Parallel Axis Theorem: rectangles and triangles

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William Haynes

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Question statement

{geogebra_applet('qqws9ffu', [['R1',R1],['R2',R2],['T1',T1],['T2',T2]])}

Determine the moment of inertia of the rectangle and the triangle about the axes specified using the parallel axis theorem when necessary. 1 grid square = 1 {units}.

Give any introductory information the student needs.

			Name	Type	Generated Value

R1	vector	vector(-5,5)
R2	vector	vector(-10,2)
T1	vector	vector(4,8)
T2	vector	vector(-2,5)
units	string	in
version	string	y
debug	boolean	false

			Name	Type	Generated Value

R_b	quantity	quantity(5, "in")
R_h	quantity	quantity(3, "in")
T_b	quantity	quantity(6, "in")
T_h	quantity	quantity(3, "in")
R_c	vector	vector(-7.5,3.5)
T_c	vector	vector(2,7)

			Name	Type	Generated Value

R_area	quantity	quantity(15, "in^2")
T_area	quantity	quantity(9, "in^2")
R_xbar	quantity	quantity(-7.5, "in")
R_ybar	quantity	quantity(3.5, "in")
T_xbar	quantity	quantity(2, "in")
T_ybar	quantity	quantity(7, "in")
R_Ibarx'	quantity	quantity(11.25, "in^4")
R_Ibary'	quantity	quantity(31.25, "in^4")
T_Ibary'	quantity	quantity(18, "in^4")
T_Ibarx'	quantity	quantity(4.5, "in^4")
R_Ix	quantity	quantity(195, "in^4")
R_Iy	quantity	quantity(875, "in^4")
T_Ix	quantity	quantity(445.5, "in^4")
T_Iy	quantity	quantity(54, "in^4")

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(random(-10..10),random(-10..10))

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: `vector`

vector(-5,5)

→ Used by:

Statement

Parts

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

Gap-fill

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

Find the area and centroidal coordinates of the rectangle.

$A$ = Gap 0. $\bar{x}$ = Gap 1. $\bar{y}$ = Gap 2.

$A = \var{R_area} \qquad \bar{x} = \var{R_xbar} \qquad \bar{y} = \var{R_ybar}$

Advice

For the x-axis

Rectangle:

$b = \var{R_b} \qquad h = \var{R_h} \qquad A = b \cdot h = \var{R_area}$

$\bar{I}_x = \frac{1}{12}b h^3 = \var{display(R_Ibarx')}$

$I_x = \bar{I}_x + A d^2$ where $d$ is the distance from the x-axis to the centroid of the rectangle: $d =\var{qty(abs(R_c[1]),units)}$

$I_x = \var{display(R_ibarx')} + (\var{R_area}) (\var{qty(abs(R_c[1]),units)})^2 = \var{display(R_Ix)}$

Triangle:

$b = \var{T_b} \qquad h = \var{T_h} \qquad A = b \cdot h/2 = \var{T_area}$

$\bar{I}_x = \frac{1}{36}b h^3 = \var{display(T_Ibarx')}$

$I_x = \bar{I}_x + A d^2$ where $d$ is the distance from the x-axis to the centroid of the triangle: $d =\var{qty(abs(T_c[1]),units)}$

$I_x = \var{display(T_ibarx')} + (\var{T_area}) (\var{qty(abs(T_c[1]),units)})^2 = \var{display(T_Ix)}$

For the y-axis

Follow a similar procedure to find the moments of inertia with respect to the y-axis.

Rectangle: $I_y = \bar{I}_y + A d^2 = \var{display(R_ibary')} + (\var{R_area}) (\var{qty(abs(R_c[0]),units)})^2 = \var{display(R_Iy)}$

Triangle: $I_y = \bar{I}_y + A d^2 = \var{display(T_ibary')} + (\var{T_area}) (\var{qty(abs(T_c[0]),units)})^2 = \var{display(T_Iy)}$

Give a worked solution to the whole question.

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There was an error which means the tests can't run:

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
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
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

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: rectangles and triangles

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Checkpoint description

William Haynes 4 years, 4 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: `vector`

Parts

Gap-fill

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Error

Parallel Axis Theorem: rectangles and triangles

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, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: vector

Parts

Gap-fill

Generated value: `vector`