The rule for subtraction is evidently, change the sign of the vector to be subtracted and then proceed as in addition. = 2.11800, the magni 87. Example.-A, B, C, and D are four vectors specified as follows. A = 2·4600, B = 1·53300, C = 2·00, and D tudes being in inches. It is required to find the value of RA + B − C + D. a Scale ст 180° 2.4 90 茶葉 D A+B-C 2-1 D R=A+B-C+D FIG. 161. 60 B cd parallel and equal to OD, then od is equal to R = A + B− C + D=re. R will be found to be equal to 0.7901 88. Resolution of Vectors.-If R is the sum or resultant of any number of vectors A, B, C, D, etc., then A, B, C, D, etc. are called the components of the vector R. The operation of finding R when A, B, C, D, etc., are given is called the summation of vectors or composition of vectors. The converse operation of breaking up a vector into a number of components is called the resolution of a vector. It is evident that a given vector may be resolved into any number of components by constructing a polygon on the given vector. a Generally when a vector has to be resolved into components only two components are required the directions of which are given. Let R or ab (Fig. 165) be a given vector and let OX and OY be two given directions; it is required to resolve R into two components P and Q whose directions shall be parallel to OX and OY respectively. Through a draw ac parallel to OX and through b draw be parallel to OY to meet ac at c, then ac and cb will be the required components P and Q. A common case in practice is that in which the angle XOY is a right angle. 89. Example.-The thick curved line shown in Fig. 166 represents the section of a vane of a water wheel or turbine which revolves about an axis at O. A jet of water impinges on the vane at A with a velocity v1. The jet is then deflected and flows over the vane leaving it at B. The wheel revolves in the direction of the arrow R. The radius of the wheel at A is r1, and the linear velocity of the vane at A is c1. The radius of the wheel at B is r, and the linear velocity of the vane at B is c Obviously 1 ་ 、 Consider the water in the jet at A where it comes in contact with the vane. This water is beginning to slide along the vane with a velocity s in a direction tangential to the vane at A. This water is also carried round with the wheel with a velocity e in the direction of the tangent to the wheel circle at A, and in order that there shall be no shock the resultant of these two velocities s and c, should be v1. Of the three velocities v1, C1, and 8, if one be known completely and the directions of the other two are given, their magnitudes are K R FIG. 166. readily found. Or if two of the velocities be known completely, the magnitude and direction of the other is readily found. Assume that the water slides along the vane with a constant velocity & relatively to the vane and consider the water in the jet at B. This water has a velocity s in the direction of the tangent to the vane at B and also a velocity c2 in the direction of the tangent to the wheel circle at B. Hence BD or v2 the resultant or absolute velocity of the water at B is the resultant of the velocities s and c. Draw BC parallel and equal to v, and join CD; then CD is the vector change v2- v, in the velocity of the water in passing over the vane. The actual path of a particle of water in passing through the wheel may be found as follows. Make the arc AE equal to a definite fraction of 8. With centre O and radius OE describe the arc EF. Make the arc EF equal to the same fraction of c, the velocity of the wheel at E, that the arc AE is of 8. Then F is a point in the actual path AFK of a particle of water which enters the wheel at A. Exercises VII 1. Three vectors, A, B, and C, acting in a horizontal plane are defined in the following table :- See also Fig. 167. The angles of 33-2° and 112° are to be set off with the protractor, and not by copying the diagram. Determine the resultant or vector sum A+ B+ C, using a scale of 1 inch to 1 unit. Measure and tabulate the results (thus completing the above table). Theorem:-A vector sum is the same in whatever sequence the vectors are added. Verify this principle by actual drawing in the following case:-Show that A+B+C=A+C+ B. [B.E.] 2. Three coplanar vectors A, B, C are given as follows: In defining direction, the vectors are supposed to act outwards from a point O (Fig. 168), and the angles are measured anti-clockwise from a fixed line OX. These angles must be set off with a protractor, and not copied from the diagram. Find A+B+C and A- B+C. Measure and record the results (in the form required to complete the foregoing table). Verify by drawing that A (B − C) = A- B+ C. 3. Find the vector sum A+B+C+D+E, having given A B = 1-600° being 1 inch. C = 2.25, D = 2.800, and E 1800, = 5 351800, the unit of magnitude 165°, 4. Four coplanar forces P, Q, R, and S, acting at a point, are specified as follows:-P = 1560, Q = 9213°, R = 10'5315°, and S 2116 the magnitudes of the forces being in pounds. Find the resultant of these forces, and specify it in the form T = t. Use a scale of 1 inch to 10 pounds. 