Centroidal Moment of inertia: T shape

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William Haynes 4 years, 4 months ago

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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

{geogebra_applet('wfhyktjn',['A':P1, 'B': P2])}

Determine the moment of inertia of the T shape with respect to a vertical axis passing through the shape's centroid. Let this axis be designated $a$ . Grid units are {units}.

Give any introductory information the student needs.

			Name	Type	Generated Value

units	string	in
debug	boolean	false
applet	ggbapplet	HTML node
P1	vector	vector(6,2)
P2	vector	vector(7,6)
b1	number	6
h1	number	4
b2	number	1
h2	number	12

			Name	Type	Generated Value

A1	number	24
A2	number	12
A	number	36

			Name	Type	Generated Value

Iy1	number	288
Iy2	number	508
ky	number	4.7022453266

			Name	Type	Generated Value

ibar1	number	72
Ibar2	number	1
d1	number	3
d2	number	6.5
Iy1'	number	288
Iy2'	number	508
Iy	number	796
Qy	number	150
xbar	number	4.1666666667
I_a	number	171

			Name	Type	Generated Value

i_a'

Definition of variable i_a' is empty.

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random('in','cm')

JME function reference

Description

width of P1

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

Can an exam override the value of this variable?

Generated value: `string`

→ Used by:

Statement
Advice
"Centroid" → "xbar" - Correct Quantity.
"Centroid" → "ybar" - Correct Quantity.
"For the $a$ axis" → "$\bar{I}_{a}$" - Correct Quantity.

Parts

Centroid Gap-fill
1. Gap xbar Engineering Accuracy with units
  Add
2. Gap ybar Engineering Accuracy with units
  Add
For the $a$ axis Gap-fill
1. Gap $\bar{I}_{a}$ Engineering Accuracy with units
  Add

Gap-fill

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

$\bar{x} = $ xbar $\qquad \bar{y} = $ ybar

Advice

Number the parts:

Part 1 horizontal rectangle: $b_1$ = {qty(b1,units)}, $h_1$ = {qty(h1,units)}.

Part 2 vertical rectangle: $b_2$ = {qty(b2,units)}, $h_2$ = {qty(h2,units)}.

Calculate the areas:

$ A_1 = b_1 h_1=\var{qty(A1,units+'^2')}$

$ A_2 = b_2 h_2 = \var{qty(A2,units+'^2')}$

Locate the centroids of the parts.

$ \bar{x}_1 = b_1/2 = \var{qty(d1,units)} \qquad \bar{x}_2 = b_1 + b_2/2 = \var{qty(d2,units)}$

Locate the centroid of the whole.

$\bar{x} = \dfrac{ A_1 \bar{x}_1 + A_2 \bar{x}_2} {A_1 + A_2} = \var{qty(siground(xbar,4),units)}$

$\bar{y} = \var{qty(0, units)} $, by symmetry

Calculate the moment of inertia with respect to the y-axis for the parts and the whole.

$\begin{align} \\{I_y}_1 &= \tfrac{1}{3} h_1 b_1^3 \\&= \tfrac{1}{3}( \var{h1})(\var{b1})^3 \\&= \var{qty(display(Iy1),units+'^4')} \\\\{I_y}_2 &= [\bar{I} + A d^2]_2\\ &= \frac{1}{12} h_2 b_2^3 + A_2 d_2^2 \\&= \var{display(Ibar2)} + (\var{display(A2)}) (\var{display(d2)})^2\\&=\var{qty(display(Iy2),units+'^4')} \\\\I_y &= {I_y}_1+{I_y}_2\\&=\var{display(Iy1)}+\var{display(Iy2)}\\&=\var{qty(display(Iy),units+'^4')}\end{align}$

Use the parallel axis theorem $I = \bar{I} + A d^2$ backwards to find the centroidal moment of inertia.

$\begin{align} \bar{I}_a &= I_y - A (\bar{x})^2 \\&= \var{display(Iy)} - \var{display(A)} (\var{display(xbar)})^2\\&=\var{qty(display(I_a),units+'^4')}\end{align}$

Give a worked solution to the whole question.

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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

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Taxonomy: mathcentre

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William Haynes 4 years, 4 months ago

William Haynes 4 years, 4 months ago

William Haynes 4 years, 6 months ago

William Haynes 4 years, 6 months ago

William Haynes 4 years, 6 months ago

Error in variable testing condition

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Generated value: `string`

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Centroidal Moment of inertia: T shape

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 4 years, 4 months ago

William Haynes 4 years, 4 months ago

William Haynes 4 years, 6 months ago

William Haynes 4 years, 6 months ago

William Haynes 4 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Gap-fill

Generated value: `string`