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11.7. Determine the locus of the center of (a) a coin rolling around and touching a smaller coin; (b) a coin rolling around and touching a larger coin; (c) a wheel moving between two parallel bars and touching both of them; (d) a wheel moving along a straight metal bar and touching it. (11.2) 11.8. Find the locus of points that are in rectangle ABCD of Fig. 11-18 and the center of a circle (a) Tangent to AD and BC (b) Tangent to AB and CD (c) Tangent to AD and EF (d) Of radius 10, tangent to BC (e) Of radius 20, tangent to AD (f) Tangent to BC at G (11.2)

The instantaneous value of a quantity is the value of that quantity at a speci c time. Often we will be interested in the average value of a time-varying quantity. But obviously, the average value of a sinusoidal function over one period is zero. For sinusoids, then, another concept, that of the root-meansquare (or rms) value, is more useful: For any time-varying function f t with period T, the average value over one period is given by 1 t0 T F0 f t dt 1:20 T t0 and the corresponding rms value is de ned as s 1 t0 T 2 F f t dt T t0 where, of course, F0 and F are independent of t0 . gathered from Example 1.12.

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The product of two complex numbers when both are in exponential form follows directly from the laws of exponents. z1 z2 r1 e j 1 r2 ej 2 r1 r2 e j 1 2

Fig. 11-18

APP. A]

1:21

11.9. Locate each of the following:

The polar or Steinmetz product is evident from reference to the exponential form. z1 z2 r1 1 r2 2 r1 r2 1 2 The rectangular product can be found by treating the two complex numbers as binomials. z1 z2 x1 jy1 x2 jy2 x1 x2 jx1 y2 jy1 x2 j 2 y1 y2 x1 x2 y1 y2 j x1 y2 y1 x2

(11.4)

(a) By KVL, VCC ICQ RC IBQ RC RF VBEQ IEQ RE Substituting (5.5) and (5.6) into (1), rearranging, and then assuming ) 1, we obtain ICQ % VCC VBEQ ICBO RC RF RE VCC VBEQ ICBO RC RF RE % 1 R RF = RC RE RC F RE RC RF RE SV RF RC RE RF = RC RE RF VCC VBEQ ICBO RC RF RE RF RC RE 2 2 1

EXAMPLE A2 If z1 5e j =3 and z2 2e j =6 , then z1 z2 5e j =3 2e j =6 10e j =6 . EXAMPLE A3 If z1 2 308 and z2 5 458, then z1 z2 2 308 5 458 10 158. EXAMPLE A4 If z1 2 j3 and z2 1 j3, then z1 z2 2 j3 1 j3 7 j9.

(a) Treasure that is buried 5 ft from a straight fence and equidistant from two given points where the fence meets the ground (b) Points that are 3 ft from a circle whose radius is 2 ft and are equidistant from two lines which are parallel to each other and tangent to the circle (c) A point equidistant from the three vertices of a given triangle (d) A point equidistant from two given points and equidistant from two given parallels (e) Points equidistant from two given intersecting lines and 5 ft from their intersection (f) A point that is equidistant from the sides of an angle and 1 in from their intersection 2 11.10. Locate the point or points which satisfy the following conditions with respect to ^ABC in Fig. 11-19: (a) Equidistant from its sides (b) Equidistant from its vertices (c) Equidistant from A and B and from AB and BC (d) Equidistant from BC and AC and 5 units from C (e) 5 units from B and 10 units from A (11.4)

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