The Language of PhysicsScalarA scalar quantity is a quantity that can becompletely described by a magnitude, thatis, by a number and a unit (p. 23).VectorA vector quantity is a quantity that needsboth a magnitude and direction tocompletely describe it (p. 23).ResultantThe vector sum of any number of vectors iscalled the resultant vector (p. 24).Parallelogram method of vector additionThe main diagonal of a parallelogram isequal to the magnitude of the sum of thevectors that make up the sides of theparallelogram (p. 25).

Sine functionThe ratio of the length of the opposite sideto the length of the hypotenuse in a righttriangle (p. 26).Cosine functionThe ratio of the length of the adjacent sideto the length of the hypotenuse in a righttriangle (p. 26).Tangent functionThe ratio of the length of the opposite sideof a right triangle to the length of theadjacent side (p. 26).Pythagorean theoremThe sum of the squares of the lengths oftwo sides of a right triangle is equal to thesquare of the length of the hypotenuse(p. 27).

Summary of Important Equations

Component of a vectorThe projection of a vector onto a specifiedaxis. The length of the projection of thevector onto the x-axis is called the x-component of the vector. The length of theprojection of the vector onto the j>-axis iscalled the ^/-component of the vector(p. 28).The addition of vectors by the componentmethodThe x-component of the resultant vector Rxis equal to the sum of the x-components ofthe individual vectors, while the y-component of the resultant vector Ry isequal to the sum of the ^-components of theindividual vectors. The magnitude of theresultant vector is then found by thePythagorean theorem applied to the righttriangle with sides Rx and Ry. The directionof the resultant vector is found bytrigonometry (p. 31).

Vector addition is commutativeR = a + b = b + aSubtraction of vectorsa - b = a + (-b)Addition of vectorsR = a + b + c + dDefinition of the sine

opposite sidesine 8 =hypotenuse

Definition of the cosineadjacent sidecosine 8hypotenuse

(2.5)

(2.6)

(2.7)

(2.8)

(2.10)

Definition of the tangentopposite side

tangent 8 = —-adjacent sidePythagorean theoremc= V"2 + b2x-component of a vectorax = a cos 8

^-component of a vectoray = a sin 8Magnitude of a vectora = \/a2x + a2

Direction of a vector

(2.12) 8 = tan-'^ax (2.26)

(2.17)x-component of resultant vectorRx = ax + bx + cx + dx (2.35)

(2.20)^-component of resultant vectorRy "■ Oj + by + Cy + dyMagnitude of resultant vector

(2.36)

(2.22) R = VJ?2. + R] (2.37)

(2.24)Direction of resultant vector

R..8 = tan"

Rx(2.39)

Questions for Chapter 21. Give an example of some quantities

that are scalars and vectors otherthan those listed in section 2.1.

2. Can a vector ever be zero? Whatdoes a zero vector mean?

t 3. Since time seems to pass from thepast to the present and then to thefuture, can you say that time has adirection and therefore could berepresented as a vector quantity?

4. Does the subtraction of two vectorsobey the commutative law?

What happens if you multiply avector by a scalar?What happens if you divide a vectorby a scalar?If a person walks around a block thatis 800 ft on each side and ends up atthe starting point, what is theperson's displacement?How can you add three vectors ofequal magnitude in a plane such thattheir resultant is zero?When are two vectors a and b equal?

tlO. If a coordinate system is rotated,what does this do to the vector? tothe components?

til. Why are all the fundamentalquantities scalars?

12. A vector equation is equivalent tohow many component equations?

13. If the components of a vector a are axand ay, what are the components ofthe vector b = —5a?

14. If | a + b | = | a - b |, what is theangle between a and b?

Problems for Chapter 2

2.7 Resolution of a Vector intoIts Components and

2.8 Determinationof a Vector fromIts Components

Chapter 2 Vectors

A strong child pulls a sled with aforce of 100 lb at an angle of 35°above the horizontal. Find thevertical and horizontal components ofthis pull.

A 50-N force is directed at an angleof 50° above the horizontal. Resolvethis force into vertical and horizontalcomponents.

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3. A girl wants to hold a 25-lb sled atrest on a snow covered hill. The hillmakes an angle of 15° with thehorizontal. What force must sheexert parallel to the slope? What isthe force perpendicular to the surfaceof the hill that presses the sledagainst the hill?

