Higher Engineering Mathematics

222 GRAPHS Alternatively, average value = sum of mid-ordinates number of mid-ordinates Problem 8. Fig. 20.11 shows a sinusoidal ...

IRREGULAR AREAS, VOLUMES AND MEAN VALUES OF WAVEFORMS 223 C Figure 20.13 Find the average value of the periodic wave- forms sho ...

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Vector geometry D 21 Vectors, phasors and the combination of waveforms 21.1 Introduction Some physical quantities are entirely d ...

226 VECTOR GEOMETRY Thus,oarepresents a vector quantity, butoais the magnitude of the vectoroa. Also, positive angles are measur ...

VECTORS, PHASORS AND THE COMBINATION OF WAVEFORMS 227 D Often it is easier to use the‘nose-to-tail’ method when more than two ve ...

228 VECTOR GEOMETRY Figure 21.9 H=7 cos 0◦+4 cos 45◦= 7 + 2. 828 = 9 .828 N Vertical component of force, V=7 sin 0◦+4 sin 45◦= 0 ...

VECTORS, PHASORS AND THE COMBINATION OF WAVEFORMS 229 D A load of 5.89 N is lifted by two strings, mak- ing angles of 20◦and 35 ...

230 VECTOR GEOMETRY Ta n−^1 ( 2. 99 − 2. 13 ) =− 54. 53 ◦, and for this to be in the second quadrant, the true angle is 180◦ dis ...

VECTORS, PHASORS AND THE COMBINATION OF WAVEFORMS 231 D Ta n−^1 ( − 6. 99 − 27. 67 ) = 14. 18 ◦, hence the required angle is 180 ...

232 VECTOR GEOMETRY opposite in direction tope. From the geometry of this vector triangle: |qp|= √ (45^2 + 552 )= 71 .06 m/s and ...

VECTORS, PHASORS AND THE COMBINATION OF WAVEFORMS 233 D (c) by determining horizontal and vertical components of lengths oa and ...

234 VECTOR GEOMETRY Ordinates are added at 15◦intervals and the resultant is shown by the broken line. The amplitude of the resu ...

VECTORS, PHASORS AND THE COMBINATION OF WAVEFORMS 235 D Using the sine rule: 3 sinφ = 4. 64 sin 135◦ from which sinφ= 3 sin 135◦ ...

236 VECTOR GEOMETRY Now try the following exercise. Exercise 96 Further problems on the combination of periodic functions Plot ...

D Vector geometry 22 Scalar and vector products 22.1 The unit triad When a vectorxof magnitudexunits and directionθ◦ is divided ...

238 VECTOR GEOMETRY 22.2 The scalar product of two vectors When vectoroais multiplied by a scalar quantity, sayk, the magnitude ...

SCALAR AND VECTOR PRODUCTS 239 D then cosθ= a•b ab (1) Let a=a 1 i+a 2 j+a 3 k and b=b 1 i+b 2 j+b 3 k a•b=(a 1 i+a 2 j+a 3 k)•( ...

240 VECTOR GEOMETRY (i) From equation (2), if p=a 1 i+a 2 j+a 3 k and q=b 1 i+b 2 j+b 3 k then p•q=a 1 b 1 +a 2 b 2 +a 3 b 3 Whe ...

SCALAR AND VECTOR PRODUCTS 241 D Note that cos^2 α+cos^2 β+cos^2 γ= 0. 8022 + 5352 + 0. 2672 =1. Practical application of scal ...