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EML 3701: Fluid Mechanics 1 Quiz 2: 20mins (Solution) 1. The standard atmosphere is equivalent to 760mm column of mercury (ρ =13,595 kg/m3). What will be the equivalent column height of water (ρ =1000 kg/m3)? [10points] a. 0.76m b. 55.90m c. 10.33m d. 32m Solution

ρwghw = ρmghm hw = ρmhm/ρw = (13,595)(0.76)/(1000) = 10.33m 2. Cooking at high altitudes takes shorter time than low altitudes. [10points] a. True b. False c. Unknown Solution Boiling occurs at lower temperature due to the lower pressure at high altitudes. Therefore cooking will take longer at higher altitudes.

3. Water from a reservoir is raised in a vertical tube of internal diameter D = 30 cm under the influence of the pulling force F of a piston. Determine the force needed to raise the water to a height of h = 1.5 m above the free surface. [10points] a. 3.19 kN b. 1.04 kN c. 0.84 kN d. 1.96 kN Solution Noting that the pressure at the free surface is Patm and hydrostatic pressure in a fluid decreases linearly with increasing height, the pressure at the piston face is  1 kN P = Patm − ρgh = 95 kPa − (1000 kg/m 3 )(9.81 m/s 2 )(1.5 m)  1000 kg ⋅ m/s 2 

Piston face area is A = πD 2 / 4 = π (0.3 m) 2 /4 = 0.07069 m 2

 1 kPa   1 kN/m 2 

  = 81.3 kPa 

A force balance on the piston yields  1 kN/m 2 F = ( Patm − P ) A = (95 − 81.3 kPa )((0.07068 m 2 )  1 kPa

 1 kPa    1 kN/m 2  = 1.04 kN 

4. Consider an aboveground swimming pool 8-m-long, 8-m-wide, and 2-m-high that is filled with water to the rim. Determine the hydrostatic force on a wall. [10points] a. 9.8 kN b. 209 kN c. 157 kN d. 311 kN Solution The average pressure on a surface is the pressure at the centroid (midpoint) of the surface, and is determined to be Pavg = PC = ρghC = ρg (h / 2)  1N = (1000 kg/m 3 )(9.81 m/s 2 )(2 / 2 m)  1 kg ⋅ m/s 2 

   

= 9810 N/m 2

Then the resultant hydrostatic force on each wall becomes FR = Pavg A = (9810 N/m 2 )(8 m × 2 m) = 156,960 N = 157 kN

5. Consider a 3-kg copper cube and a 3-kg copper ball submerged in a liquid. Will the buoyant forces acting on these two bodies be the same? [10points] a. Yes b. No c. Unknown Solution The magnitude of the buoyant force acting on a submerged body whose volume is V is expressed as FB = ρ f gV , which is independent of the shape of the body. Therefore, the buoyant forces acting on the cube and sphere made of copper submerged in water are the same since they have the same volume. (Note: The two objects have the same volume because they have the same mass and density).

6. Consider a vertical cylindrical container partially filled with water. Now the cylinder is rotated about its axis at a specified angular velocity, and rigid-body motion is established. Assuming there is no dry spot, the highest pressure occurs at the: [10points] a. Meniscus contact with side wall of container b. Midpoint of bottom surface of container

c. Bottom corners of container d. Sidewall of container at midheight Solution When a vertical cylindrical container partially filled with water is rotated about its axis and rigid body motion is established, the fluid level will drop at the center and rise towards the edges. Noting that hydrostatic pressure is proportional to fluid depth, the pressure at the midpoint will drop and the pressure at the edges of the bottom surface will rise due to the rotation. The highest pressure therefore occurs at the bottom corners of the container.