Queens College, CUNY - PHYSICS 152

Chapter 1—Introduction and Vectors

MULTIPLE CHOICE

1)The density of an object is defined as:

1. 1. the volume occupied by each unit of mass.
   2. the amount of mass for each unit of volume.
   3. the weight of each unit of volume.
   4. the amount of the substance that has unit volume and unit mass.
   5. the amount of the substance that contains as many particles as 12 grams of the carbon-12 isotope.

2. Find the average density of a red giant star with a mass of 20  10³⁰ kg (approximately 10 solar masses) and a radius of 150  10⁹ m (equal to the Earth's distance from the sun).

a.   1.41  10^?4 kg/m³

b. 0.007 kg/m³

c.   1.41 kg/m³

d. 710 kg/m³

e.   1.41  10^?3 kg/m³

3. Find the average density of a white dwarf star if it has a mass equal to that of the sun (2.0  10³⁰ kg) and a radius equal to that of the Earth (6.4  10⁶ m).

a.   9.0  10⁶ kg/m³

b. 1.8  10⁷ kg/m³

c.   1.8  10⁹ kg/m³

d. 3.6  10¹⁰ kg/m³

e.   9.0  10⁷ kg/m³

4. The term         occurs in Bernoulli's equation in Chapter 15, with ? being the density of a fluid and v

its speed. The dimensions of this term are

1. 1. M^?1L⁵T²
   2. MLT²
   3. ML^?1T^?2
   4. M?1L9T?2
   5. M?1L3T?2

5. Which of the following quantities has the same dimensions as kinetic energy,       ? Note: [a] = [g] = LT^?2; [h] = L and [v] = LT^?1.
   1. ma
   2. mvx
   3. mvt

1. 4. mgh
   5. mgt

6. The quantity with the same units as force times time, Ft, with dimensions MLT^?1 is
   1. mv
   2. mvr
   3. mv²r
   4. ma

e.

7. The equation for the change of position of a train starting at x = 0 m is given by           . The dimensions of b are
   1. T^?3
   2. LT^?3
   3. LT^?2
   4. LT^?1
   5. L?1T?1

8. One mole of the carbon-12 isotope contains 6.022  10²³ atoms. What volume in m³ would be needed to store one mole of cube-shaped children's blocks 2.00 cm long on each side?

a.  4.8  10¹⁸

b.  1.2  10²²

c.   6.0  10²³

d.  1.2  10²⁴

e.   4.8  10²⁴

9. John and Linda are arguing about the definition of density. John says the density of an object is proportional to its mass. Linda says the object's mass is proportional to its density and to its volume. Which one, if either, is correct?
   1. They are both wrong.
   2. John is correct, but Linda is wrong.
   3. John is wrong, but Linda is correct.
   4. They are both correct.
   5. They are free to redefine density as they wish.

10. Spike claims that dimensional analysis shows that the correct expression for change in velocity,

, is                    , where m is mass, t is time, and F is the magnitude of force. Carla says that can't be true because the dimensions of force are        . Which one, if either, is correct?

a.

Spike, because                   .

b.

Spike, because                  .

c.

Carla, because               .

d.

Carla, because                   .

e.

Spike, because the dimensions of force are               

11. Which one of the quantities below has dimensions equal to  ?
    1. mv
    2. mv²

c.

4. mrv

e.

12. Which quantity can be converted from the English system to the metric system by the conversion

factor                                                                ?

1. 1. feet per second
   2. feet per hour
   3. miles per second
   4. miles per hour
   5. miles per minute

13. The answer to a question is [MLT^?1]. The question is "What are the dimensions of
    1. mr?"
    2. mvr?"
    3. ma?"
    4. mat?" e.

?"

14. Which of the following products of ratios gives the conversion factor to convert miles per hour

to meters per second     ? a.

b.

c.

d.

e.

15. Which of the following products of ratios gives the conversion factors to convert meters per second

to miles per hour       ?

a.

b.

c.

d.

e.

