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Unformatted text preview: hat the xcomponent of this
equation must be zero (Ax + Bx + Cx = 0) and the
ycomponent must be zero (Ay + By + Cy = 0). These
two equations will allow us to find the magnitudes of
B and C. +y (north)
A
−x 145 units
35.0° B
65.0° +x (east) 15.0° C
−y SOLUTION The x and ycomponents of A, B, and C are given in the table below. The
plus and minus signs indicate whether the components point along the positive or negative
axes.
Vector x Component y Component A –(145 units) cos 35.0° = –119 units +(145 units) sin 35.0° = +83.2 units B
C +B sin 65.0° = +B (0.906)
–C sin 15.0° = –C (0.259) +B cos 65.0° = +B (0.423)
–C cos 15.0° = –C (0.966) A+B+C –119 units + B (0.906) – C (0.259) +83.2 units + B (0.423) – C (0.966) Setting the separate x and y components of A + B + C equal to zero gives
xcomponent (–119 units) + B (0.906) – C (0.259) = 0 ycomponent (+83.2 units) + B (0.423) – C (0.966) = 0 Solving these two equations simultaneously, we find that a. B = 178 units b. C = 164 units Chapter 1 Problems 35 56. REASONING The following table shows the components of the individual displacements
and the components of the resultant. The directions due east and due north are taken as the
positive directions.
East/West
Component Displacement
(1)
(2)
(3)
(4) North/South
Component –27.0 cm
0
–(23.0 cm) cos 35.0° = –18.84 cm –(23.0 cm) sin 35.0° = –13.19 cm
(28.0 cm) cos 55.0° = 16.06 cm –(28.0 cm) sin 55.0° = –22.94 cm
(35.0 cm) cos 63.0° = 15.89 cm
(35.0 cm) sin 63.0° = 31.19 cm
–13.89 cm Resultant –4.94 cm SOLUTION
a. From the Pythagorean theorem, we find that the
magnitude of the resultant displacement vector is
13.89 cm R = (13.89 cm) 2 + (4.94 cm) 2 = 14.7 cm R b. The angle θ is given by θ = tan −1 F4.94 cm I =
G cm J
HK
13.89 θ 4.94 cm 19 .6° , south of west 57. REASONING AND SOLUTION The following figure (not drawn to scale) shows the
geometry of the situation, when the observer is a distance r from the base of the arch.
The angle θ is related to r and h by tan θ = h / r .
Solving for r, we find
h = 192 m θ h
192 m
r=
=
= 5.5 × 10 3 m = 5.5 km
r
tanθ tan 2.0°
______________________________________________________________________________
58. REASONING AND SOLUTION In the diagram below, θ = 14.6° and h = 2830 m.
We know that sin θ = H/h and, therefore, h H = h sin θ
= (2830 m) sin 14.6° = 713 m H
θ 36 INTRODUCTION AND MATHEMATICAL CONCEPTS 59. SSM REASONING AND SOLUTION In order to determine which vector has the
largest x and y components, we calculate the magnitude of the x and y components
explicitly and compare them. In the calculations, the symbol u denotes the units of the
vectors.
Ax = (100.0 u) cos 90.0° = 0.00 u
2
Bx = (200.0 u) cos 60.0° = 1.00 × 10 u
Cx = (150.0 u) cos 0.00° = 150.0 u 2 Ay = (100.0 u) sin 90.0° = 1.00 × 10 u
By = (200.0 u) sin 60.0° = 173 u
Cy = (150.0 u) sin 0.00° = 0.00 u a. C has the largest x component. b. B has the largest y component.
______________________________________________________________________________
60. REASONING Multiplying an equation by a factor of 1 does not alter the equation; this is
the basis of our solution. We will use factors of 1 in the following forms:
1 gal
= 1 , since 1 gal = 128 oz
128 oz 3.785 ×10−3 m3
= 1 , since 3.785 × 10−3 m3 = 1 gal
1 gal
1 mL
= 1 , since 1 mL = 10−6 m3
−6 3
10 m SOLUTION The starting point for our solution is the fact that Volume = 1 oz
Multiplying this equation on the right by factors of 1 does not alter the equation, so it
follows that 1 gal 3.785 ×10−3 m3 1 mL Volume = (1 oz )(1)(1)(1) = 1 oz = 29.6 mL 128 oz 10−6 m3 1 gal ( ) Note that all the units on the right, except one, are eliminated algebraically, leaving only the
desired units of milliliters (mL). Chapter 1 Problems 37 61. REASONING AND SOLUTION The east and north components are, respectively
a. Ae = A cos θ = (155 km) cos 18.0° = b. An = A sin θ = (155 km) sin 18.0° = 147 km
47.9 km _____________________________________________________________________________________________ 62. REASONING We will use the scalar x and y components of the resultant vector to obtain
its magnitude and direction. To obtain the x component of the resultant we will add together
the x components of each of the vectors. To obtain the y component of the resultant we will
add together the y components of each of the vectors.
+y
SOLUTION
The x and y components of the resultant vector
R are Rx and Ry, respectively. In terms of
these components, the magnitude R and the
B
directional angle θ (with respect to the x axis)
A
for the resultant are R= 2
Rx 2
+ Ry and Ry
θ = tan −1 R
x 20.0°
35.0° (1) C The following table summarizes the
components of the individual vectors shown in
the drawing:
Vector +x 50.0° x component D y component A Ax = − (16.0 m ) cos 20.0° = −15.0 m Ay = (16.0 m ) sin 20.0° = 5.47 m B Bx = 0 m B y = 11.0 m C C x = − (12.0 m ) cos 35.0° = −9.83 m C y = − (12.0 m ) sin 35.0° = −6.88 m D Dx = ( 26.0 m ) cos 50.0° = 16.7 m Dy = − ( 26.0 m ) sin 50.0...
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This note was uploaded on 04/30/2013 for the course PHYS 1100 and 2 taught by Professor Chastain during the Spring '13 term at LSU.
 Spring '13
 CHASTAIN
 Physics, The Lottery

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