Lecture 9
Ch6. Friction and Centripetal Force
University Physics: Mechanics
Dr.-Ing. Erwin Sitompul
http://zitompul.wordpress.com
Homework 7: Coin On A Book
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 μs between the coin and
the book?
Hint: Draw the free-body diagram of the coin first.
Erwin Sitompul
University Physics: Mechanics
9/2
Solution of Homework 7: Coin On A Book
Forces along the y axis:
Fnet,y  may
FN  Fg cos  0• Why zero?
FN  Fg cos
FN  mg cos 
Forces along the x axis:
Fnet,x  m ax
Fg sin   fs  0• Why zero?
Fg sin   s FN  0
mg sin   s mg cos  0
sin 
h
 tan  
s 
cos 
d
So, the coefficient of static friction is:
s  tan13  0.231
Erwin Sitompul
University Physics: Mechanics
9/3
Example: Blue Block
→
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
μk = 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?
Erwin Sitompul
University Physics: Mechanics
9/4
Example: Blue Block
Forces along the y axis:
Fnet,y  may
FN  Fy  Fg  0
FN  mg  F sin 
Forces along the x axis:
Fnet,x  m ax
Fx  fk  ma
F cos  k FN  ma
F
F


a  cos   k  g  sin  
m
m


Erwin Sitompul
• What θ gives the
maximum value of a?
• da/dθ = 0
University Physics: Mechanics
9/5
Example: Blue Block
If a is given by
F
F


a  cos   k  g  sin  
m
m


then, the derivative of a with respect to θ is
da
F
F
  sin   k cos   0
d
m
m
tan   k
  tan 1 k
 tan 1 (0.4)
 21.80
Erwin Sitompul
University Physics: Mechanics
9/6
Example: Two Blocks
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.
Erwin Sitompul
University Physics: Mechanics
9/7
Example: Two Blocks
→
TW
→
TB
Block B
→
TB
→
FgB
Knot
→
TW
→
TA
→
TA
→
FNB
→
fs,max
Wall
Block A
→
FgA
→
TW
→
fs,max
Knot
→
FgA
Erwin Sitompul
University Physics: Mechanics
9/8
Example: Two Blocks
Forces along the y axis:
Fnet,y  0
TWy  FgA  0
TW sin   mA g
→
TW
TWy
θ
→
fs,max
Knot
TWx
Forces along the x axis:
Fnet,x  0
TWx  fs,max  0
TW cos  s FNB
TW cos  s mB g
mA g s mB g

sin 
cos 
sWB
WA

sin  cos 
→
FgA
WA  sWB tan 
 (0.25)(711) tan 30
 102.624 N
Erwin Sitompul
University Physics: Mechanics
9/9
Example: Multiple Objects
A block of mass m1 on a rough, horizontal surface is connected
to a ball of mass m2 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 μk.
Find the magnitude of acceleration of the two objects.
Erwin Sitompul
University Physics: Mechanics
9/10
Example: Multiple Objects
→
FN
Fy
T
Forces in m2
m2
→
Fg2
→
fk
m1
→
Fg1
→
Fnet,y  m2 a2 y
T  Fg2  m2a
T  m2 ( g  a)
F
θ
→
T
→
Fx
Forces in m1
Fnet,x  m1a1x
Fx  T  f k  m1a
F cos  T  k FN  m1a
T  F cos  m1a  k FN
Fnet,y  0
Fy  FN  Fg1  0
FN  Fg1  Fy
FN  m1g  F sin 
Erwin Sitompul
University Physics: Mechanics
9/11
Example: Multiple Objects
T  m2 ( g  a)
T  F cos  m1a  k FN
FN  m1g  F sin 
m2 ( g  a)  F cos  m1a  k (m1g  F sin  )
m1a  m2a  F cos  k F sin   k m1g  m2 g
(m1  m2 )a  F (cos  k sin  )  (k m1  m2 ) g
F (cos   k sin  )  ( k m1  m2 ) g
a
m1  m2
Erwin Sitompul
University Physics: Mechanics
9/12
Example: Trio Blocks
When the three blocks in the figure below are released from rest,
they accelerate with a magnitude of 0.5 m/s2. Block 1 has mass
M, block 2 has 2M, and block 3 has 2M.
What is the coefficient of kinetic friction between block 2 and the
table?
Erwin Sitompul
University Physics: Mechanics
9/13
Example: Trio Blocks
a
Forces in m1
Fnet,y  m1a1 y
a T1  Fg1  Ma
T1  M ( g  a)
a
→
FN
→
T1
→
T1
m1
→
fk
→
Fg1
Erwin Sitompul
→
T2
→
T2
m2
→
Fg2
m3
→
Fg3
Forces in m2
Fnet,x  m2 a2 x
T2  T1  fk  2Ma
T2  T1  k FN  2Ma
Fnet,y  m2 a2 y
FN  Fg2  0
FN  2Mg
Forces in m3
Fnet,y  m3a3 y
T2  Fg3  2M (a)
T2  2M ( g  a)
University Physics: Mechanics
9/14
Example: Trio Blocks
T1  M ( g  a)
T2  T1  k FN  2Ma
FN  2Mg
T2  2M ( g  a)
 2M ( g  a)   M ( g  a)  k  2Mg   2Ma
k  2Mg    2M ( g  a)    M ( g  a)   2Ma
2M ( g  a)    M ( g  a)   2Ma

k 
Mg  5Ma
k 
2 Mg
g  5a

2g
(9.8)  5(0.5)

2(9.8)
 0.372 m s 2
Erwin Sitompul
2Mg
University Physics: Mechanics
9/15
Homework 9
New
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 μs, between A and the table is 0.20.
(b) Block C suddenly is lifted off A. What is the acceleration of
block A if μk between A and the table is 0.15?
Erwin Sitompul
University Physics: Mechanics
9/16