My Math Forum Height of feed in a trapezoidal trough.
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 September 13th, 2015, 04:03 AM #1 Member   Joined: Jun 2015 From: Warwick Posts: 37 Thanks: 1 Height of feed in a trapezoidal trough. A trough has a cross-section in the form of a trapezium. Its base has a length of 1m, and the sides slope out at 45 degrees. to the horizontal. The trough is filled with feed to a depth of x metres. Find the value of x given that the centre of mass of the contents of the trough is 0.5m above the base. Centroid of a trapezium is (h/3)[(b+2a)/(b+a)], where h is the height and b is the longest (base) and a is the shortest base (although in this problem, it is a on the horizontal). The height of the trough is sin(45), if we take the length of the sloped edge to be 1m, as well. The answer should be 0.866
September 15th, 2015, 09:47 AM   #2
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$\displaystyle \bar{y} = \frac{h}{3} \left(\frac{2a+b}{a+b}\right)$

note from the diagram ...

$b=1$, $a-2h=1 \implies a=2h+1$

$\displaystyle \frac{1}{2} = \frac{h}{3} \left[\frac{2(2h+1)+1}{(2h+1)+1}\right]$

$\displaystyle \frac{1}{2} = \frac{h}{3} \left(\frac{4h+3}{2h+2}\right)$

$\displaystyle \frac{3}{2h} = \frac{4h+3}{2h+2}$

$\displaystyle h = \frac{\sqrt{3}}{2} \approx 0.866$
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 centroid_trap.jpg (31.2 KB, 5 views)

 September 16th, 2015, 02:54 AM #3 Newbie     Joined: Aug 2015 From: USA Posts: 29 Thanks: 0 It's near to accurate. Last edited by skipjack; September 16th, 2015 at 04:16 AM.
 September 16th, 2015, 04:28 PM #4 Member   Joined: Jun 2015 From: Warwick Posts: 37 Thanks: 1 Thanks for the reply. It is a good answer and the diagram is very clear.

 Tags centre of mass, feed, height, trapezoidal, trough

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