16. One U.S. fluid gallon contains a volume of 231 cubic inches. How many liters of gasoline would you have to buy in Canada to fill a 14-gallon tank? (Note: 1L = 10⁺³ cm³.)
    1. 53
    2. 21
    3. 14

d.   8.0

e.   4.0

17. The standard exam page is 8.50 inches by 11.0 inches. Its area in cm² is

a.   19.5

b.   36.8

c.   93.5

d.   237.

e.   603.

18. A standard exam page is 8.5 inches by 11 inches. An exam that is 2.0 mm thick has a volume of

a. 1.9  10⁴ mm³.

b. 4.7  10⁴ mm³.

c.   1.2  10⁵ mm³.

d. 3.1  10⁵ mm³.

e.   3.1  10³ mm³.

19. If you drove day and night without stopping for one year without exceeding the legal highway speed limit in the United States, the maximum number of miles you could drive would be closest to:

a.   8 700.

b.   300 000.

c.   500 000.

d. 1 000 000.

e.   32 000 000.

20. If each frame of a motion picture film is 35 cm high, and 24 frames go by in a second, estimate how many frames are needed to show a two hour long movie.

a. 1 400

b.   25 000

c.   50 000

d. 170 000

5. This cannot be determined without knowing how many reels were used.

21. One number has three significant figures and another number has four significant figures. If these numbers are added, subtracted, multiplied, or divided, which operation can produce the greatest number of significant figures?
    1. the addition
    2. the subtraction
    3. the multiplication
    4. the division
    5. All the operations result in the same number of significant figures.     
22. A rectangle has a length of 1.323 m and a width of 4.16 m. Using significant figure rules, what is the area of this rectangle?

a.   5.503 68 m²

b. 5.503 7 m²

c.   5.504 m²

d. 5.50 m²

e.   5.5 m²

23. If                      and                       , what is the magnitude of the vector            ?
    1. 42
    2. 22
    3. 64

1. 4. 90
   5. 13

24. If                      and                       , what is the direction of the vector             ?

a.   ?49

b. ?41

c.   ?90

d. +49

e.   +21

25. Given that                              and                            , what is   ? a.

b.

c.

d.

e.

26. Given that                            and                           , what is   ? a.

b.

c.

d.

e.

27. Given that                            and                           , what is   ? a.

b.

c.

d.  

e.

28. The diagram below shows 3 vectors which sum to zero, all of equal length. Which statement below is true?

a.

b.

c.

d.

e.

Exhibit 3-3

The vectors  ,    , and    are shown below.

Use this exhibit to answer the following question(s).

29. Refer to Exhibit 3-3. Which diagram below correctly represents       ?
    1. d.

1. 2. e.

c.

30. Refer to Exhibit 3-3. Which diagram below correctly represents        ?
    1. d.

1. 2. e.

c.

31. Dana says any vector  can be represented as the sum of two vectors:        . Ardis says any

vector    can be represented as the difference of two vectors:       . Which one, if either, is correct?

1. 1. They are both wrong: every vector is unique.
   2. Dana is correct: Any vector can be represented as a sum of components and not as a difference.
   3. Ardis is correct: Any vector can be represented as a difference of vector components and not as a sum.

d.

They are both correct: A difference of vectors is a sum              .

e.   They are both wrong: Vectors can be moved as long as they keep the same magnitude and

direction.

32. Anthony has added the vectors listed below and gotten the result               . What errors has he made?

1. 1. He lost the minus sign in vector    .
   2. He read the   in     as     .
   3. He lost the minus sign in vector    .
   4. All of the above are correct.
   5. Only (a) and (b) above are correct.

33. Given the statement that                    , what can we conclude?
    1. and          .
    2. .
    3. and            .
    4. Any one of the answers above is correct.
    5. Only (a) and (b) may be correct.

