Straight Lines in Engineering

CHAPTER 1

In this chapter, the applications of straight lines in engineering are introduced. It is assumed that the students are already familiar with this topic from their high school algebra course. This chapter will show, with examples, why this topic is so important for engineers. For example, the velocity of a vehicle while braking, the voltage-current relationship in a resistive circuit, and the relationship between force and displacement in a preloaded spring can all be represented by straight lines. In this chapter, the equations of these lines will be obtained using both the slope-intercept and the point-slope forms.

1.1 VEHICLE DURING BRAKING

The velocity of a vehicle during braking is measured at two distinct points in time, as indicated in Fig. 1.1.

Figure 1.1 A vehicle while braking.

The velocity satisfies the equation

where v_o is the initial velocity in m/s and a is the acceleration in m/s².

(a) Find the equation of the line v(t) and determine both the initial velocity v_o and the acceleration a.

(b) Sketch the graph of the line v(t) and clearly label the initial velocity, the acceleration, and the total stopping time on the graph.

The equation of the velocity given by equation (1.1) is in the slope-intercept form y = mx + b, where y = v(t), m = a, x = t, and b = v_o. The slope m is given by

Therefore, the slope m = a can be calculated using the data in Fig. 1.1 as

The velocity of the vehicle can now be written in the slope-intercept form as

v(t) = −3.9 t + v_o.

The y-intercept b = v_o can be determined using either one of the data points. Using the data point (t, v) = (1.5, 9.75) gives

9.75 = −3.9 (1.5) + v_o.

Solving for v_o gives

v_o = 15.6 m/s.

The y-intercept b = v_o can also be determined using the other data point (t, v) = (2.5, 5.85), yielding

5.85 = −3.9 (2.5) + v_o.

Solving for v_o gives

v_o = 15.6 m/s.

The velocity of the vehicle can now be written as

v(t) = −3.9 t + 15.6 m/s.

The total stopping time (time required to reach v(t) = 0) can be found by equating v(t) = 0, which gives

0 = −3.9 t + 15.6.

Solving for t, the stopping time is found to be t = 4.0 s. Figure 1.2 shows the velocity of the vehicle after braking. Note that the stopping time t = 4.0 s and the initial velocity v_o = 15.6 m/s are the x- and y-intercepts of the line, respectively. Also, note that the slope of the line m = −3.90 m/s² is the acceleration of the vehicle during braking.

Figure 1.2 Velocity of the vehicle after braking.

1.2 VOLTAGE-CURRENT RELATIONSHIP IN A RESISTIVE CIRCUIT

For the resistive circuit shown in Fig. 1.3, the relationship between the applied voltage V_s and the current I flowing through the circuit can be obtained using Kirchhoff's voltage law (KVL) and Ohm's law. For a closed-loop in an electric circuit, KVL states that the sum of the voltage rises is equal to the sum of the voltage drops, i.e.,

Kirchhoff's voltage law: ⇒    ∑ Voltage rise = ∑ Voltage drop.

Figure 1.3 Voltage and current in a resistive circuit.

Applying KVL to the circuit of Fig. 1.3 gives

Ohm's law states that the voltage drop across a resistor V_R in volts (V) is equal to the current I in amperes (A) flowing through the resistor multiplied by the resistance R in ohms (Ω), i.e.,

Substituting equation (1.3) into equation (1.2) gives a linear relationship between the applied voltage V_s and the current I as

The objective is to find the value of R and V when the current flowing through the circuit is known for two different voltage values given in Fig. 1.3.

The voltage-current relationship given by equation (1.4) is the equation of a straight line in the slope-intercept form y = mx + b, where y = V_s, x = I, m = R, and b = V. The slope m is given by

Using the data in Fig. 1.3, the slope R can be found as

Therefore, the source voltage can be written in slope-intercept form as

V_s = 10 I + b.

The y-intercept b = V can be determined using either one of the data points. Using the data point (V_s, I) = (10, 0.1) gives

10 = 10 (0.1) + V.

Solving for V gives

V = 9 V.

The y-intercept V can also be found by finding the equation of the straight line using the point-slope form of the straight line (y − y₁) = m(x − x₁) as

V_s − 10 = 10(I − 0.1) ⇒ V_s = 10 I − 1.0 + 10.

Therefore, the voltage-current relationship is given by

Comparing equations (1.4) and (1.5), the values of R and V are given by

R = 10 Ω, V = 9 V.

Figure 1.4 shows the graph of the source voltage V_s versus the current I. Note that the slope of the line m = 10 is the resistance R in Ω and the y-intercept b = 9 is the voltage V in volts.

