Parallel Axis Theorem: semi and quarter circles

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Question statement

{geogebra_applet('emjtbcxv',ggb_points)}

Determine the moment of inertia and radius of gyration about the {axis}-axis for the {lower(names[pts[0]])} and the {lower(names[pts[1]])} .

Grid units are [{units}].

axis	radius	area	ibar	d	I	Centroid
x	{r["A"]}	{area["A"]}	{ibarx["A"]}	{d["x"]["A"]}	{Ix["A"]}	{siground(Centroid["A"],3)}
y	{r["A"]}	{area["A"]}	{ibary["A"]}	{d["y"]["A"]}	{Iy["A"]}

axis	radius	area	ibar	d	I	Centroid
x	{r["B"]}	{area["B"]}	{ibarx["B"]}	{d["x"]["B"]}	{Ix["B"]}	{siground(Centroid["B"],3)}
y	{r["B"]}	{area["B"]}	{ibary["B"]}	{d["y"]["B"]}	{Iy["B"]}

axis	radius	area	ibar	d	I	Centroid
x	{r["C"]}	{area["C"]}	{ibarx["C"]}	{d["x"]["C"]}	{Ix["C"]}	{siground(Centroid["C"],3)}
y	{r["C"]}	{area["C"]}	{ibary["C"]}	{d["y"]["C"]}	{Iy["C"]}

Give any introductory information the student needs.

			Name	Type	Generated Value

r	dict	[ "C": 2, "A": 4, "B": 5 ]
area	dict	Dictionary with 3 entries
centroid	dict	Dictionary with 3 entries
Ibarx	dict	Dictionary with 3 entries
Iy	dict	Dictionary with 3 entries
Ibary	dict	Dictionary with 3 entries
Ix	dict	Dictionary with 3 entries
ky	dict	Dictionary with 3 entries
kx	dict	Dictionary with 3 entries
Ibar	dict	Dictionary with 2 entries
k	dict	Dictionary with 2 entries
I'	dict	Dictionary with 2 entries
d	dict	Dictionary with 2 entries
orientation	string	x

			Name	Type	Generated Value

I0	quantity	quantity(11.616862955341192, "ft^4")
I1	quantity	quantity(653.451271946677025, "ft^4")
k0	quantity	quantity(1.922956662311283, "ft")
k1	quantity	quantity(3.60555127546399, "ft")

			Name	Type	Generated Value

pts	list	[ "C", "A", "B" ]
axis	string	y
ggb_points	list	Nested 6×2 list
names	dict	Dictionary with 3 entries
centers	dict	Dictionary with 3 entries
radii	dict	Dictionary with 3 entries
units	string	ft
debug	boolean	false

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
dict(map([p,round(abs(centers[p]-radii[p]))],p,pts))

JME function reference

Description

the radii of the three shapes

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `dict`

[ "C": 2, "A": 4, "B": 5 ]

← Depends on:

centers
pts
radii

→ Used by:

Ibarx
Ibary
area

Statement
Advice

Parts

{names[pts[0]]} Gap-fill
1. Gap I Engineering Accuracy with units
  Add
2. Gap k Engineering Accuracy with units
  Add
{names[pts[1]]} Gap-fill
1. Gap I Engineering Accuracy with units
  Add
2. Gap k Engineering Accuracy with units
  Add

Gap-fill

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$I_\var{axis}$ = I $k_\var{axis}$ = k

{siground(I0,4)} {siground(k0,4)}

Advice

For the circle:

$r = \var{q(r["A"],units)} \qquad A = \pi r^2 = \var{q(area["A"],units+'^2')}$

$\bar{x} =\var{q(centroid["A"][0],units)} \qquad \bar{y} = \var{q(centroid["A"][1],units)}$

Place a {if(axis="x", "horizontal", "vertical" )} axis through the centroid of the {names["A"]}. The centroidal moment of inertia with respect to this axis is:

$\bar{I} = \dfrac{\pi r^4}{4} = \var{q(Ibar[axis]["A"],units+'^4')}$

and the distance between the parallel axes is:

$d = \var{q(abs(d[axis]["A"]),units)}$

Applying the parallel axis theorem and the definition of radius of gyration:

$I_{\var{axis}} = \bar{I} + A d^2 = \var{q(I'[axis]["A"],units+'^4')}$

$k_{\var{axis}} = \sqrt{\dfrac{I_{\var{axis}}}{A}} = \var{q(k[axis]["A"],units)}$

For the semi-circle:

$r = \var{q(r["B"],units)} \qquad A = \dfrac{\pi r^2}{2} = \var{q(area["B"],units+'^2')}$

$\bar{x} = \var{latex( offset(0,centers["B"], radii["B"]))} \var{q(centroid["B"][0],units)} \qquad \bar{y} = \var{latex( offset(1,centers["B"], radii["B"]))} \var{q(centroid["B"][1],units)}$

Place a {if(axis="x", "horizontal", "vertical" )} axis through the centroid of the {names["B"]}. The centroidal moment of inertia with respect to this axis is:

$\bar{I} =\var{ if(orientation=axis,latex("0.1098 r^4"),latex("\\dfrac\{\\pi r^4\}\{8\}")) }= \var{q(Ibar[axis]["B"],units+'^4')}$

and the distance between the parallel axes is:

$d = \var{q(abs(d[axis]["B"]),units)}$

Applying the parallel axis theorem and the definition of radius of gyration:

$I_{\var{axis}} = \bar{I} + A d^2 = \var{q(I'[axis]["B"],units+'^4')}$

$k_{\var{axis}} = \sqrt{\dfrac{I_{\var{axis}}}{A}} = \var{q(k[axis]["B"],units)}$

For the quarter-circle:

$r = \var{q(r["C"],units)} \qquad A = \dfrac{\pi r^2}{4} = \var{q(area["C"],units+'^2')}$

$\bar{x} = \var{latex( offset(0,centers["C"], radii["C"]))} \var{q(centroid["C"][0],units)} \qquad \bar{y} =\var{latex( offset(1,centers["C"], radii["C"]))} \var{q(centroid["C"][1],units)}$

Place a {if(axis="x", "horizontal", "vertical" )} axis through the centroid of the {names["C"]}. The centroidal moment of inertia with respect to this axis is:

$\bar{I} = 0.0549 r^4= \var{q(Ibar[axis]["C"],units+'^4')}$

and the distance between the parallel axes is:

$d = \var{q(abs(d[axis]["C"]),units)}$

Applying the parallel axis theorem and the definition of radius of gyration:

$I_{\var{axis}} = \bar{I} + A d^2 = \var{q(I'[axis]["C"],units+'^4')}$

$k_{\var{axis}} = \sqrt{\dfrac{I_{\var{axis}}}{A}} = \var{q(k[axis]["C"],units)}$

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This question is used in the following exams:

Chapter 10 Exercises by Michael Proudman in Engineering Statics.
24. Parallel Axis Theorem by Michael Proudman in Engineering Statics.
Chapter 10 Exercises by William Haynes in Engineering Statics.
24. Parallel Axis Theorem by William Haynes in Engineering Statics.

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Parallel Axis Theorem: semi and quarter circles

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William Haynes 2 years, 10 months ago

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Generated value: `dict`

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Parallel Axis Theorem: semi and quarter circles

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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William Haynes 2 years, 10 months ago

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Generated value: dict

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Generated value: `dict`