34. Adding vectors   and    by the graphical method gives the same result for   +        and        +          . If both additions are done graphically from the same origin, the resultant is the vector that goes from the tail of the first vector to the tip of the second vector, i.e, it is represented by a diagonal of the parallelogram formed by showing both additions in the same figure. Note that a parallelogram has 2 diagonals. Keara

says that the sum of two vectors by the parallelogram method is                                         . Shamu says it is                                                                                           . Both used the parallelogram method, but one used the wrong diagonal. Which one of the vector pairs below contains the original two vectors?

a.   ;

b. ;

c.   ;

d. ;

e.   ;

35. Given two non-zero vectors,  and   , such that                          , the sum         satisfies a.

.

b.

.

c.

.

d.

.

e.

.

NOTE: The notation    = [A, ] is a shorthand notation for                                        .

36. If     = [15, 80] and                        , what is the magnitude of      ?
    1. 15
    2. 35
    3. 32

d. 5.0

e.   23

37. If     = [10 m, 30] and        = [25 m, 130], what is the magnitude of the sum of these two vectors?
    1. 20 m
    2. 35 m
    3. 15 m
    4. 25 m
    5. 50 m

38. If     = [10 m, 30] and        = [25 m, 130], what is the direction of the sum of these two vectors?

a.   17

b. 73

c.   107

d. 163

e.   100

39. If     = [2.5 cm, 80], i.e., the magnitude and direction of are 2.5 cm and 80,        = [3.5 cm, 120], and               , what is the direction of                 (to the nearest degree)?

a.   247

b. 235

c.   243

d. 216

e.   144

                                 PTS:   3                     DIF:    Challenging

40. Vectors   and    are shown. What is the magnitude of a vector                if  ?

41. A vector,   , when added to the vector              yields a resultant vector which is in the positive y

direction and has a magnitude equal to that of                                      . What is the magnitude of                                        ? a.                                      3.2

b.   6.3

c.   9.5

4. 18
5. 5

42. If vector    is added to vector , the result is       . If     is subtracted from , the result is           . What is the magnitude of         ?

a.   5.1

b.   4.1

c.   5.4

d.   5.8

e.   8.2

43. If vector    is added to vector , the result is           . If     is subtracted from , the result is         . What is the direction of           (to the nearest degree)?

a.   225

b. 221

c.   230

d. 236

e.   206

44. A vector    is added to                . The resultant vector is in the positive x direction and has a magnitude equal to                   . What is the magnitude of  ?
    1. 11

b. 5.1

45. A vector    is added to                . The resultant vector is in the positive x direction and has a magnitude equal to that of       . What is the direction of     ?

a.   74

b. 100

c.   ?81

d. ?62

e.   106

46. If two collinear vectors  and    are added, the resultant has a magnitude equal to 4.0. If                               is subtracted from                                                                            , the resultant has a magnitude equal to 8.0. What is the magnitude of                          ?

a.   2.0

b.   3.0

c.   4.0

d.   5.0

e.   6.0

47. If two collinear vectors  and    are added, the resultant has a magnitude equal to 4.0. If                               is subtracted from                                                                            , the resultant has a magnitude equal to 8.0. What is the magnitude of                          ?

a.   2.0

b.   3.0

c.   4.0

d.   5.0

e.   6.0

48. When vector   is added to vector , which has a magnitude of 5.0, the vector representing their sum is perpendicular to and has a magnitude that is twice that of                                       . What is the magnitude of  ?

a.   2.2

b.   2.5

c.   4.5

d.   5.0

e.   7.0

49. Starting from one oasis, a camel walks 25 km in a direction 30 south of west and then walks 30 km toward the north to a second oasis. What distance separates the two oases?
    1. 15 km
    2. 48 km
    3. 28 km
    4. 53 km

1. 5. 55 km

50. Starting from one oasis, a camel walks 25 km in a direction 30 south of west and then walks 30 km toward the north to a second oasis. What is the direction from the first oasis to the second oasis?
    1. 21 N of W
    2. 39 W of N
    3. 69 N of W
    4. 51 W of N
    5. 42 W of N

Exhibit 3-1

The three forces shown act on a particle.

