Elasticity and Fluid Machanics-07-PROBLEMS

PROBLEMS 1. A container of large uniform cross-sectional area resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities and , each of height as shown in figure. The lower density liquid is open to the atmosphere having pressure . (a) A homogeneous solid cylinder of length , cross-sectional area is immersed such that if floats with its axis vertical at the liquid-liquid interface with length in the denser liquid. Determine: (i) the density of the solid (ii) the total pressure at the bottam of the container. (b) The cylinder is removed and the original arrangement is restored. A tiny hole of area is punched on the vertical side of the container at a height . Determine: (i) the initial speed of efflux of the liquid at the hole. (ii) the horizontal distance traveled by the liquid initially, and (iii) the height at which the hole should be punched so that the liquid travels the maximum distance initially. Also calculate . (Neglect the air resistance in these calculations.) 2. A thin rod of length and uniform cross-section is pivoted at its lowest point inside a stationary homogeneous and non-viscous liquid. The rod is free to rotate in a vertical plane about a horizontal axis passing through . The density of the material of the rod is smaller than the density of the liquid. The rod is displaced by small angle from its equilibrium position and then released. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters. 3. A wooden stick of length , radius and density has a small metal piece of mass (of negligible volume) attached to its one end. Find the minimum value for the mass (in terms of given parameters) that would make the stick float vertically in equilibrium in a liquid of density . 4. A uniform solid cylinder of density 0.8 g/cm3 floats in equilibrium in a combination of two non-mixing liquids and with its axis vertical. The densities of the liquids and are 0.7 g/cm3 and 1.2 g/cm3 respectively. The height of liquid is cm. the length of the part of the cylinder immersed in liquid is = 0.8 cm. (a) Find the total force exerted by liquid on the cylinder. (b) Find , the length of the part of the cylinder in air. (c) The cylinder is depressed in such a way that its top surface is just below the upper surface of liquid and is then released. Find the acceleration of the cylinder immediately after it is released. 5. A liquid of density 900 kg/m3 is filled in a cylindrical tank of upper radius 0.9 m and lower radius 0.3 m. A capillary tube of length is attached at the bottom of the tank as shown in the figure. The capillary has outer radius 0.002 m and inner radius . When pressure is applied at the top of the tank volume flow rate of the liquid is m3/s and if capillary tube is detached, the liquid comes out from the tank with a velocity 10 m/s. Determine the coefficient of viscosity of the liquid. [Given : m2 and m] 6. In a Searle’s experiment, the diameter of the wire as measured by a screw gauge of least count 0.001 cm is 0.050 cm. The length, measured by a scale of least count 0.1 cm, is 110.0 cm. When a weight of 50 N is suspended from the wire, the extension is measured to be 0.125 cm by a micrometer of least count 0.001 cm. Find the maximum error in the measurement of Young’s modulus of the material of the wire from these data. 7. A container of width is filled with liquid. A thin wire of weight per unit length is gently placed over the liquid surface in the middle of the surface as shown in the figure. As a result, the liquid surface is depressed by a distance . Determine the surface tension of the liquid. 8. A small sphere falls from rest in a viscous liquid. Due to friction, heat is produced. Find the relation between the rate of production of heat and the radius of the sphere at terminal velocity. 9. A solid sphere of radius is floating in a liquid of density with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations.

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