The values of R and V can also be determined by switching the interpretation of x and y (the independent and dependent variables). From the voltage-current relationship V_s = I R + V, the current I can be written as a function of V_s as

This is an equation of a straight line y = m x + b, where x is the applied voltage V_s, y is the current I, is the slope, and is the y-intercept. The slope and y-intercept can be found from the data given in Fig. 1.3 using the slope-intercept method as

Figure 1.4 Voltage-current relationship for the data given in Fig. 1.3.

Using the data in Fig. 1.3, the slope m can be found as

Therefore, the current I can be written in slope-intercept form as

I = 0.1 V_s + b.

The y-intercept b can be determined using either one of the data points. Using the data point (V_s, I) = (10, 0.1) gives

0.1 = 0.1 (10) + b.

Solving for b gives

b = −0.9.

Therefore, the equation of the straight line can be written in the slope-intercept form as

Comparing equations (1.6) and (1.7) gives

and

Figure 1.5 is the graph of the straight line I = 0.1V_s − 0.9. Note that the y-intercept is A and the slope is .

Figure 1.5 Straight line with I as independent variable for the data given in Fig. 1.3.

1.3 FORCE-DISPLACEMENT IN A PRELOADED TENSION SPRING

The force-displacement relationship for a spring with a preload f_o is given by

where f is the force in Newtons (N), y is the displacement in meters (m), and k is the spring constant in N/m.

Figure 1.6 Force-displacement in a preloaded spring.

The objective is to find the spring constant k and the preload f_o, if the values of the force and displacement are as given in Fig. 1.6.

Method 1 Treating the displacement y as an independent variable, the force-displacement relationship f = ky + f_o is the equation of a straight line y = mx + b, where the independent variable x is the displacement y, the dependent variable y is the force f, the slope m is the spring constant k, and the y-intercept is the preload f_o. The slope m can be calculated using the data given in Fig. 1.6 as

The equation of the force-displacement equation in the slope-intercept form can therefore be written as

f = 5y + b.

The y-intercept b can be found using one of the data points. Using the data point (f, y) = (5, 0.9) gives

5 = 5 (0.9) + b.

Solving for b gives

b = 0.5 N.

Therefore, the equation of the straight line can be written in slope-intercept form as

Comparing equations (1.8) and (1.9) gives

k = 5N/m, f_o = 0.5N.

Method 2 Now treating the force f as an independent variable, the force-displacement relationship f = ky + f_o can be written as . This relationship is the equation of a straight line y = mx + b, where the independent variable x is the force f, the dependent variable y is the displacement y, the slope m is the reciprocal of the spring constant , and the y-intercept is the negated preload divided by the spring constant . The slope m can be calculated using the data given in Fig. 1.6 as

The equation of the displacement y as a function of force f can therefore be written in slope-intercept form as

y = 0.2f + b.

The y-intercept b can be found using one of the data points. Using the data point (y, f) = (0.9, 5) gives

0.9 = 0.2 (5) + b.

Solving for b gives

b = −0.1.

Therefore, the equation of the straight line can be written in the slope-intercept form as

Comparing equation (1.10) with the expression gives

and

Therefore, the force-displacement relationship for a preloaded spring given in Fig. 1.6 is given by

f = 5y + 0.5.

1.4 FURTHER EXAMPLES OF LINES IN ENGINEERING

Solution

The velocity profile of the vehicle shown in Fig. 1.7 is a piecewise linear function with three different equations. The first linear function is a straight line passing through the origin starting at time 0 sec and ending at time equal to 2 s. The second linear function is a straight line with zero slope (cruise velocity of 10 m/s) starting at 2 s and ending at 4 s. Finally, the third piece of the trajectory is a straight line starting at 4 s and ending at 6 s. The equation of the piecewise linear function can be written as

(a) 0 ≤ t ≤ 2:

v(t) = mt + b

where b = 0 and . Therefore,

v(t) = 5t m/s.

(b) 2 ≤ t ≤ 4:

v = 10 m/s.

(c) 4 ≤ t ≤ 6:

v(t) = mt + b,

where and the value of b can be calculated using the data point (t, v(t)) = (6, 0) as

0 = −5 (6) + b   ⇒   b = 0 + 30 = 30.

The value of b can also be calculated using the point-slope formula for the straight line

v − v₁ = m(t − t₁),

where v₁ = 0 and t₁ = 6. Thus,

v − 0 = −5(t − 6).

Therefore,

v(t) = −5(t − 6).

or

v(t) = −5t + 30 m/s.

(d) t > 6:

v(t) = 0 m/s.

Solution

(a) The velocity of the vehicle for different intervals can be calculated as

(i) 0 ≤ t ≤ 3 s:

v(t) = mt + b,

where and b = 24 m/s. Therefore,

v(t) = −4t + 24 m/s.

