9781118230725.pdf

138 CHAPTER 6 FORCE AND MOTION—II

write Newton’s second law for components along the yaxis

(Fnet,ymay) as

FNcosumgm(0),

from which

FNcosumg. (6-24)

Combining results:Equation 6-24 also contains the

unknownsFNandm, but note that dividing Eq. 6-23 by

Eq. 6-24 neatly eliminates both those unknowns. Doing so,

replacing (sin u)/(cosu) with tan u, and solving for uthen

yield

tan^1. (Answer)

(20 m/s)^2

(9.8 m/s^2 )(190 m)

 12 

tan^1

v^2

gR

Radial calculation:As Fig. 6-11b shows (and as you

should verify), the angle that force makes with the ver-

tical is equal to the bank angle uof the track. Thus, the ra-

dial component FNris equal to FNsinu. We can now write

Newton’s second law for components along the raxis

(Fnet,rmar) as

. (6-23)

We cannot solve this equation for the value of ubecause it

also contains the unknowns FNandm.

Vertical calculations: We next consider the forces and ac-

celeration along the yaxis in Fig. 6-11b. The vertical com-

ponent of the normal force is FNyFNcosu, the gravita-

tional force on the car has the magnitude mg, and the

acceleration of the car along the yaxis is zero. Thus we can

F

:

g

FN sin um

v^2

R

F

:

N

Additional examples, video, and practice available at WileyPLUS

Friction When a force tends to slide a body along a surface, a
frictional forcefrom the surface acts on the body. The frictional force
is parallel to the surface and directed so as to oppose the sliding. It is
due to bonding between the atoms on the body and the atoms on the
surface, an effect called cold-welding.
If the body does not slide, the frictional force is a static
frictional force. If there is sliding, the frictional force is a kinetic
frictional force.

1.If a body does not move, the static frictional force and the
component of parallel to the surface are equal in magnitude,
and is directed opposite that component. If the component
increases,fsalso increases.
2.The magnitude of has a maximum value fs,maxgiven by

fs,maxmsFN, (6-1)

wheremsis the coefficient of static frictionandFNis the magni-

tude of the normal force. If the component of parallel to the

surface exceeds fs,max, the static friction is overwhelmed and the

body slides on the surface.

3.If the body begins to slide on the surface, the magnitude of the
frictional force rapidly decreases to a constant value fkgiven
by
fkmkFN, (6-2)
wheremkis the coefficient of kinetic friction.

Drag Force When there is relative motion between air (or
some other fluid) and a body, the body experiences a drag force
that opposes the relative motion and points in the direction in
which the fluid flows relative to the body. The magnitude of Dis
:

D

:

F:

:fs

f

:

s

F

: f

:

s

f

:

k

f

:

s

F

:

Review & Summary

related to the relative speed vby an experimentally determined

drag coefficientCaccording to

(6-14)

whereris the fluid density (mass per unit volume) and Ais the

effective cross-sectional areaof the body (the area of a cross sec-

tion taken perpendicular to the relative velocity ).

Terminal Speed When a blunt object has fallen far enough

through air, the magnitudes of the drag force and the gravita-

tional force on the body become equal. The body then falls at a

constantterminal speedvtgiven by

(6-16)

Uniform Circular Motion If a particle moves in a circle or a

circular arc of radius Rat constant speed v, the particle is said to be

inuniform circular motion.It then has a centripetal acceleration

with magnitude given by

(6-17)

This acceleration is due to a net centripetal forceon the particle,

with magnitude given by

(6-18)

wheremis the particle’s mass. The vector quantities and are

directed toward the center of curvature of the particle’s path. A

particle can move in circular motion only if a net centripetal

force acts on it.

F

:

:a

F

mv^2

R

,

a

v^2

R

.

a:

vtA

2 Fg

CrA

.

Fg

: D

:

:v

D^12 C Av^2 ,