Holooly Plus Logo
Holooly Plus Logo

Question 1.3.3: Linearization of a Square-Root Model We will see in Chapter ...

Linearization of a Square-Root Model

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.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

The truncated Taylor series for this function is
f (h) = f (h_{r}) + \frac{d(\sqrt{h} )}{d h} |_{r} (h  −  h_r)
where h_{r} = 9. This gives the linear approximation
f (h) = \sqrt{9} + \frac{1}{2} h^{−1/2} |_{r} (h  −  9) = 3 + \frac{1}{6} (h  −  9)
This equation gives the straight line shown in Figure 1.3.5.

1.3.5

Related Answered Questions

Question: 1.5.3

Fitting a Power Function with a Known Exponent Fit the power function y = bx^m to the data yi . The value of m is known. ...

Verified Answer:

The least-squares criterion is J = \sum^{n}...
Question: 1.5.2

Point Constraint Consider the data given at the beginning of this section We found that the best-fit line is y = (9/10)x + 11/6. Find the best-fit line that passes through the point x = 10, y = 11. ...

Verified Answer:

Subtracting 10 from all the x values and 11 from a...
Question: 1.5.1

Fitting Data with the Power Function Find a functional description of the following data: ...

Verified Answer:

These data do not lie close to a straight line whe...
Question: 1.4.2

Orifice Flow 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 dividin...
Question: 1.4.1

Temperature Dynamics of Water 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: 1.3.2

Linearization of the Sine Function We will see in Chapter 2 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

A Cantilever Beam Deflection Model 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). The following table gives the measured deflection x that was produced in a particular beam by the given applied force f . Plot the data to ...

Verified Answer:

The plot is shown in Figure 1.3.2. Common sense te...
Question: 1.6.6

Orifice Flow with Constrained Exponent Consider the data of Example 1.6.5. Determine the best-fit value of the coefficient b in the square-root function f = bh^1/2 ...

Verified Answer:

First obtain the flow rate data in ml/s by dividin...
Question: 1.6.5

Orifice Flow Consider again Example 1.4.2. Ahole 6 mm in diameterwas 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 ...

Verified Answer:

First obtain the flow rate data in ml/s by dividin...
Question: 1.6.4

Temperature Dynamics of Water Consider again Example 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 ...

Verified Answer:

From Example 1.4.1, we learned that the relative t...