University Physics: Mechanics
Ch6. Friction
, Drag, and Centripetal Force
Lecture
10
Dr.

Ing. Erwin Sitompul
http://zitompul.wordpress.com
2012
10/
2
Erwin Sitompul
University Physics: Mechanics
The figure below shows a coin of mass
m
at rest on a book that
has been tilted at an angle
θ
with the horizontal. By
experimenting, you find that when
θ
is increased to 13
°
, the coin
is on the verge of sliding down the book, which means that even
a slight increase beyond 13
°
produces sliding.
What is the coefficient of static friction
μ
s
between the coin and
the book?
Hint
: Draw the free

body diagram of the coin first.
Homework 8: Coin On A Book
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Erwin Sitompul
University Physics: Mechanics
Forces along the
y
axis:
Forces along the
x
axis:
•
Why zero?
•
Why zero?
So, the coefficient of static friction is:
Solution of Homework 8: Coin On A Book
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Erwin Sitompul
University Physics: Mechanics
An object is kept in rest on an inclined surface. The angle
θ
is
26
°
, which is greater than the critical angle
θ
c
(
μ
s
= tan
θ
c
).
Upon release, the object directly move and slide down to the
bottom. It requires 4.29 s to reach the bottom, which is 18 m
away from the initial point.
Determine the coefficient of kinetic friction
μ
k
between the object
and the surface.
Virtual Experiment: Determining
μ
k
θ
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Erwin Sitompul
University Physics: Mechanics
A block of mass
m
= 3 kg slides along a floor while a force
F
of
magnitude 12 N is applied to it at an upward angle
θ
. The
coefficient of kinetic friction between the block and the floor is
μ
k
= 0.4. We can vary
θ
from 0 to 90
°
(with the block remains on
the floor.
What
θ
gives the maximum value of the block’s acceleration
magnitude
a
?
→
Example: Blue Block
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Erwin Sitompul
University Physics: Mechanics
Forces along the
y
axis:
Forces along the
x
axis:
•
What
θ
gives the
maximum value of
a
?
•
da
/
dθ
= 0
Example: Blue Block
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Erwin Sitompul
University Physics: Mechanics
then, the derivative of
a
with respect to
θ
is
If
a
is given by
Example: Blue Block
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Erwin Sitompul
University Physics: Mechanics
Block
B
in the figure below weighs 711 N. The coefficient of
static friction between block and table is 0.25; angle
θ
is 30
°
.
Assume that the cord between
B
and the knot is horizontal.
Find the maximum weight of block
A
for which the system will
be stationary.
Example: Two Blocks
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Erwin Sitompul
University Physics: Mechanics
f
s,max
→
Block B
Block A
Knot
T
B
→
T
A
→
T
B
→
T
A
→
Wall
T
W
→
T
W
→
F
gA
→
Knot
T
W
→
f
s,max
→
F
gA
→
F
NB
→
F
gB
→
Example: Two Blocks
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10
Erwin Sitompul
University Physics: Mechanics
Knot
T
W
→
f
s,max
→
F
gA
→
T
W
x
T
W
y
Forces along the
y
axis:
Forces along the
x
axis:
θ
Example: Two Blocks
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Erwin Sitompul
University Physics: Mechanics
A block of mass
m
1
on a rough, horizontal surface is connected
to a ball of mass
m
2
by a lightweight cord over a lightweight,
frictionless pulley as shown in the figure below.
A force of magnitude
F
at an angle
θ
with the horizontal is
applied to the block as shown and the block slides to the right.
The coefficient of kinetic friction between the block and surface
is
μ
k
.
Find
the magnitude of acceleration of the two objects.
Example: Multiple Objects
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Erwin Sitompul
University Physics: Mechanics
f
k
→
m
1
m
2
T
→
T
→
F
g2
→
F
N
→
F
g1
→
F
→
F
x
F
y
θ
Forces in
m
2
Forces in
m
1
Example: Multiple Objects
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Erwin Sitompul
University Physics: Mechanics
Example: Multiple Objects
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Erwin Sitompul
University Physics: Mechanics
When the three blocks in the figure below are released from rest,
they accelerate with a magnitude of 0.5 m/s
2
. Block 1 has mass
M
, block 2 has 2
M
, and block 3 has 2
M
.
What is the coefficient of kinetic friction between block 2 and the
table?
Example: Trio Blocks
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Erwin Sitompul
University Physics: Mechanics
F
N
→
f
k
→
T
1
→
m
1
T
1
→
F
g1
→
m
2
F
g2
→
T
2
→
a
a
a
m
3
T
2
→
F
g3
→
Forces in
m
1
Forces in
m
3
Forces in
m
2
Example: Trio Blocks
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Erwin Sitompul
University Physics: Mechanics
Example: Trio Blocks
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Erwin Sitompul
University Physics: Mechanics
The Drag Force and Terminal Speed
A
fluid
is anything that can flow
–
generally a gas or a liquid.
When there is a relative velocity between a fluid and a body
(either because the body moves through the fluid or because
the fluid moves past the body), the body experiences a
drag
force
D
that opposes the relative motion.
→
Here we examine only cases in which air is the fluid, the body
is blunt rather than slender, and the relative motion is fast
enough so that the air becomes turbulent (breaks up into
swirls) behind the body.
In such cases, the magnitude of the drag force is related to
the relative speed by an experimentally determined
drag
coefficient
C
according to
ρ
: air specific density
A
: effective cross

