1. 6 Friction

2. Objectives Students must be able to
Utilize theory of dry friction
Describe theory of dry friction
Describe physical meanings of frictional effects
Describe and differentiate between static and kinetic coefficients of friction
Describe the angles of frictions
Add friction into the analyses of objects and structures in equilibrium

3. Objectives Students must be able to
Describe and analyze machines with frictions
Wedges
Threads, screws
Belts
Disks and clutches
Collar, pivot, thrust and journal bearings
Outline rolling resistance
Describe the physical meanings of rolling resistance
Differentiate between frictions and rolling resistance

4. Topic in textbook Section A: Frictional Phenomena
Characteristics, theory, coefficient of friction, angle of friction
We will study this Part first.
Section B: Applications
Wedges
Screws
Journal Bearings
Thrust Bearings; Disk friction
Flexible Belts
Rolling Resistance

5. Magnitude: friction’s magnitude limitation will be discussed later
Dry Friction
Force of resistance acting on a body which prevents or retards slipping of the body relative to a surface with which it is in contact.
Friction exists?
roughnesses of the contacting surfaces.
Magnitude: friction’s magnitude limitation will be discussed later
Direction: tangent to the contacting surface and
opposed to the relative motion or tendency for motion
Line of Action (Point of application): contact surface

6. Friction Model y x Frictional force F increases with force P
Equilibrium y x
In equilibrium W
a/2
a/2
FBD is correct?
modeling P h F
The DN at right side is supporting
force more than its left side. x N N
The object is toppling
(not in equilibrium)
If x > a/2 ?
Frictional force F
increases
with force P
Slipping and/or
Tipping Effect
toppling
Slipping
The application point (x) of N
x-limit
F-limit

7. Motion Slipping / Sliding
a/2
a/2
Motion P h F x N N
Slipping / Sliding
Relative sliding (translation motion) between two surfaces
Toppling / Tippling
Fall over (rotation) about the edge
Topple, tipping, rolling, tumble, trip

8. (on the verge of motion)
Experiment for determining Friction mg P P m
FBD F N
“impending motion”’
(on the verge of motion)
F=P
Static friction (no motion) F P
Object at rest (no motion)
: coefficient of static friction Fk
= kN
Kinetic friction (motion)
Object with motion (steady state)
: coefficient of kinetic friction
Fs:max
= sN
: constant on 2 certain contacting surfaces

9. Angle of Friction k = arctan(k) = angle of kinetic friction
Not depend on N
not depend on P,v,a
k = arctan(k) = angle of kinetic friction
(object in motion) mg P F  N R
s = arctan(s) = angle of (max) static friction
(object at rest)

10. Dry Friction Characteristics
Frictional force acts tangentially to the contacting surfaces, opposing the relative or tendency for motion.
Fs is independent of the area of contact, provided that the normal pressure is not very low nor great enough for deformation of the surfaces.
In equilibrium:
Impending slipping: f = fs
Slipping: f = fk
Very low velocity: fk » fs

11. Static Friction Typical Values
Dry Friction
Impending Motion
Static Friction Typical Values
Contact Materials μs
Metal / ice – 0.05
Wood / wood – 0.70
Leather / wood – 0.50
Leather / metal – 0.60
Aluminum / Aluminum – 1.70

12. Possibility of toppling?
Sample 6/1 Determine the maximum angle  which the adjustable incline may have before the block of mass m begins to slip. The coefficient of static friction between the block and the inclined surface is s.
W=mg y x
“Impending
Slip”:
H/2 x F N
Ans
(for slipping)
Three force member
Possibility of toppling?
W=mg
3 eq. , 3 unknowns

13. 6/125 A uniform block of mass m is at rest on an incline z
6/125 A uniform block of mass m is at rest on an incline z. Determine the maximum force P>0 that can be applied to the block in the direction shown before slipping begins. The coefficient of static friction between the block and the incline is z y x q mg y P N q
At this max P, object is about to move at which direction?

14. Example Friction 2 #1 Will this crate slide or topple over?
Dry Friction
Example Friction 2 #1
Will this crate slide or topple over?

