Question 1.3.3: We will see in Chapter 7 that the models of many ﬂuid system......

System Dynamics [1255330]

We will see in Chapter 7 that the models of many ﬂuid systems involve the square-root function $\sqrt{h}$ , which is nonlinear. Obtain a linear approximation of f (h) = $\sqrt{h}$ valid near h = 9.

Step-by-Step

The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.

The truncated Taylor series for this function is

f (h) = f (h_{r} )+\frac{d(\sqrt{h}) }{dh} \mid_{r} (h−h_{r} )

where $h_{r}$ =9. This gives the linear approximation

f (h) =\sqrt{9} +\frac{1}{2} h^{−1/2} \mid_{r} (h−9) =3+\frac{1}{6} (h−9)

This equation gives the straight line shown in Figure 1.3.5.

‏‏لقطة الشاشة (256)

Related Answered Questions

Question: 1.3.4

The drag force on an object moving through a liquid or a gas is a function of the velocity. A commonly used model of the drag force D on an object is D =1/2 ρ A CD v² (1) whereρ is the mass density of the ﬂuid, A is the object’s cross-sectional area normal to the relative ﬂow, v is the object’s ...

Verified Answer:

a. The Taylor series approximation of equation (2)...

Question: 1.3.2

We will see in Chapter 3 that the models of many mechanical systems involve the sine function sin θ, which is nonlinear. Obtain three linear approximations of f (θ)= sin θ, one valid near θ = 0, one near θ = π/3 rad (60°), and one near θ =2π/3 rad (120°). ...

Verified Answer:

The essence of the linearization technique is to r...

Question: 1.3.1

The deﬂection of a cantilever beam is the distance its end moves in response to a force applied at the end (Figure 1.3.1). This distance is called the deﬂection and it is the output variable. The applied force is the input. The following table gives the measured deﬂection x that was produced in a ...

Verified Answer:

The plot is shown in Figure 1.3.2. Common sense te...

Question: 1.4.1

Water in a glass measuring cup was allowed to cool after being heated to 204°F . The ambient air temperature was 70°F. The measured water temperature at various times is given in the following table. Obtain a functional description of the water temperature versus time. ...

Verified Answer:

Common sense tells us that the water temperature w...

Question: 1.4.2

A hole 6 mm in diameter was made in a translucent milk container (Figure 1.4.7). A series of marks 1 cm apart was made above the hole. While adjusting the tap ﬂow to keep the water height constant, the time for the outﬂow to ﬁll a 250-ml cup was measured (1 ml = 10^−6 m³). This was ...

Verified Answer:

First obtain the ﬂow rate data in ml/s by dividing...