Use this exhibit to answer the following question(s)..

51. Refer to Exhibit 3-1. What is the magnitude of the resultant of these three forces? a.          27.0 N

b.   33.2 N

c.   36.3 N

d.   23.8 N

e.   105 N

52. Refer to Exhibit 3-1. What is the direction of the resultant of these three forces?

a.   35

b. 45

c.   65

d. 55

e.   85

53. If vector    is added to vector , the result is a third vector that is perpendicular to  and has a magnitude equal to 3 . What is the ratio of the magnitude of           to that of      ?

a.   1.8

b.   2.2

c.   3.2

d.   1.3

e.   1.6

 3

Exhibit 3-2

A child starts at one corner of a cubical jungle gym in a playground and climbs up to the diagonally opposite corner. The original corner is the coordinate origin, and the x, y and z axes are oriented along the jungle gym edges. The length of each side is 2 m.

Use this exhibit to answer the following question(s).

54. Refer to Exhibit 3-2. The child's displacement is: a.

b.

c.

d.

e.

55. Refer to Exhibit 3-2. What is the child’s distance from her starting position?
    1. 2.8 m
    2. 3.5 m
    3. 6.0 m
    4. 6.9 m

e.   12.0 m

56. The displacement of the tip of the 10 cm long minute hand of a clock between 12:15 A.M. and 12:45

P.M. is:

a.   10 cm, 90

b. 10 cm, 180

c.   10 cm, 4 500

d. 20 cm, 180

e.   20 cm, 540

57. A student decides to spend spring break by driving 50 miles due east, then 50 miles 30 degrees south of east, then 50 miles 30 degrees south of that direction, and to continue to drive 50 miles deviating by 30 degrees each time until he returns to his original position. How far will he drive, and how many vectors must he sum to calculate his displacement?

a.   0, 0

b.   0, 8

c.   0, 12

d. 400 mi, 8

e.   600 mi, 12

58. Which statement is true about the unit vectors , and   ?
    1. Their directions are defined by a left-handed coordinate system.
    2. The angle between any two is 90 degrees.
    3. Each has a length of 1 m.
    4. If   is directed east and  is directed south,          points up out of the surface.
    5. All of the above.

59. Vectors   and    have equal magnitudes. Which statement is always true?
    1. .
    2. .
    3. is perpendicular to       .
    4. is perpendicular to       .
    5. The magnitude of                               equals the magnitude of     .

60. When three vectors,     ,           , and                                                                      are placed head to tail, the vector sum  . If the vectors all have the same magnitude, the angle between the directions of any two adjacent vectors is

a.   30

b. 60

c.   90

d. 120

e.   150

 5

Exhibit 3-4

The diagram below shows the path taken by a sailboat tacking sideways because it cannot sail directly into the wind.

Use this exhibit to answer the following question(s).

61. Refer to Exhibit 3-4. The total distance it travels is a. 1 000 m.

b.   1 732 m.

c.   2 000 m.

d.   6 298 m.

e.   8 000 m.

62. Refer to Exhibit 3-4. The total displacement of the sailboat, the vector sum of its displacements OB,

BC, CD and DE, is

1. 1. 1 732 m, East.
   2. 2 000 m, Northeast.
   3. 6 298 m, East.
   4. 8 000 m, Southeast.
   5. 8 000 m, East.

63. The vector   has components +5 and +7 along the x and y axes respectively. Along a set of axes rotated 90 degrees counterclockwise relative to the original axes, the vector's components are

a.   ?7; ?5.

b. 7; ?5.

c.   ?7; 5.

d.   7; 5.

e.   7; 0.

64. The vector   has components +5 and +7 along the x and y axes respectively. If the vector is now rotated 90 degrees counterclockwise relative to the original axes, the vector's components are now

65. The rectangular coordinates of a point are (5.00, y) and the polar coordinates of this point are (r, 67.4°). What is the value of the polar coordinate r in this case?

a.   1.92