5. A locomotive engine A represented by a point is approaching a level crossing C from the south at a speed of 15 miles an hour (22 feet per second), that is velocity of A = 22900 f.s., directions being measured anti-clockwise from the east. An engine B is approaching from the W.S.W., its velocity being 442210 f.s. They arrive together at C. (a) Show their positions one second before the collision, scale 1 inch to 10 feet. Measure their distance apart and the direction from A to B. (b) What is the relative speed of the two engines when the accident occurs? [B.E.] 6. An aeroplane is headed due west, and is propelled at 50 miles per hour relatively to a steady wind which is blowing at 20 miles per hour from the northwest. Find the actual direction and speed of flight as regards the earth. If the pilot wishes to travel westward, in what direction must he apparently steer, and what will be his speed to the west? [B.E.] 7. A ship is sailing eastwards at 10 miles an hour. It carries an instrument for recording the apparent velocity of the wind, in both magnitude and direction. (a) If the wind registered by the instrument is apparently one of 20 miles per hour from the north-east, what is the actual wind? Give the answer in miles per hour and degrees north of east of the quarter from which the wind comes. (b) If a wind of 15 miles per hour from the north-east were actually blowing, what apparent wind would the instrument on the vessel register? State this answer in miles per hour and degrees north of east as before. Use a scale of inch to 1 mile per hour, [B.E.] 8. A weight of 15 pounds is supported by two cords. One cord is inclined at 40° and the other at 55° to the horizontal. Determine the tensions in the cords. 9. A wheel weighing 100 lb. rolls at a certain speed on a horizontal rail. The wheel is out of balance to the extent that at the speed of rolling there is a radial centrifugal force of 20 lb. On a base 6 inches long, representing one revolution of the wheel, plot, for every 1-12th of a revolution, the pressure exerted by the wheel on the rail. Use a force scale of 1 inch to 20 lb. 10. A, B, C, D, E, and F are six forces acting in a plane at a point O. The magnitudes of the forces are 125, 110, 2.25, 1·75, 2·45, and 1.35 pounds respectively. The forces act outwards from O in lines inclined to a fixed line OX at the following angles, 30°, 60°, 135°, 180°, 225°, and 300° respectively. Find R the resultant of these forces. If these forces are balanced by a force P acting in the line OX and a force Q acting in a line at right angles to OX, determine P and Q. 11. A jet of water having a velocity of 353 feet per second passes over a fixed vane as shown in Fig. 169, and is deflected, leaving the vane with a velocity of 3510 feet per second. If u denote the change of velocity that has occurred, find u and ; that is, solve the vector equation 1000 The mass m of water passing per second being 2.5 units (= 2.5 × 32.2 lb.), calculate mu,, the magnitude and direction of the change of momentum per second; find mu, the magnitude and direction of the force acting on the vane. [B.E.] also 12. A stream of water flowing in the given direction AB (Fig. 170) impinges on a succession of moving vanes, one of which is shown. BT, CT are tangents to the curve of the vane at its ends. The vector V represents the velocity of the vane to a scale of 1 inch to 20 feet per second. (a) Find and measure the speed of the water along AB in order that the water may come on to the vane at B tangentially, or without shock. (b) Suppose the water to flow over the vane without change of relative speed, represent graphically to scale the absolute velocity of the water as it leaves the vane at C. (c) Show graphically the vector change U of velocity of the water that has occurred owing to its passage over the vane. Find and measure the component of U in the direction of V. (d) Determine, either the actual path of the water as it flows over the moving vane, or the line of the resultant force on the vane, due to the change of momentum of the water. [B.E.] 30° 13. A jet of water having a velocity V, of 3500 feet per second passes over a succession of curved vanes (one of which is given in Fig. 171) moving with a velocity V of 23 feet per second. Find V, V, the velocity of the water rela ၂၁ tively to the vane; that is, find s and a in the vector equation 3530° 230° s being the speed along the vane, and a the direction of the vane at entrance, the water coming on tangentially. The water leaves the vane with a velocity relatively to the latter of $161° Find U(= u), the change of velocity that has occurred. The mass m of the water flowing per second being 2.5 units (= 2.5 × 32.2 lb.), calculate mÜ, the magnitude and direction of the change of momentum per second. Find also the power developed, which is equal to the scalar product mUV. [B.E.] 14. A jet of water passes over a succession of curved vanes (one of which is shown in Fig. 172), entering tangentially with a velocity V1 of 3530° feet per second, and leaving with a velocity V2 of 8800 feet per second. The vanes have a velocity V of feet per second, and the speed s of the water along the vanes is assumed constant. 2 8f. S в 80° 35 f.s. 30 FIG. 172. -0° V=U0° Find v, the speed of the vane. Find also a and B the directions of the vane at entrance and exit. The mass m of the water flowing per second being 2:5 units (= 2.5 × 32.2 lb.), and U denoting the vector change of velocity of the water, find mU, the magnitude and direction of the change of momentum per second. Find also the power developed, which is equal to the scalar product mUV. [B.E.] |