SUBTRACTION : TRIANGLE METHOD
EXAMPLE PROBLEM 3.18
A vector equation can be written as follows:
The directions for vectors A, B, C, D, E, and F are known, and the magnitudes of vectors B, C, E, and F are also known, as shown in Figure 3.39. Analytically solve for the magnitudes of vectors A and D.
A +7 B -7 C +7 D = E +7 F
D B= 130 in./s2
E = 200 in./s2
F = 100 in./s2 C= 60 in./s2
30° 45°
60°
60° A
FIGURE 3.39 Vectors for Example Problem 3.18.
SOLUTION: 1. Use x-axis, Angle Method to Determine Vector Components
The horizontal and vertical components of each force are determined by trigonometry. For the unknown vectors, the sense is assumed and the components are found in terms of the unknown quantities. For this example, assume vector A points upward and vector D points down and to the right. The components are given in Table 3.3.
TABLE 3.3 Vector Components for Example Problem 3.18.
Vector
Reference Angle ux
h component (in./s2) ah= a cos ux
v component (in./s2) av= a sin ux
A 90° 0 +A
B 60° +65.0 +112.6
C 135° -42.4 +42.4
D 300° +.500D -.866D
E 30° +173.2 +100
F 180° -100 0
2. Use the Vector Equations to Solve for Unknown Magnitudes
The components can be used to generate algebraic equations that are derived from the original vector equation:
horizontal components:
(0) + (+65.0) - (-42.4) + (+0.500D) = (+173.2) + (-100.0) Ah + Bh - Ch + Dh = Eh + Fh
A+7 B -7 C +7 D = E +7 F
Vectors 67
PROBLEMS
Although manual drafting techniques are instructive for problems that require graphical solution, use of a CAD system is highly recommended.
Working with Triangles
3–1. Analytically determine the angle in Figure P3.1.u
A 18"
5" θ
FIGURE P3.1 Problems 1 and 2.
X R
6"
60
θ
FIGURE P3.3 Problems 3 and 4.
(a) (b)
FIGURE P3.5 Problem 5.
s
β
y s x
FIGURE P3.6 Problems 6 to 9.
3–6. Determine the angle, , and the length, , of the two identical support links in Figure P3.6 when
and .y = 275mm x = 150mm
b s vertical components:
In this case, the horizontal component equation can be solved independently for . In general, both equations are coupled and need to be solved simultaneously. In this example, the horizontal component equa-tion can be solved to obtain the following:
Substitute this value of into the vertical component equation to obtain:
3. Fully Specify the Solved Vectors
Because both values are negative, the original directions assumed for the unknown vectors were incorrect.
Therefore, the corrected results are
p
3–2. Analytically determine the length of side Figure P3.1.
3–3. Analytically determine the length of side Figure P3.3.
3–4. Calculate the angle and the hypotenuse Figure P3.3.
3–5. Calculate the angle and the hypotenuse for all triangles in Figure P3.5.
u R
R in u
X in A in
3–7. Determine the distance, , and the length, , of the two identical support links in Figure P3.6 when
and .y = 16in b = 35°
s x
a
68 CHAPTER THREE
3–17. For the farm conveyor shown in Figure P3.16, deter-mine the angle when a vertical height of 8 m is required at the end of the conveyor and ,
, and .
3–18. Determine the vertical height of the basket in
Figure P3.18 when ., ., .,
FIGURE P3.18 Problems 18 and 19.
β
x
s d
s
FIGURE P3.10 Problems 10 and 11.
3–13. For the ramp shown in Figure P3.12, determine the angle with the ground, . The trailer height is 1.5 m and the ramp is 4 m long.
3–14. The length of the ladder shown in Figure P3.14 is 12 ft and the angle with the ground, , is 70°.
Determine the vertical distance on the wall where the ladder is resting.
b b
h β
FIGURE P3.12 Problems 12 and 13.
3–8. For the folding shelf in Figure P3.6, with and ., determine the distances and . 3–9. A roof that has an 8-on-12 pitch slopes upward 8
vertical in. for every 12 in. of horizontal distance.
Determine the angle with the horizontal of such a roof.
3–10. For the swing-out window in Figure P3.10, deter-mine the length, , of the two identical support links when ,x = 850 mms d = 500 mm, and b = 35°.
y x s = 10in
b = 35°
3–11. For the swing-out window in Figure P3.10,
deter-mine the angle when ., ., and
.
3–12. If the height, , of the trailer shown in Figure P3.12 is 52 in., determine the length of ramp needed to maintain an angle, , of 30°.b
FIGURE P3.14 Problems 14 and 15.
3–15. For the ladder shown in Figure P3.14, determine the angle with the ground. The ladder is 7 m long and rests on the ground 2 m from the wall.
3–16. For the farm conveyor shown in Figure P3.16, deter-mine the required length of the support rod. The angle is and the distances are and . Also determine the vertical height of the end of the conveyor when L = 25ft.
FIGURE P3.16 Problems 16 and 17.
Vectors 69
FIGURE P3.20 Problems 20, 26, 32, 33, 38, 39.
0
FIGURE P3.21 Problems 21, 27, 34, 35, 40, 41.
