INDEX

Acceleration, 221, 226–240

Acceleration due to gravity, 221

Addition of sinusoids of same frequency, 166–173,

See also under Sinusoids

Addition of two complex numbers, 137

Angular frequency, sinusoids, 161

Angular motion of one-link planar robot, 159–162

frequency and period, relations between, 160–162

Antiderivative, 280

Armature current in DC motor, 140–141

Asphalt problem, 278–283

Average velocity, 219

Bending moment, 250

Capacitor, 314–321

current passing through, 247–250, 314–321

energy stored in, 314–321

impedance of, 135

voltage across, 247–250, 314–321

Center of Gravity (Centroid), 286–293

alternate definition of, 293–296

determination using vertical rectangles, 292

evaluation using horizontal elemental areas, 89

first moment of area, 294

of triangular section, 86

Chain rule of derivative, 241

Complementary solution, 347

Complex conjugate, 145–147

Complex numbers, 132–156

addition of two complex numbers, 137

armature current in a DC motor, 140–141

complex conjugate, 145–147

division of complex number in polar form, 139–140

examples, 141–145

exponential form, 136

impedance of a series RLC circuit, 136–137

impedance of R, L, and C as a, 134–135

multiplication of complex number in polar form, 139

polar form, 136

position of one-link robot as, 133–134

subtraction of two complex numbers, 137

Cos(θ), 61

Cramer's Rule, 188–189, 192

Current, 240

flowing in inductor, 322–326

passing through a capacitor, 314–321

Definite integral, 280

Deflection, 251

Derivatives, 218–277

applications in dynamics, 225–240

applications in electric circuits, 240–250

applications in strength of materials, 250–260,

See also Material strength

chain rule of, 241

definition, 218–221

examples of, 261–266

maxima and minima, 221–225

Differential equations (DEQ), 345–400

first-order differential equations, 348–374

leaking bucket, 345–346

linear DEQ, solution with constant coefficients, 347–348

second-order differential equations, 374–390,

See also separate entry

Direct kinematics of two-link robot, 73–74

Distributed loads, 251, 296–302

on beams, statically equivalent loading, 297–302

hydrostatic pressure on a retaining wall, 296–297

Division of complex number in polar form, 139–140

Dynamics

derivatives applications in, 225–240

integrals applications in, 302–314

Elbow-up solution for θ₁, 77–79

Electric circuits, 314–321

Electric circuits (continued)