(ii) 3 ≤ t ≤ 6 s:

v(t) = 12 m/s.

(iii) 6 ≤ t ≤ 9 s:

v(t) = mt + b,

where and b can be calculated in slope-intercept form using point (t, v(t)) = (9, 0) as

0 = −4(9) + b.

Therefore, b = 36 m/s and

v(t) = −4t + 36 m/s.

(iv) t > 9 s:

v(t) = 0 m/s.

(b) Since the acceleration of the vehicle is the slope of the velocity in each interval, the acceleration a in m/s² is given by

The plot of the acceleration is shown in Fig. 1.9.

Figure 1.9 Acceleration profile of the vehicle in Fig. 1.8.

Solution

(a) The force-load relationship F_b = CP + F_i is the equation of a straight line, y = mx + b. The slope m is the joint constant C, which can be calculated as

Therefore,

Now, the y-intercept F_i can be calculated by substituting one of the data points into equation (1.11). Substituting the second data point (F_b, P) = (600, 800) gives

600 = 0.25 × 800 + F_i.

Solving for F_i yields

F_i = 600 − 200 = 400 lb.

Therefore, F_b = 0.25 P + 400 is the equation of the straight line, where C = 0.25 and F_i = 400 lb. Note that the joint constant C is dimensionless!

(b) The plot of the force F_b in the bolt as a function of the external load P is shown in Fig. 1.11.

Figure 1.11 Plot of the bolt force F_b as a function of the external load P.

Solution

The voltage-current relationship V = R I + R I_o is the equation of a straight line y = mx + b, where the slope m = R can be found from the data given in Fig. 1.12 as

Therefore,

The y-intercept b = 10 I₀ can be found by substituting the second data point (2.2, 0.2) in equation (1.12) as

2.2 = 100 × 0.2 + 10 I₀.

Solving for I₀ gives

10 I₀ = 2.2 − 2 = 0.2,

which gives

I₀ = 0.02 A.

Therefore, V = 10 I + 0.2; and R = 10 Ω and I₀ = 0.02 A.

Solution

(a) The input-output relationship is the equation of a straight line, y = mx + b, where the slope can be found from the data given in Fig. 1.13 as

Solving for R gives R = 50 Ω. Therefore,

The y-intercept b = 3 v_b can be found by substituting the first data point (v₀, v_in) = (5, 5) in equation (1.13) as

5 = −2 × 5 + 3 v_b.

Solving for v_b yields

3 v_b = 5 + 10 = 15,

which gives v_b = 5 V. Therefore, v_o = −2 v_in + 15, R = 50 Ω, and v_b = 5 V. The x-intercept can be found by substituting v_o = 0 in the equation v_o = −2 v_in + 15 and finding the value of v_in as

0 = −2 v_in + 15,

which gives v_in = 7.5 V. Therefore, the x-intercept occurs at V_in = 7.5 V.

(b) The plot of the output voltage of the Op–Amp as a function of the input voltage if v_b = 5 V is shown in Fig. 1.14.

Figure 1.14 An Op–Amp circuit as a summing amplifier.

Solution

(a) The input-output relationship F = kV is the equation of a straight line y = mx, where the slope m = k can be found from the given data as

Therefore, the equation of the straight line representing the actuator force F as a function of applied voltage V is given by

Thus, the force produced by the actuator when supplied with 7.3 volts is found by substituting V = 7.3 in equation (1.14) as

(b) The voltage needed to achieve a force of 6.0 N can be found by substituting F = 6.0 N in equation (1.14) as

(c) The plot of force F as a function of voltage V can now be drawn as shown in Fig. 1.16.

Figure 1.16 Plot of the actuator force verses the applied voltage.

Solution

(a) The input-output relationship A = mF + b is the equation of a straight line y = mx + b, where the slope m can be found from the EMG data given in the table (Fig. 1.17) as

The y-intercept b can be found by substituting the first data point (A, F) = (0.0005, 110) in equation (1.16) as

0.0005 = 4.55 × 10⁻⁶(110) + b.

Solving for b yields

b = 5 × 10⁻⁷ ≈ 0.

Therefore, the equation of the straight line representing the RMS amplitude as a function of applied force is given by

(b) The plot of the RMS amplitude as a function of the applied muscle force is shown in Fig. 1.18.

Figure 1.18 Plot of the RMS amplitude verses the applied muscle force.

(c) The RMS value of the amplitude for a muscle force of 200 N can be found by substituting F = 200 N in equation (1.17) as

A = 4.55 × 10⁻⁶ × (200) = 0.91 × 10⁻³V.

