Chapter(6(
Circular(mo0on(and(resis0ve(forces(

!!v!
v2
∑F = macentripetal = m r !!(towards!center!of!circle)
!

!!a!
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!!v!
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NOTE:((It(is(the(sum$of$forces(ac0ng(on(the(
object(gives(a(net(radially(inward(force(for(
uniform(circular(mo0on.(
Example(6C1:(
(
A(1(kg(ball(is(rota0ng(on(a(flat(surface(with(a(
circle(of(radius,(R.((It(is(connected(to(a(2(kg(
block(that(is(suspended(at(the(center(of(the(
table.((If(the(speed(of(the(ball(is(1.8(m/s,(
what(is(the(radius(of(the(circle?(
rota0on(
Example(6C2:((Flat(curve(
(
A(car(enters(a(curve(with(a(radius,(R(=(75(m.((If(the(coefficient(of(sta0c(
fric0on(between(the(0res(and(road(is(0.2,(what(is(the(maximum(safe(
speed(to(enter(the(curve?(
QuickCheck(8.2(
A toy car moves around a circular track at constant
speed. It suddenly doubles its speed — a change of a
factor of 2. As a result, the centripetal acceleration
changes by a factor of
A.
B.
C.
D.
E.
1/4.
1/2.
No change since the radius doesn t change.
2.
4.
Slide(8C38(
QuickCheck(8.4(
An(ice(hockey(puck(is(0ed(by(a(string(to(
a(stake(in(the(ice.(The(puck(is(then(
swung(in(a(circle.(What(force(is(
producing(the(centripetal(accelera0on(
of(the(puck?(
(
A. (((((Gravity(
B. (((((Air(resistance(
C. (((((Fric0on(
D. (((((Normal(force(
E. (((((Tension(in(the(string(
Slide(8C46(
Example(6C3:(Conical(pendulum(
(
A(mass,(m,(hangs(at(the(end(of(a(string(of(
length,(L(=(1.2(m.((The(mass(is(rota0ng(in(a(
conical(pendulum(with(θ(=(22˚.((Find(the(
velocity(of(the(mass.(
θ(
r(
L(
Ver0cal(circles:((Hills,(roller(coaster(
r(
At(the(bobom:(
acentripetal(
N$
(
mg$
v$
v2
Fr = mac = m = N − mg
∑
r
!
Ver0cal(circles:((Hills,(roller(coaster(
r(
v$
N$$$$$$$mg$
At(the(top:(
acentripetal(
v2
Fr = mac = m = N + mg
∑
r
!
Example(6C4:(Circular(mo0on(and(gravity(2(
(
A(250(kg(roller(coaster(car(enters(a(20(m(
diameter(loop(at(15(m/s.((Find(the(normal(
force(on(the(car(and(the(minimum(velocity(
the(car(must(have(at(the(top(of(the(loop(in(
order(to(remain(on(the(track.(
v$
r(
N$$$$$$$mg$
acentripetal(
At(the
iClicker(ques0ons(–(forces(in(circular(mo0on(
QuickCheck(8.5(
A car turns a corner on a
banked road. Which of the
diagrams could be the car’s
free-body diagram?
Slide(8C56(
QuickCheck(8.9(
A physics textbook swings back
and forth as a pendulum. Which is
the correct free-body diagram
when the book is at the bottom
and moving to the right?
Slide(8C78(
QuickCheck(8.11(
A roller coaster car does a loopthe-loop. Which of the free-body
diagrams shows the forces on
the car at the top of the loop?
Rolling friction can be neglected.
Slide(8C82(
Resis0ve(forces(
1
Drag:( D = C ρ Av 2
2
C !$!drag!coefficient
ρ !$!density!of!medium!(air!=!1.2!kg/m3 )
A!$!cross$sectional!area
!v!$!velocity
dv
= D − mg
dt
dv C ρ Av 2
⇒a= =
− g!!!(differential!equation!in!v)
dt
2m
2mg
In!the!limit:!a→ 0!!!!!!!!!!!!!!!v terminal =
C ρA
!
∑ F = ma = m
When an object moves through the air, the
magnitude of the drag force on it
A.  Increases as the object s speed increases.
B.  Decreases as the object s speed increases.
C.  Does not depend on the object s speed.
A(sky(diver((with(mass(m)(jumps(from(a(plane(and(quickly(reaches(
a(terminal(velocity.(As(the(sky(diver(is(falling(with(this(terminal(
velocity,(what(is(the(net(force(on(the(sky(diver?(
((
a.  The(net(force(is(zero.(
b.  mg(directed(downwards.(
c.  mg(directed(upwards.(
d.  We(do(not(have(enough(informa0on(to(know.(
((