# Problem set 8

```MEC E 230
Problem set 8, Fall 2015
Due date: 10 am Wednesday, Nov. 25 (return problem set to the 4th floor assignment box).
Instructions:
1. Write your name and the problem set number on all sheets, number the sheets and staple them
together.
2. All questions (or parts of questions) will be assigned full, half or zero marks according to the
amount of work presented. To receive full credit, your solution must be correct and the associated
methodology must be sound and well-described.
3. If you collaborate with other students, solutions must be written up independently and the names
of those that you’ve collaborated with must be provided.
Suggested reading: Introduction to Thermal and Fluids Engineering by Kaminski & Jensen, sections
4.2.5, 4.3, 4.4 and 4.5.
Problem 1 [2 points] (from the Winter 2015 Final Exam): The semicircular plate shown
schematically below extends a distance 4 ft into the page and acts as a gate in a channel. Determine the magnitude of the resultant force, FR , associated with the water pushing on the plate. Also,
compute the components of the reaction force at the hinge (Point B). Neglect the weight of the semicircular plate. Hints: (i) ρwater × g = 62.4 lbf/ft3 , (ii) the moment of inertia of a rectangle is given by
Ixx,c = base × height3 /12, (iii) relative to Point B, the semicircle centroid is located r units up and
4r/(3π) units to the left where r = 3 ft is the radius of the semicircular plate.
Point A (smooth support)
Air
Water
Air
Hinge
Semicircular
plate
Point B
Figure 1: A semicircular plate acting as a gate in a channel.
Problem 2 [3 points]: When floating in water (specific gravity of 1.0), an equilateral triangular body
having a specific gravity of 0.9 might take on one of the following two orientations: apex up (∆) or apex
down (∇). Show that the former and latter orientations are unstable and stable, respectively. Assume
1
that the triangular body has a large length into the page (think of the package that holds a Toblerone
bar). Hint: The following webpages may prove helpful: http://www.efunda.com/math/areas/triangle.cfm, http://www.efunda.com/math/areas/isostrapezoid.cfm.
Problem 3 [1 points]: Exercise P4-32 from Kaminski & Jensen.
Problem 4 [2 points]: Exercise P4-34 from Kaminski & Jensen.
2
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