Solution

(a) The equation of elevation given by equation (1.18) is a straight line in the slope-intercept form y = mx + b, where the slope m can be found from the elevation data given in Fig. 1.19 as

The y-intercept E₁ can be found by substituting the first data point (E, l) = (428.4, 0) in equation (1.18) as

428.4 = 0.0483 × (0) + E₁

Solving for E₁ yields

E₁ = 428.4 m.

Therefore, the equation of the straight line representing the elevation as a function of distance l is given by

(b) The elevation E* of the cornerstone can be found by substituting l = 565 m in equation (1.19) as

E* = 0.0483 × (565) + 428.4 = 455.7 m.

(c) The plot of the elevation as a function of the length is shown in Fig. 1.20.

Figure 1.20 Elevation along a uniform grade.

PROBLEMS

1. 1-1. A constant force F = 2 N is applied to a spring and the displacement x is measured as 0.2 m. If the spring force and displacement satisfy the linear relation F = kx, find the stiffness k of the spring.

   

   Figure P1.1 Displacement of a spring in problem P1-1.

2. 1-2. The spring force F and displacement x for a close-wound tension spring are measured as shown in Fig. P1.2. The spring force F and displacement x satisfy the linear equation F = kx + F_i, where k is the spring constant and F_i is the preload induced during manufacturing of the spring.

   (a) Using the given data in Fig. P1.2, find the equation of the line for the spring force F as a function of the displacement x, and determine the values of the spring constant k and preload F_i.

   (b) Sketch the graph of F as a function of x. Use appropriate axis scales and clearly label the preload F_i, the spring constant k, and both given data points on your graph.

   

   Figure P1.2 Close-wound tension spring for problem P1-2.

3. 1-3. The spring force F and displacement x for a close-wound tension spring are measured as shown in Fig. P1.3. The spring force F and displacement x satisfy the linear equation F = kx + F_i, where k is the spring constant and F_i is the preload induced during manufacturing of the spring.

   (a) Using the given data, find the equation of the line for the spring force F as a function of the displacement x, and determine the values of the spring constant k and preload F_i.

   (b) Sketch the graph of F as a function of x and clearly indicate both the spring constant k and preload F_i.

   

   Figure P1.3 Close-wound tension spring for problem P1-3.

4. 1-4. In a bolted connection shown in Fig. P1.4, the force in the bolt F_b is given in terms of the external load P as F_b = CP + F_i.

   (a) Given the data in Fig. P1.4, determine the joint constant C and the preload F_i.

   

   Figure P1.4 Bolted connection for problem P1-4.

   (b) Plot the bolt force F_b as a function of the load P and label C and F_i on the graph.

5. 1-5. Repeat problem P1-4 for the data given in Fig. P1.5.

   

   Figure P1.5 Bolted connection for problem P1-5.

6. 1-6. The velocity v(t) of a ball thrown upward satisfies the equation v(t) = v_o + at, where v_o is the initial velocity of the ball in ft/s and a is the acceleration in ft/s².

   (a) Given the data in Fig. P1.6, find the equation of the line representing the velocity v(t) of the ball, and determine both the initial velocity v_o and the acceleration a.

   (b) Sketch the graph of the line v(t), and clearly indicate both the initial velocity and the acceleration on your graph. Also determine the time at which the velocity is zero.

   

   Figure P1.6 A ball thrown upward with an initial velocity v_o(t) in problem P1-6.

7. 1-7. The velocity v(t) of a ball thrown upward satisfies the equation v(t) = v_o + at, where v_o is the initial velocity of the ball in m/s and a is the acceleration in m/s².

   (a) Given the data in Fig. P1.7, find the equation of the line representing the velocity v(t) of the ball, and determine both the initial velocity v_o and the acceleration a.

   (b) Sketch the graph of the line v(t), and clearly indicate both the initial velocity and the acceleration on your graph. Also determine the time at which the velocity is zero.

   

   Figure P1.7 A ball thrown upward with an initial velocity v_o(t) in problem P1-7.

8. 1-8. A model rocket is fired in the vertical plane. The velocity v(t) is measured as shown in Fig. P1.8. The velocity satisfies the equation v(t) = v_o + at, where v_o is the initial velocity of the rocket in m/s and a is the acceleration in m/s².

   (a) Given the data in Fig. P1.8, find the equation of the line representing the velocity v(t) of the rocket, and determine both the initial velocity v_o and the acceleration a.

   

   Figure P1.8 A model rocket fired in the vertical plane in problem P1-8.

   (b) Sketch the graph of the line v(t) for 0 ≤ t ≤ 8 seconds, and clearly indicate both the initial velocity and the acceleration on your graph. Also determine the time at which the velocity is zero (i.e., the time required to reach the maximum height).