4. A boy wants to hold a 68.0-N sled atrest on a snow-covered hill. The hillmakes an angle of 27.5° with thehorizontal, (a) What force must heexert parallel to the slope? (b) Whatis the force perpendicular to thesurface of the hill that presses thesled against the hill?

5. A displacement vector, at an angle of35° with respect to a specifieddirection, has a ^-component equal to150 ft. What is the magnitude of thedisplacement vector?

6. A plane is traveling northeast at 200km/hr. What is (a) the northwardcomponent of its velocity, and (b) theeastward component of its velocity?

7. While taking off, an airplane climbsat an 8° angle with respect to theground. If the aircraft's speed is 200km/hr, what are the vertical andhorizontal components of its velocity?

8. A car that weighs 3200 lb is parkedon a hill that makes an angle of 23°with the horizontal. Find thecomponent of the car's weightparallel to the hill and perpendicularto the hill.

9. A car that weighs 8900 N is parked'ona hill that makes an angle of 43°with the horizontal. Find thecomponent of the car's weightparallel to the hill and perpendicularto the hill.

10. A girl pushes a lawn mower with aforce of 90 N. The handle of themower makes an angle of 40° withthe ground. What are the verticaland horizontal components of thisforce and what are their physicalsignificances? What effect doesraising the handle to 50° have?

11. A missile is launched with a speed of1000 m/s at an angle of 73° abovethe horizontal. What are thehorizontal and vertical componentsof the missile's velocity?

12. When a ladder leans against asmooth wall, the wall exerts ahorizontal force F on the ladder, as

shown in the diagram. If F is equalto 50 N and 8 is equal to 63°, findthe component of the forceperpendicular to the ladder and thecomponent parallel to the ladder.

2.9 The Addition of Vectors bythe Component Method

13. Find the resultant of the followingthree displacements; 3 mi due east,6 mi east-northeast, and 7 minorthwest.

14. A girl drives 3 km north, then 12 kmto the northwest, and finally 5 kmsouth-southwest. How far has shetraveled? What is her displacement?

15. An airplane flies due east at 200 mphstraight from city A to city B. Anortheast wind of 40 mph is blowing.(Note that all winds are defined interms of the direction from which thewind blows. Hence, a northeast windblows out of the northeast and blowstoward the southwest.) What is theresultant velocity of the plane withrespect to the ground?

16. An airplane flies due north at 380km/hr straight from city A to city B.A southeast wind of 75 km/hr isblowing. (Note that all winds aredefined in terms of the direction fromwhich the wind blows. Hence, asoutheast wind blows out of thesoutheast and blows toward thenorthwest.) What is the resultantvelocity of the plane with respect tothe ground?

17. Find the resultant of the followingforces: (a) 30 N at an angle of 40°with respect to the x-axis, (b) 120 Nat an angle of 135°, and (c) 60 N atan angle of 260°.

18. Find the resultant of the followingset of forces, (a) Fi of 200 N at anangle of 53° with respect to the x-axis. (b) F2 of 300 N at an angle of150° with respect to the x-axis.(c) F3 of 200 N at an angle of 270°with respect to the x-axis. (d) F4 of350 N at an angle of 310° withrespect to the x-axis.

Addi t ional Problems19. A heavy trunk weighing 150 lb is

pulled along a smooth stationplatform by a 50 lb force making anangle of 37° above the horizontal.Find (a) the horizontal component ofthe force, (b) the vertical componentof the force, and (c) the resultantdownward force on the floor.

20. A heavy trunk weighing 800 N ispulled along a smooth stationplatform by a 210-N force makingan angle of 53° above the horizontal.Find (a) the horizontal component of

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the force, (b) the vertical cofnponentof the force, and (c) the resultantdownward force on the floor.

21. Vector A has a magnitude of 50 ftand points in a direction of 50° northof east. What are the magnitudesand directions of the vectors, (a) 2A,(b) 0.5A, (c) -A, (d) -5A, (e) A +4A, (f) A - 4A?

22. Given the two force vectorsF, = 20.0 N at an angle of 30.0°with the positive x-axis andF2 = 40.0 N at an angle of 150.0°with the positive x-axis, find themagnitude and direction of a thirdforce that when added to F| and F2gives a zero resultant.

23. When vector A, of magnitude 5.00m/s at an angle of 120° with respectto the positive x-axis, is added to asecond vector B, the resultant vectorhas a magnitude R = 8.00 m/s andis at an angle of 85.0° with thepositive x-axis. Find the vector B.

