TBIL-Cal Density, Mass, and Center of Mass (AI5)

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Section 6.5 Density, Mass, and Center of Mass (AI5)

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Learning Outcomes

Set up integrals to solve problems involving density, mass, and center of mass.

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Subsection 6.5.1 Activities

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Activity 6.5.1.

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Consider a rectangular prism with a 10 meters

\times

10 meters square base and height 20 meters. Suppose the density of the material in the prism increases with height, following the function

δ (h) = 10 + h

kg/m

^{3},

where

h

is the height in meters.

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(a)

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If one were to cut this prism, parallel to the base, into 4 pieces with height 5 meters, what would the volume of each piece be?

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(b)

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Consider the piece sitting on top of the slice made at height

h = 5 .

Using a density of

δ (5) = 15

kg/m

^{3},

and the volume you found in (a), estimate the mass of this piece.

$500 \cdot 5 = 2500$ kg
$500 \cdot 15 = 7500$ kg
$500 \cdot 15 \cdot 5 = 37500$ kg

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(c)

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Is this estimate the actual mass of this piece?

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Activity 6.5.2.

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Consider all 4 slices from Activity 6.5.1.

described in detail following the image — Figure 137. $10 \times 10 \times 20$ prism sliced into 4 pieces.

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(a)

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Fill out the following table.

\begin{array}{cccc} h_{i} & δ (h_{i}) & Volume & Estimated Mass \\ h_{4} = 15 m & δ (15) = 25 {kg/m}^{3} & 500 m^{3} \\ h_{3} = 10 m & δ (10) = 20 {kg/m}^{3} & 500 m^{3} \\ h_{2} = 5 m & δ (5) = 15 {kg/m}^{3} & 500 m^{3} & 7500 kg \\ h_{1} = 0 m & δ (0) = 10 {kg/m}^{3} & 500 m^{3} \end{array}

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(b)

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What is the estimated mass of the rectangular prism?

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Activity 6.5.3.

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Suppose instead that we sliced the prism from Activity 6.5.1 into 5 pices of height 4 meters.

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(a)

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Fill out the following table.

\begin{array}{cccc} h_{i} & δ (h_{i}) & Volume & Estimated Mass \\ h_{5} = 16 m & δ (16) = 26 {kg/m}^{3} & 400 m^{3} \\ h_{4} = 12 m & δ (12) = 22 {kg/m}^{3} & 400 m^{3} \\ h_{3} = 8 m & δ (8) = 18 {kg/m}^{3} & 400 m^{3} \\ h_{2} = 4 m & δ (4) = 14 {kg/m}^{3} & 400 m^{3} \\ h_{1} = 0 m & δ (0) = 10 {kg/m}^{3} & 400 m^{3} \end{array}

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(b)

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What is the estimated mass of the rectangular prism?

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Activity 6.5.4.

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Which of the estimates computed in Activity 6.5.2 and Activity 6.5.3 is a better estimate of the mass of the prism?

Activity 6.5.2, 4 pieces is a better estimate.
Activity 6.5.3, 5 pieces is a better estimate.

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Activity 6.5.5.

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Suppose now that we slice the prism from Activity 6.5.1 into slices of height

Δ h

meters.

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(a)

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Consider the piece sitting atop the slice made at height

h_{i} .

Using

δ (h_{i}) = 10 + h_{i}

as the estimate for the density of this piece, what is the mass of this piece?

$(10 + h) 100 \cdot h_{i}$
$(10 + Δ h) 100 \cdot h_{i}$
$(10 + h_{i}) 100 \cdot Δ h$
$(10 + h_{i}) 100 \cdot h$

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Activity 6.5.6.

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Consider a cylindrical cone with a base radius of 15 inches and a height of 60 inches. Suppose the density of the cone is

δ (h) = 15 + \sqrt{h}

oz/in

^{3} .

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(a)

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Let

r_{2}

be the radius of the circular cross section of the cone, made at height 30 inches. Recall that

Δ A B C, Δ A B^{'} C^{'}

are similar triangles, what is

r_{2} ?

15 inches.
7.5 inches.
30 inches.
60 inches.

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(b)

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What is the volume of a cylinder with radius

r_{1} = 15

inches and height

30

inches?

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(c)

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What is the volume of a cylinder with radius

r_{2}

inches and height

30

inches?

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Activity 6.5.7.

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Suppose that we estimate the mass of the cone from Activity 6.5.6 with 2 cylinders of height 30 inches.