sectional area of the body
C
: drag coefficient
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Erwin Sitompul
University Physics: Mechanics
The Drag Force and Terminal Speed
10/
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Erwin Sitompul
University Physics: Mechanics
The Drag Force and Terminal Speed
When a blunt body falls from rest through air, the drag force D
is directed upward. This upward force
D
opposes the
downward gravitational force
F
g
on the body.
→
→
If the body falls long enough,
D
eventually equals
F
g
. This
means that
a
= 0, and so the body’s speed no longer
increases. The body then falls at a
constant speed
, called the
terminal speed
v
t
.
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Erwin Sitompul
University Physics: Mechanics
The Drag Force and Terminal Speed
•
Cyclists and downhill
speed skiers try to
maximize terminal
speeds by reducing
effective cross

sectional area
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21
Erwin Sitompul
University Physics: Mechanics
If a falling cat reaches a first terminal speed of 97 km/h while it
is wrapped up and then stretches out, doubling
A
, how fast is it
falling when it reaches a new terminal speed
Example: Falling Cat
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22
Erwin Sitompul
University Physics: Mechanics
A raindrop with radius
R
=
1.5 mm falls from a cloud that is at
height
h
=
1200 m above the ground. The drag coefficient
C
for
the drop is 0.6. Assume that the drop is spherical throughout its
fall. The density of water
ρ
w
is 1000 kg/m
3
, and the density of air
ρ
a
is 1.2 kg/m
3
.
(a)
What is the terminal speed of the drop?
(b)
What would be the drop’s speed just before impact if there
were no drag force?
Example: Raindrop
(a)
(b)
10/
23
Erwin Sitompul
University Physics: Mechanics
Homework 9A
In the next figure, blocks
A
and
B
have weights of 44 N and
22 N, respectively.
(a)
Determine the minimum weight of block
C
to keep
A
from
sliding if
μ
s
, between
A
and the table is 0.20.
(b)
Block
C
suddenly is lifted off
A
. What is the acceleration of
block
A
if
μ
k
between
A
and the table is 0.15?
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Erwin Sitompul
University Physics: Mechanics
Homework 9B
1.
The figure shows a 1.0

kg University Physics book
connected to a 500

g tea mug. The book is pushed up
the slope and reach a speed of 3.0 m/s before being
released. The coefficients of friction are
μ
s
= 0.50 and
μ
k
= 0.20.
(a)
How far will the book slide upwards?
(b)
After the book reaches the highest point, will the
book stick to the surface, or will it slide back down?
2.
In downhill speed skiing, a skier is retarded by both
the air drag force on the body and the kinetic
frictional force on the skis. Suppose the slope angle
is
θ
= 40.0
°
, the snow is dry with a coefficient of
kinetic friction
μ
k
= 0.04, the mass of the skier and
equipment is
m
= 85.0 kg, the cross

sectional area
of the (tucked) skier is
A
= 1.30 m
2
, the drag
coefficient is
C
= 0.150, and the air density is 1.20
kg/m
3
.
(a) What is the terminal speed? (b) If a skier can
vary
C
by a slight amount
dC
by adjusting, say, the
hand positions, what is the corresponding variation
in the terminal speed?
D
f
k
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