15. Two possibilities 4 Unknowns: P, F, N, x 1) about to slip
1st Eq.
4 Unknowns: P, F, N, x
2nd Eq.
Two possibilities
3rd Eq.
1) about to slip
2) about to tip
4th Eq.
4th Eq.
It’s time consuming,
Better to know it
exactly
Check
condition:
Check
condition:

16. The block of mass m is homogeneous, moving at a constant velocity
The block of mass m is homogeneous, moving at a constant velocity. The coefficient of kinetic friction is
Determine
a) the greatest value that h may have so that the block will not tip over
b) The location of point C on the bottom face of the block through with the resultant of the friction and normal force act if h = H/2.

17. The block of mass m is homogeneous, moving at a constant velocity
The block of mass m is homogeneous, moving at a constant velocity. The coefficient of kinetic friction is
Determine
a) the greatest value that h may have so that the block will not tip over
Moving
With no acceleration
Solution 1
= ?
on the verge of tipping
Moving
The box is in Equilibrium (no acceleration) P
Three force member in Statics
h=?
Concurrent at one point OR parallel mg F F N N
Ans
on the verge of tipping

18. Ans The block of mass m is homogeneous.
The block is moving at a constant velocity.
Determine
b) The location of point C on the bottom face of the block through with the resultant of the friction and normal force act if h = H/2.
Moving
with constant velocity
Solution 1
=H/2
C =? N
Moving
The box is in Equilibrium (no acceleration)
Three force member P
Concurrent at one point OR parallel mg
h=H/2 F x
Ans N

20. + Dry Friction’s Problem Kinetic motion is known. Inequality:
hard to deal
Dry Friction’s Problem
static friction + or
Equilibrium eq.
kinetic friction
fricitional eq.
Friction
Kinetic motion is known.
Moving at constant vel. +
Kinetic friction
Equilibrium eq.

21. + If In static equilibrium Static equilibrium? Assume:
Static friction
not assuming
“impending motion”
In static equilibrium
friction can support equilibrium
Static
equilibrium?
No. of unknown
must =
No. of Equilibrium eq.
Assume:
static equilibrium.
Equilibrium eq.
Solve for F (and also N)
using only equilibrium eq.
Only equilibrium eq. can be used to determine unknown values.
Assumption Checking
Equilibrium eq.
Friction is determined by equilibrium eq’s.
Static friction If
O.K.
However,
F must <= mN
Assumption is not true!
Motion occurs. F = kN
Static friction

22. + Find min q to hold equilibrium Find min P to initiate the motion
total unknown
<
equilibrium +
frictional eq. +
Equilibrium eq.
Static friction
Find min q to hold equilibrium
Find min P to initiate the motion
Impending motions occur at both point in the same time.
impending motions do not occur in both point at the same time.
Both can be used together to determine unknown values.
several possibilities to slip
total unknown =
equilibrium +
frictional eq.
Equilibrium eq.
Assumption needs to be make and
that assumption should be checked later.
static friction

23. 500N and
100N
Equilibrium?
Sample 6/3 Determine the friction force acting on the block shown if P = 500N and P = 100N. The block is initially at rest.
Equilibrium state is unknown
Assume: Body in equilibrium
(not assuming the impending motion)
2 Eq ,
2 unknown
impending motion
OK !
P=500N
Friction must be enough to maintain equilibrium
Assumption Checking:
Correct?
P=100N
Object not in Equilibrium
Contradict!
Not valid!
still valid!

24. Dry Friction
Example Hibbeler Ex 8-1 #1
The uniform crate has a mass of 20 kg. If a force P = 80 N is applied to the crate, determine whether it remains in equilibrium. The coefficient of static friction = 0.3.
Equilibrium State
is not known

25. Example Hibbeler Ex 8-1 #2 Assume: Equilibrium |x| < 0.4 m
(physical boundary
of toppling)

26. Dry Friction
Example Bedford Ex 9.5 #1
Suppose that a = 10° and the coefficient of friction between the surface of the wedge and the log are ms = 0.22 and mk = Neglect the weight of the wedge.
If the wedge is driven into the log at a constant rate by vertical force F, what are the magnitude of the normal forces exerted on the log by the wedge?
Will the wedge remain in place in the log when the force is removed?

