Ann's copy of Moment of inertia: semicircle and hole

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Ann Smith 6 years, 6 months ago

Created this as a copy of Moment of inertia: semicircle and hole.

Name	Status	Author	Last Modified
Moment of inertia: semicircle and hole	Ready to use	William Haynes	30/03/2021 14:36
Ann's copy of Moment of inertia: semicircle and hole	draft	Ann Smith	10/10/2018 17:46
Centroidal Moment of inertia: T shape	Ready to use	William Haynes	22/03/2021 14:12

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Question statement

Determine the moment of inertia and radius of gyration of the composite shape with respect to the x- and y-axes knowing that $b$ = {b}, $h$ = {h}, $r_o$ = {r_o}, and $r_i$ = {display(r_i)}.

{geogebra_applet('fqywxznp',[['A',vector(scalar(b),scalar(h))],['R_o',scalar(r_o)],['R_i',scalar(r_i)]])}

Give any introductory information the student needs.

			Name	Type	Generated Value

b	quantity	quantity(4, "cm")
h	quantity	quantity(5, "cm")
r_o	quantity	quantity(2, "cm")
r_i	quantity	quantity(1.5, "cm")
units	string	cm
debug	boolean	false

			Name	Type	Generated Value

Ix1	quantity	quantity(166.6666666666666666666666666666666666667, "cm^4")
Ibarx2	quantity	quantity(1.7568, "cm^4")
Ix2	quantity	quantity(216.6967110940284661234880709732615932979, "cm^4")
d2	quantity	quantity(5.848827080130337398154013307486548743365, "cm")
Ibarx3	quantity	quantity(3.97607484375, "cm^4")
Ix3	quantity	quantity(180.69051234375, "cm^4")
Ix	quantity	quantity(202.6728654169451327901547376399282599646, "cm^4")
kx	quantity	quantity(3.247746336771080841279590169114970696209, "cm")

			Name	Type	Generated Value

A1	quantity	quantity(20, "cm^2")
A2	quantity	quantity(6.28318, "cm^2")
A3	quantity	quantity(7.0685775, "cm^2")
A	quantity	quantity(19.2146025, "cm^2")

			Name	Type	Generated Value

Iy1	quantity	quantity(106.6666666666666666666666666666666666667, "cm^4")
Ibary2	quantity	quantity(6.28318, "cm^4")
Iy2	quantity	quantity(45.553055, "cm^4")
Ibary3	quantity	quantity(3.97607484375, "cm^4")
Iy3	quantity	quantity(48.15468421875, "cm^4")
Iy	quantity	quantity(104.0650374479166666666666666666666666667, "cm^4")
ky	quantity	quantity(2.327216139797851409739338344139715817568, "cm")

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
qty(random(4..12#2),units)

JME function reference

Description

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

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Generated value: `quantity`

quantity(4, "cm")

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Engineering Accuracy with units
  Add
2. Gap Gap 1. Engineering Accuracy with units
  Add
Part b) Gap-fill
1. Gap Gap 0. Engineering Accuracy with units
  Add
2. Gap Gap 1. Engineering Accuracy with units
  Add

Gap-fill

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Properties with respect to the x-axis.

$I_x$ = Gap 0. {display(Ix)}

$k_x$ = Gap 1. {display(kx)}

Advice

Determine the moment of inertia of the composite shape about the x- and y-axes knowing that b = {b}, h = {h}, $r_o$ = {r_o}, and $r_i$ = {display(r_i)}.

{geogebra_applet('fqywxznp',[['A',vector(scalar(b),scalar(h))],['R_o',scalar(r_o)],['R_i',scalar(r_i)]])}

Number the parts:

Part 1 rectangle: $b$ = {b}, $h$ = {h}.

Part 2 semicircle: $r_o$ = {r_o}.

Part 3 circular hole: $r_i$ = {display(r_i)}.