current and voltage in a capacitor, 314–321

derivatives applications in, 240–250

integrals applications in, 314–321

Energy stored in a capacitor, 314–321

Equivalent resistance, 37–38

Euler's formula, 134

Exponential form, complex numbers, 136

Factoring, 33

First-order differential equations, 348–374

Force-displacement relationship, 6–7

Free-body diagram (FBD), 117, 119

Graphical method solution, for two-loop circuit, 185–186

Homogeneous solution, 347

Hydrostatic pressure on a retaining wall, 296–297

Impedance

of capacitor, 135

of inductor, 134–135, 137–140

of resistor, 134, 137–140

of a series RLC circuit, 136–137

Indefinite integrals, 281

Inductor

as a circuit element, 243

current and voltage in, 243–247, 322–326

impedance of, 134–135

Integrals, 278–344, See also Distributed loads

Asphalt problem, 278–283

concept of work, 283–286, See also Work

definite integral, 280

in dynamics, 302–314

in electric circuits, 314–321

examples, 327–333

indefinite integrals, 281

inductor, current and voltage in, 322–326

inverse operations, 281

in statics, 286–296, See also Statics

Inverse kinematics of one-link robot, 68–72

Inverse kinematics of two-link robot, 75–79

Inverse operations, 281

Kirchhoff's voltage law (KVL), 3, 184

Linear DEQ, solution with constant coefficients, 347–348

steady-state solution, 348

transient solution, 347

Linear frequency, sinusoids, 161

Low-pass filter, 373

Material strength, derivatives applications in, 250–260

bending moment, 250

deflection, 251

distributed load, 251

maximum stress under axial loading, 256–260

maximum value, 259

minimum value, 259

moment, 255

shear force, 251, 255

slope, 252

Matrix algebra method solution

tension in cables, 191–192

for two-loop circuit, 186–188

Maxima, 221–225

Maximum stress under axial loading, 256–260

inclined section of the rectangular bar, 256

normal and shear stresses acting on the inclined cross section, 258

rectangular bar under axial loading, 256

Method of undetermined coefficients, 348

Minima, 221–225

Moment, 255, 298

Multiplication of complex number in polar form, 139

Newton's first law, 117

Newton's second law, 375

Ohm's law, 3, 184

One-link planar robot

angular motion of, 159–162

as a sinusoid, 157–159

Particular solution, 348

Period, one-link planar robot, 161

Phase angle, 162–164

Phase shift, 162–164

Polar form

complex numbers, 136

position vector in, 107–110

Position, 226–240

Position of one-link robot as a complex number, 133–134

Position vector

in polar form, 107–110

in rectangular form, 107

Power, 240

Quadratic equations, 32–59

current in a lamp, 36–37

equivalent resistance, 37–38

examples, 38–50

pipeline through parabolic hill, 48–50

projectile in a vertical plane, 32–36

resistors in parallel, 37

Quadratic formula, 33

Rectangular form, position vector in, 107

Reference angle, 65

Relative velocity, 114

Repeated roots, 347

Resistive circuit, voltage-current relationship in, 3–5

Resistor, impedance of, 134

Resistors in parallel, 37

Resonance, 383

Second-order differential equations, 374–390

forced vibration of a spring-mass system, 379–386

free vibration of a spring-mass system, 374–379

second-order LC circuit, 386–390

Second-order LC circuit, 386–390

Shear force, 251, 255

Sin(θ), 61

Sinusoids, 157–183

addition of sinusoids of same frequency, 166–173

amplitude of, 160

general form of, 164–166

one-link planar robot as, 157–159

phase angle, 162–164

phase shift, 162–164

sine function, two cycles of, 159

time shift, 162–164

Slope, 252

Slope-intercept method, 5

Speed at impact, 219

Spring-mass system

forced vibration of, 379–386

free vibration of, 374–379

Static equilibrium, 117, 119

Statically equivalent loading, 297–302

Statics, integrals application in, 286–296

Center of Gravity (Centroid), 286–293, See also individual entry

Steady-state solution, 348

Straight lines, 1–31

examples, 8–18

force-displacement relationship, 6–7

slope-intercept method, 5

vehicle during braking, 1–3

voltage-current relationship in resistive circuit, 3–5

Substitution method solution

tension in cables, 190–191

for two-loop circuit, 185

Subtraction of two complex numbers, 137

Systems of equations, 184–217

examples, 193–205

solution of a two-loop circuit, 184–189, See also Tension in cables

Tension in cables, 190–192

Cramer's Rule, 192

matrix algebra method, 191–192

substitution method, 190–191

Time constant, 351, 357

Time shift, 162–164

Transient solution, 347

Triangular section, centroid of, 86

Trigonometry, 60–105

examples, 89–96

one-link planar robot, 60–72

reference angle, 65

sine and cosine functions, values of, 64

trigonometric functions in four quadrants, 64

two-link planar robot, 72–89

Two-dimensional vectors, 106–131

free-body diagram (FBD), 117, 119

position vector

relative velocity, 114

static equilibrium, 117, 119

vector addition, 110–123

Two-loop circuit, solution of, 184–189

Vector addition, 110–123

Velocity, 226–240

Voltage, 240

in capacitor, 314–321

in inductor, 322–326

Voltage-current relationship in resistive circuit, 3–5

Work, 283–286

as area under a constant force curve, 284

as area under a variable force curve, 284