24. A car travels 100 mi due west andthen 45 mi due north. How far is thecar from its starting point? Solvegraphically and analytically.

25. Find the resultant of the followingforces graphically and analytically:5 lb at an angle of 33° above thehorizontal and 20 lb at an angle of97° counterclockwise from thehorizontal.

26. Find the resultant of the followingforces graphically and analytically:25 N at an angle of 53° above thehorizontal and 100 N at an angle of117° counterclockwise from thehorizontal.

t27. The velocity of an aircraft is 200km/hr due west. A northwest wind of50 km/hr is blowing, (a) What is thevelocity of the aircraft relative to theground? (b) If the pilot's destinationis due west, at what angle should hepoint his plane to get there? (c) If hisdestination is 400 km due west, howlong will it take him to get there?

28. A plane flies east for 50.0 km, thenat an angle of 30.0° north of east for75.0 km. In what direction should itnow fly and how far, such that it willbe 200 km northwest of its originalposition?

t29. The current in a river flows north at5 mph. A boat starts straight acrossthe river at 8 mph relative to thewater, (a) What is the speed of theboat relative to the land? (b) If theriver is 2 mi wide, how long does ittake the boat to cross the river?(c) If the boat sets out straight forthe opposite side, how far north willit reach the opposite shore? (d) If wewant to have the boat go straightacross the river, at what angle shouldthe boat be headed?

Mechanics

:30. The current in a river flows south at

7 km/hr. A boat starts straightacross the river at 19 km/hr relativeto the water, (a) What is the speed ofthe boat relative to the land? (b) Ifthe river is 1.5 km wide, how longdoes it take the boat to cross theriver? (c) If the boat sets out straightfor the opposite side, how far southwill it reach the opposite shore?(d) If we want to have the boat gostraight across the river, at whatangle should the boat be headed?

t31. Show that if the angle betweenvectors a and b is an acute angle,then the sum a + b becomes themain diagonal of the parallelogramand the difference a — b becomes theminor diagonal of the parallelogram.Also show that if the angle is obtusethe results are reversed.

32. Find the resultant of the followingthree vectors. The magnitudes of thevectors are a = 5.00 km, b = 10.0km, and c = 20.0 km.

33. Find the resultant of the followingthree forces. The magnitudes of theforces are F, = 2.00 N, F2 = 8.00N, and F3 = 6.00 N.

t34. Show that for three nonparallelvectors all in the same plane, any oneof them can be represented as alinear sum of the other two.

t35. A unit vector is a vector that has amagnitude of one unit and is in aspecified direction. If a unit vector iis defined to be in the x-direction,and a unit vector j is defined to be inthe ^-direction, show that any vectora can be written in the form

a = ax\ + ay)

t36. Prove that |a + b| < | a | + |b|.37. An airplane flies due east at 200 mph

straight from city A to city B adistance of 200 mi. A wind of 40mph from the northwest is blowing.If the pilot doesn't make anycorrections, where will the plane bein 1 hr?

38. Given vectors a and b, where a = 50,8, = 33°, b = 80, and02 = 128°,find (a) a + b, (b) a — b, (c) a —2b, (d) 3a + b, (e) 2a - b, and(f) 2b - a.

39. In the accompanying figure thetension Fin the cable is 200 N. Findthe vertical component Ty and thehorizontal component Tx of thistension.

t41. In the accompanying diagram wi =2 N, w2 = 5 N, and 8 = 65°. Findthe angle <fi such that the componentsof the two forces parallel to theinclines are equal.

t42. In the accompanying diagram w =50 lb, and 8 = 10°. What must bethe value of F such that w will beheld in place? What happens if theangle is doubled to 20°?

Boom

t43. In projectile motion in twodimensions the projectile is locatedby the displacement vector ri at thetime ti and by the displacementvector r2 at t2, as shown in thediagram. If/-, = 20 m, 8, = 60°, r2= 25 m, and 82 = 25°, find themagnitude and direction of the vectorr2 - r,.

Mastf

f40. In the accompanying diagram wt is5 lb and w2 is 3 lb. Find the angle 8such that the component of wtparallel to the incline is equal to w2.

Interactive Tutorialsy 44. A 50.0-N force is directed at an

angle of 50° above the horizontal.Resolve this force into vertical andhorizontal components.

y 45. Find the resultant of any number offorce vectors (up to five vectors).

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