Centripetal Force
Angular acceleration vs. Centripetal acceleration


Angular acceleration arises when a
spinning object is speeding up or
slowing down its spin. (Example: A car
tire as a car is speeding up)
Centripetal acceleration has to do with
the fact that the tangential velocity of
an object is always changing even if the
speed of the spin is not. (Example:
Earth’s orbit around the sun)
The centripetal force (which is the net
force causing circular motion) always
points towards the center of the circle.
Circular Motion
An object moving in circular motion
is inherently accelerating (centripetal
acceleration).
"F = mac
It must therefore have a net force
continually acting on it. We call this net
force the Centripetal Force.
!
Q
A car rounds a curve while
maintaining a constant speed. Is
there a net force on the car as it
rounds the curve?
(Q7-2)
A car rounds a curve while
maintaining a constant speed. Is
there a net force on the car as it
rounds the curve?
(Q7-2)
1.
1.
No – its speed is constant
No – its speed is constant
2. Yes
2. Yes
3. It depends on the sharpness of the curve and the speed of the car
3. It depends on the sharpness of the curve and the speed of the car
4. No – its curve isn’t horizontal.
4. No – its curve isn’t horizontal.
1
It’s hearsay…
You may have heard that the force that pushes you
to the right side of the car when making a left hand
turn is called the centrifugal force.
This is a myth that must be busted!
The centriFugal Force is a Fictitious Force!
2
Centripetal Acceleration: Equation
2
2
" d % " 2(r %
$ ' $
'
v 2 # t & # T & 4( 2 r
ac = =
=
= 2
r
r
r
T
Centripetal Acceleration: Period



The centripetal acceleration has to do
with the period of the spin.
Period (T): the time it takes for one
object to complete one full rotation or
revolution
Remember one full revolution or
rotation is either 360o or 2π radians
4" 2 r
ac = 2
T
!
Centripetal acceleration = 4 x pi squared x the radius
divided by the period squared
!
Linear/tangential velocity
(m/s)
Example:

A child 1.35 m from the center of a
merry-go-round is moving with a period
of 2 minutes. What is her centripetal
acceleration in m/sec2?
4" 2 r= 4" (1.35)
ac = 2 ) # 60 sec &,
T +2 min%
(.
2
2
*
!
!
= 0.0037
m
s2
$ 1min '-
!
ΣF = m ac
F centripetal = F friction
On a day when the roads are icy, the
coefficient of friction between a typical
tire and the pavement is 0.200. What is
the maximum speed a car can make a turn
of radius 200.m?
mv 2
= µ mg
R
v2 = µ g R
v = µ g R = (0.2)(9.8)(200)
v = 19.8 m / s
3
Roads can be banked,
to reduce the need
for friction to make a
turn.
N
Take a cross
section of
the road…
If a car of mass, m, is
going to drive at a
linear velocity, v, on a
road. At what angle,
θ, must the road be
banked?
N
N cos θ
X
N sin θ
Banked turns
N cos θ
X
θ
N sin θ
θ
"Fy = 0
θ
From the y axis..
!
ΣFy = 0
r
r
N cos" ! mg = 0
θ
mg
From the x axis..
r
r
mg
N=
cos !
& mg #
mv 2
$$ cos ' !! sin ' = R
%
"
r
v2
g tan ! =
R
v2
tan ! = r
gR
A girl on a merry-go-round is standing 5
m from the center, and holds a yoyo. If
the string makes an angle of 3o with the
vertical, how fast is she going?
Note that the
banking angle
does not depend
upon mass.
r
"F
y
Tcos3
v2
tan ! = r
gR
mg
r
mv 2
N sin ! =
R
r
T
X
3o
=0
r
r
T cos # ! mg = 0
r mgr
T=
cos!
Tsin3
mg
4
A girl on a merry-go-round is standing 5
m from the center, and holds a yoyo. If
the string makes an angle of 3o with the
vertical, how fast is she going?
r
r
o
! F = T sin 3
x
3o
mv 2
r
v2
R g tan 3o =
R
mv 2
=
r
mg
sin 3o =
cos 3o
R
Tsin3
A daredevil drives a motorcycle
on a loop-the-loop track. What is the
slowest he can go and still make it?
(R = 4.0m)
4m
r
r
v = Rg tan 3o
r
v = 5(9.8) tan 3o
= 1.6 m / s
Remember, that mg
is constant, but the
normal force varies
with the speed.
Of course the tricky part is at the top.
Radius = 4.0 m
"F = ma
c
r r
mg + N = mac
0
N
mg
N
mg
!
!
r mv 2
mg =
R
r
2
v = gR
r
v = gR = (9.8)(4)
= 6.26 m / s
HIDDEN
So why aren’t
real loop-theloop tracks
circular?
A ball moves at a constant speed along a horizontal
circle inside a friction-free cone. The weight of the ball is mg.
What other significant force(s) act on the ball? What is the
direction of the net force on the ball?
N
mg
This is the unbalanced
force that is the
centripetal force.
5
A rider in a amusement park ride finds
herself stuck with her back to the carpeted wall.
Which diagram correctly shows the forces acting
on her?
Lift
1
2
mg
3
4
5
Floor
Q
A rider in a amusement park ride finds
herself stuck with her back to the carpeted wall.
Which diagram correctly shows the forces acting
on her?
Friction
Q7-4
Normal Force
1
2
3
4
5
(Centripetal Force)
Floor
mg
6
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