Las Matematicas

Chapter 3. Technical Measurement and Vectors

Unit Conversions

3-1. A soccer field is 100 m long and 60 m across. What are the length and width of the field in feet?

L = 328 ft

W = 197 ft

3-2. A wrench has a handle 8 in. long. What is the length of the handle in centimeters?

L = 20.3 cm

3-3. A 19-in. computer monitor has a viewable area that measures 18 in. diagonally. Express this distance in meters. .

L = 0.457 m

3-4. The length of a notebook is 234.5 mm and the width is 158.4 mm. Express the surface area in square meters.

A = 0.0371 m2

3-5. A cube has 5 in. on a side. What is the volume of the cube in SI units and in fundamental USCS units? .

V = 0.00205 m3

V = 0.0723 ft3

3-6. The speed limit on an interstate highway is posted at 75 mi/h. (a) What is this speed in kilometers per hour? (b) In feet per second?

(a) 121 km/h (b) = 110 ft/s

3-7. A Nissan engine has a piston displacement (volume) of 1600 cm3 and a bore diameter of 84 mm. Express these measurements in cubic inches and inches. Ans. 97.6 in.3, 3.31 in.

(a) = 97.6 in.3 (b) = 3.31 in.

3-8. An electrician must install an underground cable from the highway to a home located 1.20 mi into the woods. How many feet of cable will be needed?

L = 6340 ft

3-9. One U.S. gallon is a volume equivalent to 231 in.3. How many gallons are needed to fill a tank that is 18 in. long, 16 in. wide, and 12 in. high? Ans. 15.0 gal.

V = (18 in.)(16 in.)(12 in.) = 3456 in.3

V = 15.0 gal

3-10. The density of brass is 8.89 g/cm3. What is the density in kg/m3?

 = 8890 kg/m3

Addition of Vectors by Graphical Methods

3-11. A woman walks 4 km east and then 8 km north. (a) Use the polygon method to find her resultant displacement. (b) Verify the result by the parallelogram method.

Let 1 cm = 1 km; Then: R = 8.94 km,  = 63.40

3-12. A land-rover, on the surface of Mars, moves a distance of 38 m at an angle of 1800. It then turns and moves a distance of 66 m at an angle of 2700. What is the displacement from the starting position?

Choose a scale, e.g., 1 cm = 10 m

Draw each vector to scale as shown.

Measure R = 7.62 cm or R = 76.2 m

Measure angle  = 60.10 S of W

 = 1800 + 60.10 = 240.10 R = 76.2 m, 240.10

3-13. A surveyor starts at the southeast corner of a lot and charts the following displacements: A = 600 m, N; B = 400 m, W; C = 200 m, S; and D = 100 m, E. What is the net displacement from the starting point? .

Choose a scale, 1 cm = 100 m

Draw each vector tail to tip until all are drawn.

Construct resultant from origin to finish.

R = 500 m,  = 53.10 N of E or  = 126.90.

3-14. A downward force of 200 N acts simultaneously with a 500-N force directed to the left. Use the polygon method to find the resultant force.

Chose scale, measure: R = 539 N,  = 21.80 S. of E.

3-15. The following three forces act simultaneously on the same object. A = 300 N, 300 N of E; B = 600 N, 2700; and C = 100 N due east. Find the resultant force

using the polygon method. Choose a scale, draw and measure: R = 576 N,  = 51.40 S of E

3-16. A boat travels west a distance of 200 m, then north for 400 m, and finally 100 m at 300 S of E. What is the net displacement? (Set 1 cm = 100 N)

Draw and measure: R = 368 N,  = 108.00

3-17. Two ropes A and B are attached to a mooring hook so that an angle o 60 exists between the two ropes. The tension in rope A is 80 lb and the tension in rope B is 120 lb. Use the parallelogram method to find the resultant force on the hook.

Draw and measure: R = 174 lb

3-18. Two forces A and B act on the same object producing a resultant force of 50 lb at 36.90 N of W. The force A = 40 lb due west. Find the magnitude and direction of force B?

Draw R = 50 lb, 36.90 N of W first, then draw 40 lb, W.

F = 30 lb, 900

Trigonometry and Vectors

3-19. Find the x and y-components of: (a) a displacement of 200 km, at 340. (b) a velocity of 40 km/h, at 1200; and (c) A force of 50 N at 330o.

¬¬¬¬¬ (a) Dx = 200 cos 340 = 166 km

Dy = 200 sin 340 = 112 km

(b) vx = -40 cos 600 = -20.0 km/h

vy = 40 sin 600 = +34.6 km/h

(c) Fx = 50 cos 300 = 43.3 N; Fy = - 50 sin 300 = -25.0 N

3-20. A sled is pulled with a force of 540 N at an angle of 400 with the horizontal. What are the horizontal and vertical components of this force?

Fx = 540 cos 400 = 414 N Fy = 540 sin 400 = 347 N

3-21. The hammer in Fig. 3-26 applies a force of 260 N at an angle of 150 with the vertical. What is the upward component of the force on the nail?

F = 260 lb,  = 750; Fxy = 260 sin  Fy = 251 N.

3-22. A jogger runs 2.0 mi west and then 6.0 mi north. Find the magnitude and direction of the resultant displacement.

6.32 mi ;  = 71.60 N of W

* 3-23. A river flows south with a velocity of 20 km/h. A boat has a maximum speed of 50 km/h in still water. In the river, at maximum throttle, the boat heads due west. What is the resultant speed and direction of the boat?

* 3-24. A rope, making an angle of 300 with the horizontal, drags a crate along the floor. What must be the tension in the rope, if a horizontal force of 40 lb is required to drag the crate?

Fx = F cos 300; F = 46.2 N

* 3-25. An vertical lift of 80 N is needed to lift a window. A long pole is used to lift the window. What force must be exerted along the pole if it makes an angle of 340 with the wall?

Fy = F sin 300; F = 96.5 N

* 3-26. The resultant of two forces A and B is 400 N at 2100. If force A is 200 N at 2700, what are the magnitude and direction of force B? ( = 210 - 1800 = 300)

B = -400 N cos 300 = -346 N: B = 346 N, 1800

The Component Method of Vector Addition

3-27. Find the resultant of the following perpendicular forces: (a) 400 N, 00; (b) 820 N, 2700; and (b) 500 N, 900. Draw each vector, then find R: Ax = +400 N; Bx = 0; Cx = 0:

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