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(a)

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Fill out the following table.

\begin{array}{cccc} h_{i} & δ (h_{i}) & Volume & Estimated Mass \\ h_{2} = 30 m & δ (30) = 15 + \sqrt{30} {oz/in}^{3} & π (7.5)^{2} \cdot 30 {in}^{3} \\ h_{1} = 0 m & δ (0) = 15 {oz/in}^{3} & π (15)^{2} \cdot 30 {in}^{3} \end{array}

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(b)

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What is the estimated mass of the cone?

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Activity 6.5.8.

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Suppose that we estimate the mass of the cone from Activity 6.5.6 with 3 cylinders of height 20 inches.

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(a)

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Fill out the following table.

\begin{array}{cccc} h_{i} & δ (h_{i}) & Volume & Estimated Mass \\ h_{2} = 40 m & δ (40) = 15 + \sqrt{40} {oz/in}^{3} & π (5)^{2} \cdot 20 {in}^{3} \\ h_{2} = 20 m & δ (20) = 15 + \sqrt{20} {oz/in}^{3} & π (10)^{2} \cdot 20 {in}^{3} \\ h_{1} = 0 m & δ (0) = 15 {oz/in}^{3} & π (15)^{2} \cdot 20 {in}^{3} \end{array}

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(b)

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What is the estimated mass of the cone?

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Activity 6.5.9.

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Suppose that we estimate the mass of the cone from Activity 6.5.6 with cylinders of height

Δ h .

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(a)

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Consider the piece sitting atop the slice made at height

h_{i} .

Using

δ (h_{i}) = 15 + \sqrt{h_{i}}

as the estimate for the density of this cylinder, what is the mass of this cylinder?

$(15 + \sqrt{h}) π r_{i}^{2} \cdot Δ h$
$(15 + \sqrt{h_{i}}) π r_{i}^{2} \cdot Δ h$
$(15 + Δ h) π r_{i}^{2} \cdot Δ h_{i}$
$(15 + \sqrt{h_{i}}) π r^{2} \cdot Δ h$

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Activity 6.5.10.

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Consider a solid where the cross section of the solid at

x = x_{i}

has area

A (x_{i}),

and the density when

x = x_{i}

δ (x_{i}) .

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(a)

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If we used prisms of width

Δ x

to approximate this solid, what is the mass of the slice associated with

x_{i} ?

$A (x) δ (x) Δ x$
$π A (x)^{2} δ (x_{i}) Δ x$
$A (x_{i}) δ (x_{i}) Δ x$
$A (x_{i}) δ (x_{i}) Δ x_{i}$

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Fact 6.5.11.

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Consider a solid where the cross section of the solid at

x = x_{i}

has area

A (x_{i}),

and the density when

x = x_{i}

δ (x_{i}) .

Suppose the interval

[a, b]

represents the

x

values of this solid. If one slices the solid into

n

pieces of width

Δ x = \frac{b - a}{n},

then one can approximate the mass of the solid by

\sum_{i = 1}^{n} δ (x_{i}) A (x_{i}) Δ x .

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We can then find actual mass by taking the limit as

n \to \infty :

lim_{n \to \infty} (\sum_{i = 1}^{n} δ (x_{i}) A (x_{i}) Δ x) = \int_{a}^{b} δ (x) A (x) d x .

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Activity 6.5.12.

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Consider that for the prism from Activity 6.5.1, a cross section of height

h

A (h) = 10^{2} = 100

^{2} .

Also recall that the density of the prism is

δ (h) = 10 + h

kg/m

^{3},

where

h

is the height in meters.

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Use Fact 6.5.11 to find the mass of the prism.

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Activity 6.5.13.

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Consider that for the cone from Activity 6.5.6, a cross section of height

h

A (h) = π r^{2}

^{2},

where

r

is the radius of the circular cross-section at height

h

inches. Also recall that the density of the cone is

δ (h) = 15 + \sqrt{h}

oz/in

^{3},

where

h

is the height in inches.

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(a)

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When the height is

h

inches, what is

r ?

Hint.

Use similar triangles:

a 15 by 60 triangle, with a similar r by 60-h triangle. — Figure 147. The right triangles in this figure are similar.

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(b)

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Find

A (h)

as a function of

h

using this information.

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(c)

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Use Fact 6.5.11 to find the mass of the cone.

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Activity 6.5.14.

Consider a pyramid with a

8 \times 8

ft square base and a height of 16 feet. Suppose the density of the pyramid is

δ (h) = 10 + \cos (π h)

lb/ft

^{3}

where

h

is the height in feet.