9. 1-9. A model rocket is fired in the vertical plane. The velocity v(t) is measured as shown in Fig. P1.9. The velocity satisfies the equation v(t) = v_o + at, where v_o is the initial velocity of the rocket in ft/s and a is the acceleration in ft/s².

   (a) Given the data in Fig. P1.9, find the equation of the line representing the velocity v(t) of the rocket, and determine both the initial velocity v_o and the acceleration a.

   (b) Sketch the graph of the line v(t) for 0 ≤ t ≤ 10 seconds, and clearly indicate both the initial velocity and the acceleration on your graph. Also determine the time at which the velocity is zero (i.e., the time required to reach the maximum height).

   

   Figure P1.9 A model rocket fired in the vertical plane in problem P1-9.

10. 1-10. The velocity of a vehicle is measured at two distinct points in time as shown in Fig. P1.10. The velocity satisfies the relationship v(t) = v_o + at, where v_o is the initial velocity in m/s and a is the acceleration in m/s².

    (a) Find the equation of the line v(t), and determine both the initial velocity v_o and the acceleration a.

    (b) Sketch the graph of the line v(t), and clearly label the initial velocity, the acceleration, and the total stopping time on the graph.

    

    Figure P1.10 Velocity of a vehicle during braking in problem P1-10.

11. 1-11. The velocity of a vehicle is measured at two distinct points in time as shown in Fig. P1.11. The velocity satisfies the relationship v(t) = v_o + at, where v_o is the initial velocity in ft/s and a is the acceleration in ft/s².

    (a) Find the equation of the line v(t), and determine both the initial velocity v_o and the acceleration a.

    (b) Sketch the graph of the line v(t), and clearly label the initial velocity, the acceleration, and the total stopping time on the graph.

    

    Figure P1.11 Velocity of a vehicle during braking in problem P1-11.

12. 1-12. The velocity v(t) of a vehicle during braking is given in Fig. P1.12. Determine the equation for v(t) for

    (a) 0 ≤ t ≤ 2 s

    (b) 2 ≤ t ≤ 4 s

    (c) 4 ≤ t ≤ 6 s

    

    Figure P1.12 Velocity of a vehicle during braking in problem P1-12.

13. 1-13. A linear trajectory is planned for a robot to pick up a part in a manufacturing process. The velocity of the trajectory of one of the joints is shown in Fig. P1.13. Determine the equation of v(t) for

    (a) 0 ≤ t ≤ 1 s

    (b) 1 ≤ t ≤ 3 s

    (c) 3 ≤ t ≤ 4 s

    

    Figure P1.13 Velocity of a robot trajectory.

14. 1-14. The acceleration of the linear trajectory of problem P1-13 is shown in Fig. P1.14. Determine the equation of a(t) for

    (a) 0 ≤ t ≤ 1 s

    (b) 1 ≤ t ≤ 3 s

    (c) 3 ≤ t ≤ 4 s

    

    Figure P1.14 Acceleration of the robot trajectory.

15. 1-15. The temperature distribution in a well-insulated axial rod varies linearly with respect to distance when the temperature at both ends is held constant as shown in Fig. P1.15. The temperature satisfies the equation of a line T(x) = C₁ x + C₂, where C₁ and C₂ are constants of integration with units of °F/ft and °F, respectively

    (a) Find the equation of the line T(x), and determine both constants C₁ and C₂.

    (b) Sketch the graph of the line T(x) for 0 ≤ x ≤ 1.5 ft, and clearly label C₁ and C₂ on your graph. Also, clearly indicate the temperature at the center of the rod (x = 0.75 ft).

    

    Figure P1.15 Temperature distribution in a well-insulated axial rod in problem P1-15.

16. 1-16. The temperature distribution in a well-insulated axial rod varies linearly with respect to distance when the temperature at both ends is held constant as shown in Fig. P1.16. The temperature satisfies the equation of a line T(x) = C₁ x + C₂, where C₁ and C₂ are constants of integration with units of °C/m and °C, respectively.

    (a) Find the equation of the line T(x), and determine both constants C₁ and C₂.

    (b) Sketch the graph of the line T(x) for 0 ≤ x ≤ 0.5 m, and clearly label C₁ and C₂ on your graph. Also, clearly indicate the temperature at the center of the rod (x = 0.25 m).

    

    Figure P1.16 Temperature distribution in a well-insulated axial rod in problem P1-16.

17. 1-17. The voltage-current relationship for the circuit shown in Fig. P1.17 is given by Ohm's law as V = I R, where V is the applied voltage in volts, I is the current in amps, and R is the resistance of the resistor in ohms.

    (a) Sketch the graph of I as a function of V if the resistance is 5 Ω.

    

    Figure P1.17 Resistive circuit for problem P1-17.

