Centroid by integration: upper and lower bounds are functions

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William Haynes

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William Haynes commented 3 years, 5 months ago

Changed gaps to engineering accuracy with units, where units are specified as "". Should edit question to use actual units.

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William Haynes 3 years, 11 months ago

Gave some feedback: Ready to use

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William Haynes 3 years, 11 months ago

Gave some feedback: Has some problems

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William Haynes 4 years, 2 months ago

Published this.

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William Haynes 4 years, 5 months ago

Gave some feedback: Ready to use

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William Haynes 4 years, 5 months ago

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Question statement

{applet()}

Determine the coordinates of the centroid of the shaded area bounded by the $y$ -axis, a parabola passing throught the origin and point $A$ , and a straight line passing through points $A$ and $B$ .

Given: $A = ( \var{A[0]}, \var{A[1]} ), B = (0, \var{b})$

Give any introductory information the student needs.

			Name	Type	Generated Value

area	number	7.3333333333
Qy	number	5.6666666667
xbar	number	0.7727272727
Qx	number	26.5333333333
ybar	number	3.6181818182

			Name	Type	Generated Value

A	vector	vector(2,7)
b	integer	5
k	number	1.75
m	number	1
x	number	2

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
- k x^3/3 + m x^2/2 + b x

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: `number`

7.3333333333

← Depends on:

→ Used by:

xbar
ybar

This variable doesn't seem to be used anywhere.

Parts

centroid Gap-fill
1. Gap xbar Engineering Accuracy with units
  Add
2. Gap ybar 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 units

$\bar{y} = $ ybar units

Advice

Bounding Functions

Parabola: $f(x) = k x^2$

at $x=\var{A[0]}, f(x)=\var{A[1]}$ , so $k = \dfrac{\var{A[1]}}{\var{A[0]}^2}$

so, $f(x) = \var[fractionNumbers]{A[1]/A[0]^2} x^2$

Line: $g(x) = m x + b$

where $m = \dfrac{(\var{A[1]}-\var{b})}{\var{x}}$, and $b = \var{b}$

so, g(x) = $\var[fractionNumbers]{(A[1]-b) /x} x + \var{b}$

Strip Properties for Vertical strips

$dA = [g(x) - f(x)]\; dx =\left(\simplify[all]{- {A[1]}/{A[0]}^2} x^2 +\simplify{({A[1]}-{b}) /{x}} x + \simplify{{b}} \right )\; dx $

$\bar{x}_{el} = x$

$\bar{y}_{el} = \dfrac{g(x) + f(x)}{2} $

Integrals

$A = \int_0^\var{x}\; dA = \var{siground(area,4)} \text{ units}^2$

$Q_y = \int_0^\var{x} \bar{x}_{el} \; dA = \var{siground(Qy,4)} \text{ units}^3$

$Q_x = \int_0^\var{x} \bar{y}_{el} \; dA = \var{siground(Qx,4)}\text{ units}^3$

Centroid

$\bar{x} = \dfrac{Q_y}{A} = \var{siground(xbar,4)}$ units

$\bar{y} = \dfrac{Q_x}{A} = \var{siground(ybar,4)}$ units

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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

This question is used in the following exams:

Chapter 7 Exercises by William Haynes in Engineering Statics.
Chapter 07 Exercises by Michael Proudman in Engineering Statics.
20. Centroids by Integration by Michael Proudman in Engineering Statics.
21. Centroids by Integration by William Haynes in Engineering Statics.

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Centroid by integration: upper and lower bounds are functions

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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William Haynes commented 3 years, 5 months ago

William Haynes 3 years, 11 months ago

William Haynes 3 years, 11 months ago

William Haynes 4 years, 2 months ago

William Haynes 4 years, 5 months ago

William Haynes 4 years, 5 months ago

Error in variable testing condition

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Generated value: `number`

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Gap-fill

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Centroid by integration: upper and lower bounds are functions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes commented 3 years, 5 months ago

William Haynes 3 years, 11 months ago

William Haynes 3 years, 11 months ago

William Haynes 4 years, 2 months ago

William Haynes 4 years, 5 months ago

William Haynes 4 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`