4. (a) Find a differential equation to model the velocity v of a falling mass m as a function of time. Assume that air resistance is proportional to the instantaneous velocity, with a constant of proportionality k > 0 (this is called the drag coefficient). Take the downward direction to be positive. (b) Solve the differential equation subject to the initial condition v(t = 0) = vo. (c) Determine the terminal velocity of the mass.
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Given, mass of the object = m.
Let 'v' be the instantaneous velocity of the object free falling.
Air resistance is assumed proportional to instantaneous velocity as: ; where is the proportionality constant.
(a) Free body diagram of the falling mass is drawn below.
Here, ; where acceleration of the mass,
------------------------------ (1)
Equation (1) represents the differential equation to model velocity (v) of falling mass (m).
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- When a sphere falls freely through a homogeneous fluid, it reaches a terminal velocity at which the weight of the sphere is balanced by the buoyant force and the frictional resistance of the fluid. Make a dimensional analysis of this problem and indicate how experimental data for this problem could be correlated. Neglect compressibility effects and the influence of surface roughness.In the field of air pollution control, one often needs to sample the quality of a moving airstream. In such measurements a sampling probe is aligned with the flow as sketched in Fig. A suction pump draws air through the probe at volume flow rate V· as sketched. For accurate sampling, the air speed through the probe should be the same as that of the airstream (isokinetic sampling). However, if the applied suction is too large, as sketched in Fig, the air speed through the probe is greater than that of the airstream (super iso kinetic sampling). For simplicity consider a two-dimensional case in which the sampling probe height is h = 4.58 mm and its width is W = 39.5 mm. The values of the stream function corresponding to the lower and upper dividing streamlines are ?l = 0.093 m2/s and ?u = 0.150 m2/s, respectively. Calculate the volume flow rate through the probe (in units of m3/s) and the average speed of the air sucked through the probe.The wind flutter on the wing of a newly proposed jet fighter is given by the following 1st order differential equation: With the Boundary Condition: y(0) = 1 (remember this means that y = 1 when x = 0) Determine the vertical motion (y) in terms of the span (x) of the wing. The frequency of fluctuations of the wing at mach 2 is given by the non-homogenous 2nd order differential equation: With the boundary conditions: y(0) = 1 and y(1) = 0 (i.e., y = 1 when x = 0 and y = 0 when x = 1) By solving the homogenous form of this equation, complete the analysis and determine the amplitude (y) of vibration of the wing tip at mach 2. Critically evaluate wing flutter and fluctuation frequency amplitude determined by solving the two differential equations above.
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- The following equation may be used to estimate the take-off ground run for an aircraft: Equation has been attached as an image. Calculate the take-off ground run, from a runway at ISA-SL conditions, for a twin engine aircraft for which the following data may be assumed Aircraft lift-off speed 155 knots Max take-off gross weight 220 tonnes Wing planform area (S) 358 m Wing CL (t/o flaps deployed, a = 0) 1.1 Wing span 53.18 m Oswald efficiency factor, e 0.7 KGE = CD(IGE) / Co(OGE) 0.4 Co sum (fuselage, wing, tailplane and nacelle) 0.015 Co for undercarriage 0.021 Co for flaps at taking-off setting 0.0073 Coefficient of rolling friction, u 0.02 Engine thrust (assumed constant) 310 kN per engine It may be assumed that 1knot = 0.51444 m/s It may be assumed that 1knot = 0.51444 m/s#4 1.11 For a small particle of styrofoam (1 lbm/ft) (spherical, with diameter d = 0.3 mm) falling in standard air at speed V, the drag is given by FD-3mVd, where is the air viscosity. Find the maximum speed starting from rest, and the time it takes to reach 95 percent of this speed. Plot the speed as a function of time. s) Answer: (Vmax=0.0435",t=0.0133 S3.1 Two-dimensional incompressible flow can also be simulated using the vorticity w and streamfunction , instead of using the velocity and pressure variables. The vorticity field can be related to the velocity variables through the curl operation Əv ди Əx ду and the velocity field (u, v) can be related to the streamfunction with მს ду 11== v== Using the continuity equation, show that 2² 2² + əx² Əy² მს əx =-w. (3.252) (3.253)