    (b) Find the current I if the applied voltage is 10 V.

18. 1-18. A voltage source V_s is used to apply two different voltages (12V and 18V) to the single-loop circuit shown in Fig. P1.18. The values of the measured current are shown in Fig. P1.18. The voltage and current satisfy the linear relation V_s = IR + V, where R is the resistance in ohms, I is the current in amps, and V_s is the voltage in volts.

    (a) Using the data given in Fig. P1.18, find the equation of the line for V_s as a function of I, and determine the values of R and V.

    (b) Sketch the graph of V_s as a function of I and clearly indicate the resistance R and voltage V on the graph.

    

    Figure P1.18 Single-loop circuit for problem P1-18.

19. 1-19. Repeat problem P1-18 for the data shown in Fig. P1.19.

    

    Figure P1.19 Single-loop circuit for problem P1-19.

20. 1-20. Repeat problem P1-18 for the data shown in Fig. P1.20.

    

    Figure P1.20 Single-loop circuit for problem P1-20.

21. 1-21. A linear model of a diode is shown in Fig. P1.21, where R_d is the forward resistance of the diode and V_ON is the voltage that turns the diode ON. To determine the resistance R_d and voltage V_ON, two voltage values are applied to the diode and the corresponding currents are measured. The applied voltage V_S and the measured current I are given in Fig. P1.21. The applied voltage and the measured current satisfy the linear equation V_s = I R_d + V_ON.

    (a) Find the equation of the line for V_s as a function of I and determine the resistance R_d and the voltage V_ON.

    

    Figure P1.21 Linear model of a diode for problem P1-21.

    (b) Sketch the graph of V_S as a function of I, and clearly indicate the resistance R_d and the voltage V_ON on the graph.

22. 1-22. Repeat problem P1-21 for the data given in Fig. P1.22.

    

    Figure P1.22 Linear model of a diode for problem P1-22.

23. 1-23. The output voltage, v_o, of the Op–Amp circuit shown in Fig. P1.23 satisfies the relationship , where R is the unknown resistance in kΩ and v_b is the unknown voltage in volts. Fig. P1.23 gives the values of the output voltage for two different values of the input voltage.

    

    Figure P1.23 An Op–Amp circuit as a summing amplifier for problem P1.23.

    (a) Determine the equation of the line for v_o as a function of v_in and find the values of R and v_b.

    (b) Plot the output voltage v_o as a function of the input voltage v_in. On the plot, clearly indicate the value of the output voltage when the input voltage is zero (y-intercept) and the value of the input voltage when the output voltage is zero (x-intercept).

24. 1-24. The output voltage, v_o, of the Op–Amp circuit shown in Fig. P1.24 satisfies the relationship , where R is the unknown resistance in kΩ, v_in is the input voltage, and v₂ is the unknown voltage. Fig. P1.24 gives the values of the output voltage for two different values of the input voltage v_in.

    (a) Find the equation of the line for v_o as a function of v_in and determine the values of R and v₂.

    (b) Plot the output voltage v_o as a function of the input voltage v_in. Clearly indicate the value of the output voltage when the input voltage is zero (y-intercept) and the value of the input voltage when the output voltage is zero (x-intercept).

    

    Figure P1.24 An Op–Amp circuit for problem P1-24.

25. 1-25. A DC motor is driving an inertial load J_L shown in Fig. P1.25. To maintain a constant speed, two different values of the voltage e_a are applied to the motor. The voltage e_a and the current i_a flowing through the armature winding of the motor satisfy the relationship e_a = i_a R_a + e_b, where R_a is the resistance of the armature winding in ohms and e_b is the back-emf in volts. Figure P1.25 gives the values of the current for two different values of the input voltage applied to the armature of the DC motor.

    (a) Find the equation of the line for e_a as a function of i_a and determine the values of R_a and e_b.

    (b) Plot the applied voltage e_a as a function of the current i_a. Clearly indicate the value of the back-emf e_b and the winding resistance R_a.

    

    Figure P1.25 Voltage-current data of a DC motor for problem P1-25.

26. 1-26. Repeat problem P1-25 for the data shown in Fig. P1.26.

    

    Figure P1.26 Voltage-current data of a DC motor in problem P1-26.

27. 1-27. In the active region, the output voltage v_o of the n-channel enhancement-type MOSFET (NMOS) circuit shown in Fig. P1.27 satisfies the relationship v_o = V_DD − R_D i_D, where R_D is the unknown drain resistance and V_D is the unknown drain voltage. Fig. P1.27 gives the values of the output voltage for two different values of the drain current. Plot the output voltage v_o as a function of the input drain current i_D. On the plot, clearly indicate the values of R_D and V_DD.

    

    Figure P1.27 n-channel enhancement-type MOSFET.