27. Neglect the weight of the wedge.
= 10° ms = 0.22 and mk = 0.20.
Neglect the weight of the wedge.
Moving down at constant rate
symmetric/mirror
concept

28. Think of minimum friction coefficient that still “self-lock” the wedge
= 10° ms = 0.22 and mk = 0.20.
Neglect the weight of the wedge.
Will the wedge remain in place in the log when the force is removed?
Impending Motion
(On the verge of slipping)
Think of minimum friction coefficient that still “self-lock” the wedge
symmetric/mirror
concept

29. Dry Friction
Example Bedford 9.20 #1
The coefficient of static friction between the two boxes and between the lower box and the inclined surface is ms. What is the largest angle a for which the lower box will not slip.

30. The coefficient of static friction between the two boxes and between the lower box and the inclined surface is ms. What is the largest angle a for which the lower box will not slip.
Impending Motion
at same time ?

31. Bedford 9.30 (ex)
The cylinder has weight W. The coefficient of static friction between the cylinder and the floor and between the cylinder and the wall is ms. What is the largest couple M that can be applied to the stationary cylinder without causing it to rotate.
Impending Motion
at same time
Impending
rotation

32. Bedford 9.33 (Ex)
The disk of weight W and radius R is held in equilibrium on the circular surface by a couple M. The coefficient of static friction between the disk and the surface is ms. Find that the largest value M can have without causing the disk to slip.

33. Dry Friction
Example Bedford #1
Each of the uniform 1-m bars has a mass of 4 kg. The coefficient of static friction between the bar and the surface at B is 0.2. If the system is in equilibrium, what is the magnitude of the friction force exerted on the bar at B.

34. Example Bedford #2
Dry Friction

35. Example Bedford #3
Dry Friction

36. Dry Friction
Example Bedford #4
symmetry?

38. Dry Friction
Example Hibbeler Ex 8-3 #1
The rod with weight W is about to slip on rough surfaces at A and B. Find coefficient of static friction.
Direction of N?

39. Example Hibbeler Ex 8-3 #2
Dry Friction

40. Example Hibbeler Ex 8-3 #3
Dry Friction

41. 6/9 For a 20 jaw opening, what is the minimum coefficient of static friction between the jaws and the tube which will enable tongs to grip the tube with out slipping
Solution 1 y x

42. 6/9 For a 20 jaw opening, what is the minimum coefficient of static friction between the jaws and the tube which will enable tongs to grip the tube with out slipping
Solution 2 y
The object is in equilibrium
Two force systems x

44. Sample 6/5 Find the maximum value which P may have before any slipping takes place.?
Possible ways to slip
1) Middle object is going to move lonely.
2) Middle + buttom object is going to move together. y x T P

45. Assume: 1) Middle object is going to move lonely.
Friction F3 can support the equilibrium of 40-kg object
Check the Assumption
Assume:
1) Middle object is going to move lonely.
Impending motion y x T
5 unknowns
3 equilibrium eq
2 friction eq P
Impossible, the assumption is wrong

46. Assume: Ans 2) Middle + bottom object is going to move together.
F2 can support the 50-kg and 40-kg as one body
Assume:
2) Middle + bottom object is going to move together.
Impending motion y x T P OK
Ans

47. Assume: 3) 40-kg object moves lonely --– possible?x T
Equation without unknown
(little chance to be true) P

48. 8-61 The uniform 50-N slender rod rests on the top center of the 20-N block. Determine the largest couple moment M which can be applied to the rod without causing motion of the rod. y x +
Motion Possibility?
1. Slip at C, B
2. Slip at C, A
3. Slip at A, B ?
4. other… x

49. 8-61 The uniform 50-N slender rod rests on the top center of the 20-N block. Determine the largest couple moment M which can be applied to the rod without causing motion of the rod. y x +
Assume
2. Slip at C, A
8 unknowns,
6+2 equation.
Don’t solve 8 eq paralleling!
Solve only 6 eq. excluding
Assumption
Checking x

50. 8-56 The uniform 50-N slender rod rests on the top center of the 20-N block. Determine the largest couple moment M which can be applied to the rod without causing motion of the rod. y x +
Assume
2. Slip at C, A
Block is tipping
= 0.12 m
= 62.5 N
= 9.5 N-m
= 42.5 N
= 25 N
Assumption
Checking
= 7.5 N
Correct?

