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Suppose that a1, a2, a3, , an are nonzero real numbers, and b1, b2, b3, , bn are real stand-alone constants Let x be the variable in the following equation: (a1x + b1)(a2x + b2)(a3x + b3) (anx + bn) = 0 This is the binomial factor form for an nth-degree equation in the variable x

gear-driven devices, which wear out with time, is the optical encoder, also called the optical shaft encoder An optical encoder consists of two LEDs, two photodetectors, and a device called a chopping wheel The LEDs shine on the photodetectors through the wheel The wheel has radial bands, alternately transparent and opaque (Fig 30-5) The wheel is attached to the tuning shaft, which is attached to a large knob As the tuning knob is rotated, the light beams are interrupted Each interruption causes the frequency to change by a specified increment The difference between frequency up and frequency down (clockwise and counterclockwise shaft rotation, respectively) is determined according to which photodetector senses each sequential beam interruption first

Question 26-7

Table 30-2

How many real roots does the equation stated in Answer 25-5 have What is that root, or what are they What is the real solution set X

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A sensor employs one or more transducers to detect and/or measure phenomena such as temperature, humidity, barometric pressure, texture, proximity, and the presence of certain substances

Answer 26-7

The logic NOT, also called negation or inversion, involves only a single logic variable It changes the value of a bit from 1 to 0, or from 0 to 1, as shown in the truth table for logic operations (Table 30-3) It is represented by a minus sign in front of a capital letter, usually from the end of the alphabet (W, X, Y, or Z) It might also be represented by a squiggly dash before the letter, an apostrophe after the letter, or a line over the letter

There are n real roots They are the solutions to the equations we obtain when we set the binomials equal to 0: a1x + b1 = 0 a2x + b2 = 0 a3x + b3 = 0 anx + bn = 0 When we subtract the b-constant from both sides in each of these equations and then divide through by the a-coefficient (which is okay because we know that a1, a2, a3, , an are all nonzero), we get these roots: x = b1/a1 or x = b2/a2 or x = b3/a3 or x = bn/an

Capacitive Pressure Sensor A capacitive pressure sensor is shown in Fig 30-6 Two metal plates are separated by a layer of dielectric (electrically insulating) foam, forming a capacitor This component is connected in parallel with an inductor The resulting inductance/capacitance (LC ) circuit determines the frequency of an oscillator If an object strikes the sensor, the plate spacing momentarily decreases This increases the capacitance, causing a drop in the oscillator frequency When the object moves away from the transducer, the foam springs back, the plates return to their original spacing, and the oscillator frequency returns to normal The output of a capacitive pressure sensor can be converted to digital data using an analog-todigital converter (ADC) This signal can be sent to a microcomputer such as a robot controller Pressure sensors can be mounted in various places on a mobile robot, such as the front, back, and sides Then, for example, physical pressure on the sensor in the front of the robot can send a signal to the controller, which tells the machine to move backward A capacitive pressure sensor can be fooled by massive conducting or semiconducting objects in its vicinity If such a mass comes near the transducer, the capacitance may change even if direct contact is not made This phenomenon is known as body capacitance When the effect must be avoided, an elastomer device can be used for pressure sensing

The logic AND, or conjunction, involves two variables The resultant is high if, but only if, both variables are high If either variable is low, or if both are low, then the resultant is

The real solution set is therefore X = { b1/a1, b2/a2, b3/a3, bn/an}

Question 26-8

560 Basic digital principles low This operation gets its name from the fact that it corresponds to the function of the word and in propositional calculus It is usually represented as multiplication See Table 30-3

sensor When force is applied, the spacing between the plates decreases, causing the capacitance to increase and the oscillator frequency to go down

What are the real roots of the following equation What is the multiplicity of each root What is the real solution set X What is the degree of the equation (x + 4)(2x 8)2(3x)5 = 0

Answer 26-8

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