28. 1-28. Repeat problem P1-27 for the data given in Fig. P1.28.

    

    Figure P1.28 NMOS for P1-28.

29. 1-29. An actuator used in a prosthetic arm can produce different amounts of force by changing the voltage of the power supply. The force and voltage satisfy the linear relation F = kV, where V is the voltage applied and F is the force produced by the prosthetic arm. The maximum force the arm can produce is 30.0 N when supplied with 10 V.

    (a) Find the force produced by the actuator when supplied with 6.0 V.

    (b) What voltage is needed to achieve a force of 5.0 N?

    (c) Using the results of parts (a) and (b), sketch the graph of F as a function of voltage V. Use the appropriate scales and clearly label the slope and the results of parts (a) and (b).

30. 1-30. The following two measurements of maximum heart rate R (in beats per minute, bpm) were recorded in an exercise physiology laboratory.

    

    The maximum heart rate R and age A satisfy the linear equation

    R = mA + B

    where R is the heart rate in beats per minute and A is the age in years.

    (a) Using the data provided, find the equation of the line for R.

    (b) Sketch R as a function of A.

    (c) Using the relationship developed in part (a), find the maximum heart rate of a 60-year-old person.

31. 1-31. The electrical activity of muscles can be monitored with an electromyogram (EMG). The RMS amplitude measurements of the EMG signal when a person is using the hand grip muscle to tighten the lid on a jar is given in the table below:

    

    The RMS amplitude of the EMG signal satisfies the linear equation R = mA + B, where A is the RMS amplitude in volts, F is the applied muscle force in N, and m is the slope of the line.

    (a) Using the data provided in the table, find the equation of the line for A(F).

    (b) Sketch A as a function of F.

    (c) Using the relationship developed in part (a), find the RMS amplitude for a muscle force of 200 N.

32. 1-32. A civil engineer needs to establish the elevation of the cornerstone for a building located between two benchmarks, B1 and B2, of known elevations, as shown in Fig. P1.32.

    

    Figure P1.32 Elevations along a uniform grade for exercise P1-32.

    The elevation E along the grade satisfies the linear relationship

    

    where E₁ is the elevation of B1, l is the taped distance from B1 along the grade, and m is the rate of change of E with respect to l.

    (a) Find the equation of the line E and determine the slope m of the linear relationship.

    (b) Using the equation of the line from part (a), find the elevation of the cornerstone E* if it is located at a distance l = 300 m from B1.

    (c) Sketch the graph of E as a function of l and clearly indicate both the slope m and elevation E₁ of B1.

33. 1-33. A thermocouple is a temperature measurement device, which produces a voltage V proportional to the temperature at the junction of two dissimilar metals. The voltage across a thermocouple is calibrated using the boiling point of water (100°C) and the freezing point of Zinc (420°C), as shown in Fig. P1.33.

    

    Figure P1.33 Thermocouple to measure temperature in Celsius.

    The junction temperature T and the voltage across the thermocouple V satisfy the linear equation , where α is the thermocouple sensitivity in mV/°C and T_R is the reference temperature in °C.

    (a) Using the calibration data given in Fig. P1.33, find the equation of the line for the measured temperature T as a function of the voltage V and determine the value of the sensitivity α and the reference temperature T_R.

    (b) Sketch the graph of T as a function of V and clearly indicate both the reference temperature T_R and the sensitivity α on the graph.

34. 1-34. The voltage across a thermocouple is calibrated using the boiling point of water (373°K) and the freezing point of Silver (1235°K), as shown in Fig. P1.34.

    

    Figure P1.34 Thermocouple to measure temperature in Kelvin.

    The junction temperature T and the voltage across the thermocouple V satisfy the linear equation , where α is the thermocouple sensitivity in mV/°K and T_R is the reference temperature in °K.

    (a) Using the calibration data given in Fig. P1.34, find the equation of the line for the measured temperature T as a function of the voltage V and determine the value of the sensitivity α and the reference temperature T_R.

    (b) Sketch the graph of T as a function of V and clearly indicate both the reference temperature T_R and the sensitivity α on the graph.

35. 1-35. An iron-costantan thermocouple is calibrated by inserting its junction in boiling water (100°C) and measuring a voltage V = 5.27 mV, and then inserting the juction in silver chloride at its melting point (455°C) and measuring V = 24.88 mV.

    (a) Find the equation of the line for the measured temperature T as a function of the voltage V and determine the value of the sensitivity α and the reference temperature T_R.

    (b) Sketch the graph of T as a function of V and clearly indicate both the reference temperature T_R and the sensitivity α on the graph.