Calculate the areas:

$\begin{align} A_1 &= b h & A_2 &= \tfrac{1}{2}\pi r_o^2 &A_3 &= \pi r_i^2&A&=A_1 +A_2 - A_3\\&=\var{display(A1)} &&=\var{display(A2)} &&=\var{display(A3)}&&=\var{display(A)}\\&&\end{align}$

Calculate the moment of inertia with respect to the x-axis.

$\begin{align} \\{I_1}_x &= \tfrac{1}{3} b h^3 \\&= \tfrac{1}{3}( \var{b})(\var{h})^3 \\&= \var{display(Ix1)} \\\\{I_x}_2 &= [\bar{I} + A d^2]_2\\ &= 0.1098 r_o^4 + \left(\frac{\pi r_o^2}{2}\right) \left(h + \frac{4 r_o}{3 \pi}\right)^2\\&= \var{display(Ibarx2)} + (\var{display(A2)}) (\var{display(d2)})^2\\&=\var{display(Ix2)} \\\\ {I_x}_3 &= [\bar{I} + A d^2]_3 \\ &= \frac{\pi r_i^4}{4} + \left(\pi r_i^2\right) \left(h \right)^2\\&= \var{display(Ibarx3)} + (\var{display(A3)}) (\var{display(h)})^2\\&=\var{display(Ix3)}\\\\I_x &= {I_x}_1+{I_x}_2-{I_x}_3\\&=\var{display(Ix1)}+\var{display(Ix2)}-\var{display(Ix3)}\\&=\var{display(Ix)}\end{align}$

Calculate the moment of inertia with respect to the y-axis.

$\begin{align} \\{I_y}_1 &= \tfrac{1}{3} h b^3 \\&= \tfrac{1}{3}( \var{h})(\var{b})^3 \\&= \var{display(Iy1)} \\\\{I_y}_2 &= [\bar{I} + A d^2]_2\\ &= \frac{\pi r_o^4}{8} + \left(\frac{\pi r_o^2}{2}\right) \left(\frac{b}{2}\right)^2\\&= \var{display(Ibary2)} + (\var{display(A2)}) (\var{display(b/2)})^2\\&=\var{display(Iy2)} \\\\ {I_y}_3 &= [\bar{I} + A d^2]_3 \\ &= \frac{\pi r_i^4}{4} + \left(\pi r_i^2\right) \left(\frac{b}{2} \right)^2\\&= \var{display(Ibary3)} + (\var{display(A3)}) (\var{display(b/2)})^2\\&=\var{display(Iy3)}\\\\I_y &= {I_y}_1+{I_y}_2-{I_y}_3\\&=\var{display(Iy1)}+\var{display(Iy2)}-\var{display(Iy3)}\\&=\var{display(Iy)}\end{align}$

Calculate the radius of gyration with respect to the x- and y- axes.

$\begin{align}\\k_x &= \sqrt{\dfrac{I_x}{A}}&k_y &= \sqrt{\dfrac{I_y}{A}}\\ &= \sqrt{\dfrac{\var{display(Ix)}}{\var{display(A)}}}&&= \sqrt{\dfrac{\var{display(Iy)}}{\var{display(A)}}}\\ &= \var{display(kx)} &&=\var{display(ky)} \end{align}$

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Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Engineering Accuracy with units
Engineering Accuracy with units		Hasn't run yet
Engineering Accuracy with units
Engineering Accuracy with units		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Engineering Accuracy with units
Engineering Accuracy with units		Hasn't run yet
Engineering Accuracy with units
Engineering Accuracy with units		Hasn't run yet

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Ann's copy of Moment of inertia: semicircle and hole

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

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Checkpoint description

Ann Smith 6 years, 6 months ago

Error in variable testing condition

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Warning

Generated value: `quantity`

Parts

Gap-fill

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Error

Ann's copy of Moment of inertia: semicircle and hole

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Ann Smith 6 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: quantity

Parts

Gap-fill

Generated value: `quantity`