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(a)

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When the height is

h

feet, what is the area of the square cross section at that height,

A (h) ?

Hint.

Use similar triangles:

a 8 by 60 triangle, with a similar r by 16-h triangle. — Figure 148. The triangles in this figure are similar.

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(b)

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Use Fact 6.5.11 to find the mass of the pyramid.

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Activity 6.5.15.

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Consider a board sitting atop the

x

-axis with six

1 \times 1

blocks each weighing 1 kg placed upon it in the following way: two blocks are atop the 1, three blocks are atop the 2, and one block is atop the 6.

Six 1 by 1 blocks on the x-axis, each weighing a kilogram. Two blocks atop the 1, three blocks atop the 2, and one block atop the 6. — Figure 149. Six 1 kg blocks atop the $x$ -axis.

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Which of the following describes the

x

-value of the center of gravity of the board with the blocks?

$\frac{1 + 6}{2} = 3.5 .$
$\frac{1 + 2 + 6}{3} = 3 .$
$\frac{2 \cdot 1 + 3 \cdot 2 + 1 \cdot 6}{6} \approx 2.3333 .$

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Activity 6.5.16.

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Consider a board sitting atop the

x

-axis with six

1 \times 1

blocks each weighing 1 kg placed upon it in the following way: two blocks are atop the 1, three blocks are atop the 2, and one block is atop the 8.

Six 1 by 1 blocks on the x-axis, each weighing a kilogram. Two blocks atop the 1, three blocks atop the 2, and one block atop the 8. — Figure 150. Six 1 kg blocks atop the $x$ -axis.

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Find the

x

-value of the center of gravity of the board with the blocks.

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Fact 6.5.17.

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Consider a solid where the cross section of the solid at

x = x_{i}

has area

A (x_{i}),

and the density when

x = x_{i}

δ (x_{i}) .

Suppose the interval

[a, b]

represents the

x

values of this solid. Since each slice has approximate mass

δ (x_{i}) A (x_{i}) δ (x_{i}),

we can approximate the center of mass by taking the weighted “average” of the

x_{i}

-values weighted by the associated mass:

\frac{\sum_{i = 1}^{n} x_{i} δ (x_{i}) A (x_{i}) Δ x}{\sum_{i = 1}^{n} δ (x_{i}) A (x_{i}) Δ x} .

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We can then find actual center of mass by taking the limit as

n \to \infty :

lim_{n \to \infty} (\frac{\sum_{i = 1}^{n} x_{i} δ (x_{i}) A (x_{i}) Δ x}{\sum_{i = 1}^{n} δ (x_{i}) A (x_{i}) Δ x}) = \frac{\int_{a}^{b} x δ (x) A (x) d x}{\int_{a}^{b} δ (x) A (x) d x} = \frac{\int_{a}^{b} x δ (x) A (x) d x}{The Total Mass} .

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Activity 6.5.18.

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Consider that for the prism from Activity 6.5.12, a cross section of height

h

A (h) = 10^{2} = 100

^{2} .

Also recall that the density of the prism is

δ (h) = 10 + h

kg/m

^{3},

where

h

is the height in meters, and that we found the total mass to be 40000 kg.

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Use Fact 6.5.17 to find the height where the center of mass occurs.

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Activity 6.5.19.

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Consider that for the prism from Activity 6.5.13, a cross section of height

h

A (h) = π \cdot {(\frac{60 - h}{4})}^{2}

^{2} .

Also recall that the density of the cone is

δ (h) = 15 + \sqrt{h}

oz/in

^{3},

where

h

is the height in inches, and that we found the total mass to be about 142492.6 oz.

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Use Fact 6.5.17 to find the height where the center of mass occurs.

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Activity 6.5.20.

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Consider that for the pyramid from Activity 6.5.14, a cross section of height

h

A (h) = π \cdot {(\frac{16 - h}{2})}^{2}

^{2} .

Also recall that the density of the pyramid is

δ (h) = 10 + \cos π h

lb/feet

^{3},

where

h

is the height in feet, and that we found the total mass to be about 3414.14.6 lbs.

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Use Fact 6.5.17 to find the height where the center of mass occurs.

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Subsection 6.5.2 Videos

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Figure 153. Video: Set up integrals to solve problems involving density, mass, and center of mass

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Subsection 6.5.3 Exercises

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Exercises available at https://tbil.org/calculus/preview/exercises/#/bank/AI5/

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