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Chapter 4: Problem 42

As is indicated in Fig. \(P 4.42,\) the speed of exhaust in a car's cxhaust pipe varies in time and distance because of the periodic nature of the engine's operation and the damping effect with distance from the engine. Assume that the speed is given \\[ \text { by } \quad V=V_{0}\left[1+a e^{-b x} \sin (\omega t)\right], \quad \text { where } \quad V_{0}=8 \text { fps, } a=0.05 \\] \(b=0.2 \mathrm{ft}^{-1},\) and \(\omega=50 \mathrm{rad} / \mathrm{s} .\) Calculate and plot the fluid acceleration at \(x=0,1,2,3,4,\) and 5 ft for \(0 \leq t \leq \pi / 25\) s.

Short Answer

Expert verified
The fluid acceleration at different points within the exhaust pipe (from 0 to 5 ft) over the duration of \(0 \leq t \leq \pi / 25\) s can be calculated by differentiating the given speed function, substituting the specified values into the resulting acceleration function, and solving it for each point over the given time range. These results can then be plotted to visually represent the fluid acceleration at each point within the exhaust pipe over time.

Step by step solution

01

Find the Fluid Acceleration Function

The fluid acceleration is the second derivation of the position with respect to time, which in this case equates to the second derivation of the speed (V) with respect to time (t). So differentiation of the speed function is needed. Using the chain rule for differentiation, the derivative of \(V = V_{0}[1+ a e^{-bx}\sin(\omega t)]\) with respect to time [\(a(t)\)] can be obtained. This will give us the fluid acceleration function.
02

Calculate Fluid Acceleration at Specified Points

Substitute the given values of \(V_0\), \(a\), \(b\), and \(\omega\) into the previously derived acceleration function. Then substitute the specified distances (\(x = 0, 1, 2, 3, 4,\) and 5 ft) into the acceleration function and solve it for \(0 \leq t \leq \pi / 25\) s. This will give us the fluid acceleration at every foot within the exhaust pipe over the given duration of time.
03

Plot the Fluid Acceleration

Plot the calculated fluid acceleration values against the corresponding time values for each point within the exhaust pipe (i.e., for each \(x\) value). The x-axis will represent time, and the y-axis will represent fluid acceleration. This will provide a visual representation of the fluid acceleration at each foot within the exhaust pipe over the given duration of time.

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Most popular questions from this chapter

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