51. 8-61 The uniform 50-N slender rod rests on the top center of the 20-N block. Determine the largest couple moment M which can be applied to the rod without causing motion of the rod. y x +
Possibility?
1. Slip at C, B
2. Slip at C, A
3. Slip at A, B ?
4. C-Slip, x-limit
4. other…
5. B-Slip, x-limit x
6. A-Slip, x-limit

52. 8-61 The uniform 50-N slender rod rests on the top center of the 20-N block. Determine the largest couple moment M which can be applied to the rod without causing motion of the rod. y x +
Assume
4. C-Slip, x-limit
8 unknowns,
6+2 equation.
Don’t solve 8 eq paralleling!
Solve only 6 eq. excluding
Assumption
Checking x

53. 8-61 The uniform 50-N slender rod rests on the top center of the 20-N block. Determine the largest couple moment M which can be applied to the rod without causing motion of the rod. y x +
Assume
4. C-Slip, x-limit
= N
= 5.78 N-m
= N
Ans
21.21 <=0.6*43.64=26.18
= N
Assumption
Checking
21.21 <=0.4*63.64=25.46
= 6.36 N

55. 6/28 Determine the minimum angle at which slipping does not occur at either contact point A or C. The coefficient of static friction at both A and C is (Note: you should assume the mass of each bar – why?)

56. 6/28 Determine the minimum angle at which slipping does not occur at either contact point A or C. The coefficient of static friction at both A and C is 0.05
2 force
A or C is going to slip first?

57. However value P are, BC will not slip at C.
P applies at
pin B, AB or BC?
8-61 The end C of the two-bar linkage rests on the top center of the 50-kg cylinder. Determine the largest vertical force P which can be applied at B without causing motion. Neglect the mass of the bars. P
P applies at pin B
2 force
From solving
equilibrium eqs.
2 force
However value P are, BC will not slip at C.

58. Max P without causing motion
8-61 The end C of the two-bar linkage rests on the top center of the 50-kg cylinder. Determine the largest vertical force P which can be applied at B without causing motion. Neglect the mass of the bars.
Max P without causing motion
Slipping or tipping? x
Slipping?
Tipping (x=0.05)?
Ans
minimum value

60. (1) friction force F acting at each side when P=0
6/23 The 10-kg cylinder is initially placed at V-Block. If , find
(1) friction force F acting at each side when P=0
(2) The P to start sliding the cylinder up y
Assume: body in equilibrium
(incomplete) y x y P z
Assumption Checking OK
No tendency to move in z-axis direction !! =N
(2) Find P min to move
impending motion
N = 601

61. Think of Equilibrium to find out the relation of F and N
Determine the minimum coefficient of static friction at each point of contact so that the pile does not collapse. A B
weight W, radius r C D
Think of Equilibrium to find out the relation of F and N

62. Yes: C not moving No, C is moving. Posibility? 1) A B and C don’t move
6/108 Describe what should happen
Posibility?
1) A B and C don’t move
2) A move, B and C don’t move
3) B move (so do A), C don’t move
Many possibilities?
4) C move (so do A and B)
Yes: C not moving
No, C is moving.
C may or may not moving depends on friction at A and B.
C not moving
B may or may not moving depends on friction at A.
A may or may not moving depends on friction at A. y x
= 56.85
= 75.81
= 99.50 = =
Always in equilibrium,
however A B C are moving or not. =

63. So, A and B move and C don’t move
6/108 Describe what should happen
Assume
1) A and B don’t move, C don’t move = = y x = =
Assumption Checking
Assumption is wrong !
So, A and B move and C don’t move y x

64. Recommend Problem
6/16 6/125 6/23 6/41 6/40 6/37 6/108