0
FIGURE P3.22 Problems 22, 28, 36, 37, 42, 43.
3–23. For the vectors shown in Figure P3.23, graphically determine the resultant,R = A +7 B +7 C.
FIGURE P3.23 Problems 23, 29, 44, 45, 52, 53.
0 20°
FIGURE P3.24 Problems 24, 30, 46, 47, 54, 55.
0
FIGURE P3.25 Problems 25, 31, 48, 49, 56, 57.
3–19. For the lift described in Problem 3–18, determine the vertical height of the basket when the hydraulic cylinder is shortened to 50 in.
Graphical Vector Addition
3–20. For the vectors shown in Figure P3.20, graphically determine the resultant,R = A +7 B.
3–21. For the vectors shown in Figure P3.21, graphically determine the resultant,R = A +7 B.
3–22. For the vectors shown in Figure P3.22, graphically determine the resultant,R = A +7 B.
3–24. For the vectors shown in Figure P3.24, graphically determine the resultant,R = A +7 B +7 C +7 D.
3–25. For the vectors shown in Figure P3.25, graphically determine the resultant,
s D+7 E
R = A +7 B+7 C+7
Analytical Vector Addition
3–26. For the vectors shown in Figure P3.20, analytically determine the resultant, .
3–27. For the vectors shown in Figure P3.21, analytically determine the resultant, .
3–28. For the vectors shown in Figure P3.22, analytically determine the resultant, .
3–29. For the vectors shown in Figure P3.23, analytically
determine the resultant, .
3–30. For the vectors shown in Figure P3.24, analytically
determine the resultant, .
3–31. For the vectors shown in Figure P3.25, analytically determine the resultant,
.
Graphical Vector Subtraction
3–32. For the vectors shown in Figure P3.20, graphically determine the vector, .
3–33. For the vectors shown in Figure P3.20, graphically determine the vector, .
3–34. For the vectors shown in Figure P3.21, graphically determine the vector, .
3–35. For the vectors shown in Figure P3.21, graphically determine the vector, .
3–36. For the vectors shown in Figure P3.22, graphically determine the vector, .
3–37. For the vectors shown in Figure P3.22, graphically determine the vector,K = B -7 A.
70 CHAPTER THREE
FIGURE P3.58 Problems 58 and 61.
0
FIGURE P3.59 Problems 59 and 62.
Analytical Vector Subtraction
3–38. For the vectors shown in Figure P3.20, analytically determine the vector, .
3–39. For the vectors shown in Figure P3.20, analytically determine the vector, .
3–40. For the vectors shown in Figure P3.21, analytically determine the vector, .
3–41. For the vectors shown in Figure P3.21, analytically determine the vector, .
3–42. For the vectors shown in Figure P3.22, analytically determine the vector, .
3–43. For the vectors shown in Figure P3.22, analytically determine the vector, .
General Vector Equations (Graphical)
3–44. For the vectors shown in Figure P3.23, graphically
determine the vector, .
3–45. For the vectors shown in Figure P3.23, graphically
determine the vector, .
3–46. For the vectors shown in Figure P3.24, graphically
determine the vector, .
3–47. For the vectors shown in Figure P3.24, graphically
determine the vector, .
3–48. For the vectors shown in Figure P3.25, graphically
det-ermine the vector, .
3–49. For the vectors shown in Figure P3.25, graphically
det-ermine the vector, .
3–50. Using the vector diagram in Figure P3.50:
a. Generate an equation that describes the vector diagram.
b. Rewrite the equations to eliminate the negative terms.
c. Roughly scale the vectors and rearrange them according to the equation generated in part b.
K = B -7 D +7 A -7 C +7 E
FIGURE P3.50 Problem 50.
3–51. Using the vector diagram in Figure P3.51:
a. Generate an equation that describes the vector diagram.
b. Rewrite the equations to eliminate the negative terms.
c. Roughly scale the vectors and rearrange them according to the equation generated in part b.
3–52. For the vectors shown in Figure P3.23, analytically determine the vector,J = C +7 A -7 B.
B E C
D
A
FIGURE P3.51 Problem 51.
3–53. For the vectors shown in Figure P3.23, analytically
determine the vector, .
3–54. For the vectors shown in Figure P3.24, analytically
determine the vector, .
3–55. For the vectors shown in Figure P3.24, analytically
determine the vector, .
3–56. For the vectors shown in Figure P3.25, analytically
det-ermine the vector, .
3–57. For the vectors shown in Figure P3.25, analytically
det-ermine the vector, .
Solving for Vector Magnitudes (Graphical)
3–58. A vector equation can be written as . The directions of all vectors and magnitudes of A, B, and D are shown in Figure P3.58. Graphically (using either manual drawing techniques or CAD) determine the magni-tudes of vectors C and E.
A+7B -7C = D -7E
3–59. A vector equation can be written as
. The directions of all vectors and magni-tudes of A, B, C, and E are shown in Figure P3.59.
Graphically (using either manual drawing techniques or CAD) determine the magnitudes of vectors D and F.
D = E + F A+7 B + C-7
Vectors 71