frac{3 R}{8}) (1 + \cos α))$

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Find the specified quantities for the solids described below:
The center of mass of the region from Exercise $55$ , assuming that the density of the region is constant.

Short Answer

Step by step solution

Given Information

Evaluation of center of mass of x&y coordinate

Evaluation of center of mass of z coordinate

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Find the specified quantities for the solids described below:The center of mass of the region from Exercise 55, assuming that the density of the region is constant.

Short Answer

Step by step solution

Given Information

Evaluation of center of mass of x&y coordinate

Evaluation of center of mass of z coordinate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Decision Maths

Mechanics Maths

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Find the specified quantities for the solids described below:
The center of mass of the region from Exercise $55$ , assuming that the density of the region is constant.