MATH3503 Online Assignment 10

• Make sure to use the link on the Learning Hub, which contains a user name and a user ID unique to you, otherwise your mark would not be recorded. You may use the URL without '?uname=...&uid=...' if you just want to practice. You will get different numbers every time you open the page.
• Numerical answers are evaluated as correct if the difference from the correct value is less than `0.001 times` (correct value) unless otherwise specified. That means, you need to keep at least 5 significant digits (e.g., 12.34, 0.01234, etc.) in your final answer, and a couple of more digits in the intermediate steps.

Hydrostatic force

1.  
   An isosceles triangle with a base length of 1 m and a height of 3 m is submerged so that the top of the triangle is at a depth of 2 m. Compute the hydrostatic force on the triangle. `F = `Newtons
   
   
2.  

   A vertical dam has a gate in the shape of a parabolic arch with a maximum height of 1 m and a maximum width of 1 m, as shown in the figure. If the dam is 34 m high and the water level is 4 m below the top of the dam, find the hydrostatic force against the gate. `F = `Newtons

   34 m 1 m 1 m 4 m Fig. 1

   *Hint: Try the substitution `u = `inside square root.
   
   
3.  

   A horizontally placed trough has two semi-circular ends (See Fig. 2) of radius 0.1 m. Find the force on each end plate if the trough is filled with water. `F = `Newtons

   Fig. 2

   
   
4.  

   Shown in Fig. 3 is a gravity dam (outlined in red). Compute the moment of force about the bottom of the dam due to the water in the dam. `M = `N`cdot`m

   89 m 72 m 130 m 159 m Fig. 3

   
   

   Center of Mass, Centroid

   Compute the center of mass or the centroid of each of the following objects.
5.  
   Compute the center of mass for a concrete beam with a mass distribution described by linear density `lambda(x) = 0.012x^2+2` for `0\le x\le 13`, where `lambda` in tons/metre and `x` in metres.
   `bar{x} = `metres
   
   
6.  
   Compute the centroid of the area bounded by \(\displaystyle \sqrt{5x}+\sqrt{3y}=1\) and the coordinate axes (i.e., the `x`- and `y`- axes).
   `(bar{x}, bar{y})` = (, )
   
   
7.  
   Compute the centroid for the region bounded by the curve `y = 4-x^2` and the lower half of \(\displaystyle x^{2}+y^{2} = 4\).
   `(bar{x}, bar{y})` = (, )
   * Hint: 1) Utilize the symmetry. 2) You already know the area of a semi-circle.
   
   

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