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Unformatted text preview: The University of Michigan
Department of Mechanical Engineering
ME 320 Profs. Akhavan and Hasselbrink
Midterm Exam, March 8 2004
Open Book, One crib sheet (2 sides) allowed (10 points) Problem 1: A tank of densitystratiﬁed salt solution, having a depth h = 0.1m7 is
prepared in the laboratory. The density of the liquid increases linearly with depth, as indicated
in the ﬁgure, starting from the density of pure water pw at the top surface and increasing to
a value of 1.6pw at the bottom of the tank. A rectangular block of wood of height h = 0.1771
and density p1, = 0.5pw is placed in the tank. Calculate the depth b below the surface of the
liquid to which the bottom surface of the block sinks when it is floating in equilibrium. (25 points) Problem 2: A water tank of crosssectional area A is placed on a cart and
mounted on frictionless rollers as shown in the ﬁgure. Water in the tank discharges to the
atmosphere through a short nozzle of area Aj << A. The water inside the tank is maintained
at. a constant height hoabove the nozzle by an external supply which discharges in the tank
at a volumetric flow rate Q. The cart is restrained from moving by a cable attached to an
immovable Wall. The mass of the tank, cart, and water is initially Mo. In terms of the given parameters 13, Aj, Ma, and the water density p,
(5 points) (a) Find the volumetric flow rate Q.
(5 points) (b) Find the tensile force T in the cable. (10 points) (c) At time t = 0, the external supply of water is shut off and the cable holding
the tank is cut. The cart begins to move to the left at a velocity Vc(t). Derive the differential
equation governing Vc(t) for all times t Z 0. Assume the free surface of the ﬂuid remains
horizontal. (5 points) (d) How would your answer change if you considered the effect of acceleration
on the free surface? Assume no sloshing; that is, assume the free surface results from quasi
hydrostatic equilibrium at any given moment. Q
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