    (c) If the thermocouple is mounted in a chemical reactor and the voltage is observed to go from 10.0 mV to 13.6 mV, what is the change in temperature of the reactor?

36. 1-36. Strain is a measure of the deformation of an object. It can be measured using a foil strain gauge shown in Fig. P1.36.

    The strain being measured () and resistance of the sensor satisfy the linear equation , where R_o is the initial resistance (measured in ohms, Ω) of the sensor with no strain, and is the gauge factor (a multiplier with NO units).

    (a) Using the given data, find the equation of the line for the sensor's resistance R as a function of the strain , and determine the values of the gauge factor and initial resistance R_o.

    (b) Sketch the graph of R as a function of , and clearly indicate R_o on your graph.

    

    Figure P1.36 Foil strain gauge to measure strain in problem P1-36.

37. 1-37. Repeat problem P1-36 for the data shown in Fig. P1.37.

38. 1-38. To determine the concentration of a purified protein sample, a graduate student used spectrophotometry to measure the absorbance given in the table below:

    

    The concentration-absorbance relationship for this protein satisfies a linear equation a = mc + a_i, where c is the concentration of a purified protein, a is the absorbance of the sample, m is the the rate of change of absorbance a with respect to concentration c, and a_i is the y-intercept.

    

    Figure P1.38 Foil strain gauge to measure strain in problem P1-38.

    (a) Find the equation of the line that describes the concentration-absorbance relationship for this protein and determine the slope m of the linear relationship.

    (b) Using the equation of the line from part (a), find the concentration of the sample if this sample had an absorption of 0.486.

    (c) If the sample is diluted to a concentration of 0.00419 μg/ml, what would you expect the absorption to be? Would this value be accurate?

    (d) Sketch the graph of absorbance a as a function of concentration c and clearly indicate both the slope m and the y-intercept.

39. 1-39. Repeat problem P1-38 if the absorption-concentration data for another sample is obtained as

    

40. 1-40. A chemistry student is performing an experiment to determine the temperature-volume behavior of a gas mixture at constant pressure and quantity. Due to technical difficulties, he could only obtain values at two temperatures as shown in the table below:

    

    The student knows that the gas volume directly depends on temperature, that is, V(T) = mT + K, where V is the volume in L, T is the temperature in °C, K is the y-intercept in L, and m is the slope of the line in L/°C.

    (a) Find the equation of the line that describes the temperature-volume relationship of the gas mixture and determine the slope m of the linear relationship.

    (b) Using the equation of the line from part (a), find the temperature of the gas mixture if the volume is 1.15 L.

    (c) Using the equation of the line from part (a), find the volume of the gas if the temperature is 70°C.

    (d) Sketch the graph of the volume-temperature relationship for the gas mixture from −300°C to 100°C and clearly indicate both the slope m and the y-intercept. What is the significance of the temperature when V = 0 L?

41. 1-41. To obtain the linear relationship between the Fahrenheit and Celsius temperature scales, the freezing and boiling point of water is used as given in the table below:

    

    The relationship between the temperatures in Fahrenheit and Celsius scales satisfies the linear equation T (°F) = aT (°C) + b.

    (a) Using the given data, find the equation of the line relating the Fahrenheit and Celsius scales.

    (b) Sketch the graph of T (°F) as a function of T (°C), and clearly indicate a and b on your graph.

    (c) Using the graph obtained in part (b), find the temperature interval in °C if the temperature is between 20°F and 80°F.

42. 1-42. A thermostat control with dial marking from 0 to 100 is used to regulate the temperature of an oil bath. To calibrate the thermostat, the data for the temperature T (°F) versus the dial setting R was obtained as shown in the table below:

    

    The relationship between the temperature T in Fahrenheit and the dial setting R satisfies the linear equation T (°F) = a R + b.

    (a) Using the given data, find the equation of the line relating the temperature to the dial setting.

    (b) Sketch the graph of T (°F) as a function of R, and clearly indicate a and b on your graph.

    (c) Calculate the thermostat setting needed to obtain a temperature of 320°F.

43. 1-43. In a pressure-fed journal bearing, forced cooling is provided by a pressurized lubricant flowing along the axial direction of the shaft (the x-direction) as shown in Fig. P1.43. The lubricant pressure satisfies the linear equation

    ,

    where p_s is the supply pressure and l is the length of the bearing.

    

    Figure P1.43 Pressure-fed journal bearing.

    (a) Using the data given in the table (Fig. P1.43), find the equation of the line for the lubricant pressure p(x) and determine the values of the supply pressure p_s and the bearing length l.

    (b) Calculate the lubricant pressure p(x) if x is 1.0 in.

    (c) Sketch the graph of the lubricant pressure p(x), and clearly indicate both the supply pressure p_s and the bearing length l on the graph.