Centroid of a triangle by integration

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William Haynes

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Question statement

{applet}

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			Name	Type	Generated Value

applet

ggbapplet

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xxxxxxxxxx
 
geogebra_applet("jyqxv2uf",[B: vector(random(-1,1,2,4,5),3.5)])

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This variable is an interactive HTML node. Interactive HTML nodes can not be relied upon to work correctly when resuming a session - for example, attached event callbacks will be lost, and mathematical notation will likely also break.

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Generated value: `ggbapplet`

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Statement

Parts

Questions Gap-fill
1. Gap dA Mathematical expression
  Add
2. Gap $\bar{y}_{el} Match text pattern
  Add
3. Gap lower Number entry
  Add
4. Gap upper Match text pattern
  Add
5. Gap Qx Mathematical expression
  Add
6. Gap ybar Mathematical expression
  Add

Gap-fill

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

Use horizontal strips to prove by integration that for a triangle with base $b$ and height $h$, $\bar{y} = h/3$.

1. Derive expressions for $\bar{y}_{el}$ and $dA$ in terms of $y$.

$\bar{y}_{el} =$ $\bar{y}_{el} $\qquad dA=$ dA

2. Determine the limits of the integration.

Lower limit lower Upper limit upper

3. Evaluate $Q_x $ between your limits.

$Q_x = \int \bar{y}_{el} dA = $ Qx

4. Evaluate $\bar{y}$ for the triangle.

$\bar{y} = \dfrac{Q_x}{A} = $ ybar

Advice

Use similar triangles to express the width of the strip as a function of $y$:

$\dfrac{w}{h-y} = \dfrac{b}{h}$

$w = b\dfrac{(h-y)}{h}$

Therefore,

$dA = w\; dy = b\dfrac{(h-y)}{h} dy$

With this you can evaluate $Q_x$

$\begin{align} Q_x &= \int y\; dA\\ &=\int_0^h y\;b\frac{(h-y)}{h}\; dy\\&= \frac{b}{h}\int_0^h (hy - y^2)\; dy\\& = \frac{b}{h}\left. \left(\frac{h y^2}{2} - \frac{y^3}{3}\right)\right |_0^h \\ &= bh^2\left( \frac{1}{2} - \frac{1}{3}\right)\\ &= \frac{bh^2}{6}\end{align}$

Finaly, find $\bar{y}$ knowing the area of the triangle is $\dfrac{bh}{2}$

$\begin{align} \bar{y} &= \frac{Q_x}{A}\\ &= \frac{{bh^2}/{6}}{{bh}/{2}}\\ &= \frac{h}{3}\end{align}$

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This question is used in the following exam:

21. Centroids by Integration by William Haynes in Engineering Statics.

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William Haynes 2 years, 5 months ago

William Haynes 2 years, 5 months ago

William Haynes 2 years, 5 months ago

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Generated value: `ggbapplet`

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Centroid of a triangle by integration

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 2 years, 5 months ago

William Haynes 2 years, 5 months ago

William Haynes 2 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: ggbapplet

Parts

Gap-fill

Generated value: `ggbapplet`