Moment of inertia: semicircle and hole

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William Haynes 5 years, 8 months ago

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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

{geogebra_applet('fqywxznp',{ggbArgs})}

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)}.

Give any introductory information the student needs.

			Name	Type	Generated Value

b	quantity	quantity(6, "in")
h	quantity	quantity(4, "in")
r_o	quantity	quantity(3, "in")
r_i	quantity	quantity(2.25, "in")
units	string	in
debug	boolean	false
ggbArgs	list	Nested 3×2 list

			Name	Type	Generated Value

Ix1	quantity	quantity(128, "in^4")
Ibarx2	quantity	quantity(8.8938, "in^4")
Ix2	quantity	quantity(402.0066111635191097501583593021368160708, "in^4")
d2	quantity	quantity(5.273240620195506097231019961229823115047, "in")
Ibarx3	quantity	quantity(20.128878896484375, "in^4")
Ix3	quantity	quantity(274.597668896484375, "in^4")
Ix	quantity	quantity(255.4089422670347347501583593021368160708, "in^4")
kx	quantity	quantity(3.389381271468487462979613845787817637748, "in")

			Name	Type	Generated Value

A1	quantity	quantity(24, "in^2")
A2	quantity	quantity(14.137155, "in^2")
A3	quantity	quantity(15.904299375, "in^2")
A	quantity	quantity(22.232855625, "in^2")

			Name	Type	Generated Value

Iy1	quantity	quantity(288, "in^4")
Ibary2	quantity	quantity(31.80859875, "in^4")
Iy2	quantity	quantity(159.04299375, "in^4")
Ibary3	quantity	quantity(20.128878896484375, "in^4")
Iy3	quantity	quantity(163.267573271484375, "in^4")
Iy	quantity	quantity(283.775420478515625, "in^4")
ky	quantity	quantity(3.572644176215881945187219118078008415672, "in")

			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.

Can an exam override the value of this variable?

Generated value: `quantity`

quantity(6, "in")

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

For the x-axis Gap-fill
1. Gap I Engineering Accuracy with units
  Add
2. Gap k Engineering Accuracy with units
  Add
For the y axis Gap-fill
1. Gap I Engineering Accuracy with units
  Add
2. Gap k Engineering Accuracy with units
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$I_x$ = I $k_x$ = k

{display(Ix)} {display(kx)}

Advice

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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Taxonomy: mathcentre

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William Haynes 5 years, 8 months ago

William Haynes 5 years, 11 months ago

William Haynes 5 years, 11 months ago

William Haynes 6 years, 6 months ago

William Haynes 6 years, 6 months ago

William Haynes 6 years, 6 months ago

Error in variable testing condition

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Generated value: `quantity`

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Moment of inertia: semicircle and hole

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 5 years, 8 months ago

William Haynes 5 years, 11 months ago

William Haynes 5 years, 11 months ago

William Haynes 6 years, 6 months ago

William Haynes 6 years, 6 months ago

William Haynes 6 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: quantity

Parts

Gap-fill

Generated value: `quantity`