Page 40 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 2] VECTORS 25

SOLVED PROBLEM 2.12
◦
A car weighing 12.0 kN is on a hill that makes an angle of 20 with the horizontal. Find the components
of the car’s weight parallel and perpendicular to the road.
The weight w of an object is the gravitational force with which the earth attracts it, and this force always acts
vertically downward (Fig. 2-16). Because w is vertical and F 2 is the component perpendicular to the road, the angle
θ between w and F 2 is the same as the angle θ between the road and the horizontal. Hence the components of w
parallel and perpendicular to the road are
◦
F 1 = w sin θ = (12.0kN)(sin 20 ) = 4.1kN
◦
F 2 = w cos θ = (12.0kN)(cos 20 ) = 11.3kN

Fig. 2-16

VECTOR ADDITION: COMPONENT METHOD
When vectors to be added are not perpendicular, the method of addition by components described below can be
used. There do exist trigonometric procedures for dealing with oblique triangles (the law of sines and the law of
cosines), but these are not necessary since the component method is entirely general in its application.
To add two or more vectors A, B, C,... by the component method, follow this procedure:
1. Resolve the initial vectors into components in the x, y, and z directions.
2. Add the components in the x direction to give R x , add the components in the y direction to give R y , and
add the components in the z direction to give R z . That is, the magnitudes of R x , R y , and R z are given
by, respectively,

R x = A x + B x + C x +· · ·
R y = A y + B y + C y +· · ·
R z = A z + B z + C z + ···
3. Calculate the magnitude of the resultant R from its components R x , R y , and R z by using the Pythagorean
theorem:

2
2
R = R + R + R 2
x y z
If the vectors being added all lie in the same plane, only two components need to be considered.
SOLVED PROBLEM 2.13

Use the component method of vector addition to solve Prob. 2.2.