Chapter 4: Problem 171
Inflate a toy balloon with air and release it. Watch as the balloon darts about the room. Explain what causes the phenomenon you see.
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Chapter 4: Problem 171
Inflate a toy balloon with air and release it. Watch as the balloon darts about the room. Explain what causes the phenomenon you see.
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\( \mathrm{A}\) disk, of mass \(M,\) is constrained horizontally but is free to move vertically. A jet of water strikes the disk from below. The jet leaves the nozzle at initial speed \(V_{0}\), Obtain a differential equation for the disk height, \(h(t),\) above the jet exit plane if the disk is released from large height, \(H .(\mathrm{You}\) will not be able to solve this ODE, as it is highly nonlinear!) Assume that when the disk reaches equilibrium, its height above the jet exit plane is \(h_{0}\) (a) Sketch \(h(t)\) for the disk released at \(t=0\) from \(H>h_{0}\) (b) Explain why the sketch is as you show it.
\(\mathrm{A}\) rocket sled traveling on a horizontal track is slowed by a retro- rocket fired in the direction of travel. The initial speed of the sled is \(U_{0}=500 \mathrm{m} / \mathrm{s}\). The initial mass of the sled is \(M_{0}=1500 \mathrm{kg}\). The retro-rocket consumes fuel at the rate of 7.75 kgls, and the exhaust gases leave the nozzle at atmospheric pressure and a speed of \(2500 \mathrm{m} / \mathrm{s}\) relative to the rocket. The retro- rocket fires for 20 s. Neglect acrodynamic drag and rolling resistance. Obtain and plot an algebraic expression for sled speed \(U\) as a function of firing time. Calculate the sled speed at the end of retro-rocket firing.
You turn on the kitchen faucet very slightly, so that a very narrow stream of water flows into the sink. You notice that it is "glassy" (laminar flow) and gets narrower and remains "glassy" for about the first \(50 \mathrm{mm}\) of descent. When you measure the flow, it takes three min to fill a 1 -L bottle, and you estimate the stream of water is initially \(5 \mathrm{mm}\) in diameter. Assuming the speed at any cross section is uniform and neglecting viscous effects, derive expressions for and plot the variations of stream speed and diameter as functions of \(z\) (take the origin of coordinates at the faucet exit). What are the speed and diameter when it falls to the 50 -mm point?
The tank shown rolls with negligible resistance along a horizontal track. It is to be accelerated from rest by a liquid jet that strikes the vane and is deflected into the tank. The initial mass of the tank is \(M_{0}\). Use the continuity and momentum equations to show that at any instant the mass of the vehicle and liquid contents is \(M=M_{0} V /(V-U) .\) Obtain a general expression for \(U\) / \(V\) as a function of time.
A cylindrical holding water tank has a \(3 \mathrm{m}\) ID and a height of \(3 \mathrm{m}\). There is one inlet of diameter \(10 \mathrm{cm},\) an exit of diameter \(8 \mathrm{cm},\) and a drain. The tank is initially empty when the inlet pump is turned on, producing an average inlet velocity of \(5 \mathrm{m} / \mathrm{s}\). When the level in the tank reaches \(0.7 \mathrm{m}\) the exit pump turns on, causing flow out of the exit; the exit average velocity is \(3 \mathrm{m} / \mathrm{s}\). When the water level reaches \(2 \mathrm{m}\) the drain is opened such that the level remains at \(2 \mathrm{m}\). Find (a) the time at which the exit pump is switched on, (b) the time at which the drain is opened, and (c) the flow rate into the drain \(\left(\mathrm{m}^{3} / \mathrm{min}\right)\)
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