unit 3 mm9400 ver1.1(2014)

Friction 3 - 1

Statics & Dynamics (MM9400) Version 1.1

3. FRICTION

Objectives: At the end of this unit, students should be able to: Understand Friction on Horizontal and Inclined Planes

Define the force of "friction." Sketch the free body diagram of a block moving or tending to move on a horizontal

surface. State the laws of friction. Define: (a) normal reaction (b) coefficient of static friction (c) coefficient of kinetic friction (e) angle of friction Determine analytically the force required to move a body: (a) along a horizontal plane (b) along an inclined plane

Charles Augustin de Coulomb spent 9 years as a military engineer in the West Indies. When the French Revolution began, he retired to France to do scientific research. He experimented on Friction but is best known for his work on electricity and magnetism. He also wrote on structural analysis,

the fracture of beams and columns, the thrust of arches and the thrust of the soil. The SI Unit of electric charge is named after him.

Charles A Coulomb (1736 – 1806)

3 - 2 Friction

Version 1.1 Statics & Dynamics (MM9400)

3.1 Introduction When a body moves or tends to move over a rough surface, there is always some resistance to the motion or tendency to move. This resistance is due to the force of friction. Friction is due to irregularities on the surfaces in contact. We rely on friction between our shoes and the ground to push us forward when walking. Without friction, vehicles would not be able to start and stop, as well as nails and screws would not hold in place. Friction in the moving parts of machinery can be reduced by lubrication. However, we shall confine our study to Dry Friction, i.e. cases where the surfaces in contact are clean, dry and non-lubricated. 3.2 Static Friction & Kinetic Friction Consider a block resting on a rough horizontal surface.

Initially, the only forces acting on the block are its weight, W, and the normal reaction Rn from the contact surfaces. When a force F pulls the block and creates a tendency to slide, friction comes into play in the opposing direction. As the pulling force increases, friction will increase to equal the pull in order to maintain equilibrium. Eventually, when the friction reaches a maximum value Fs ( known as static friction ), the block is at the point of sliding, i.e. at impending motion . At this point, Fs = P.

A pull Q ( >P ) moves the block and the friction

reduces slightly to a value Fk , known as the kinetic friction.

Fig. 3.1 No tendency to move

Fig. 3.2 Tendency to move

W

Rn

P

W

Fs

Rn

Fig. 3.3 Impending motion

Fig. 3.4 Motion in progress

direction of motion

Q

W

Fk

Rn

F

W

F (friction)

Rn

Friction 3 - 3


3.3 Laws of Friction for Dry Surfaces The laws of friction were first discovered by Leonado da Vinci. These same laws were verified by Coulomb in 1781, who further distinguished between static and kinetic friction. 1. Friction tends to oppose motion, i.e. it acts opposite to the direction of motion. 2. The force of friction is parallel to flat surfaces or tangential to curve surfaces in contact. 3. Friction is independent of the area of the surfaces in contact. 4. The static friction Fs is proportional to the normal reaction between the surfaces in contact (this is also known as Coulomb friction, named after the French scientist,

Charles Coulomb). Fs Rn 5. If motion has started, the kinetic friction Fk is proportional to the normal reaction, but

its value is slightly less than the static value. Fk Rn 3.4 Coefficient of Friction

At the point of impending motion, the ratio of the friction to the normal reaction between the two sliding surfaces is effectively constant & known as the coefficient of static friction, s.

i.e. n

s

R

F = s

A high coefficient means the contact surface is rougher than one with a lower coefficient. After motion has begun, the relative motion or rubbing between surfaces smoothen the surfaces slightly; hence the ratio of the kinetic friction to the normal reaction is slightly lower and known as the coefficient of kinetic friction, k.

i.e. n

k

R

F = k

3.5 Steps in Solving Problems Involving Friction

1. Sketch a free body diagram of the body in question. 2. Determine if the body is in equilibrium or not. If it is, resolve all the forces acting on it

into rectangular components; the algebraic sum of components will be zero. 3. The value of the friction force depending on the body’s state of motion: F < sRn ............... (before motion starts)

Fs = sRn ............... (for impending motion)

Fk = kRn ............... (when motion is in progress)

Note: In this chapter, we deal only with problems involving impending motion & uniform

motion, i.e. the forces are in equilibrium. Non-equilibrium cases will be considered in Unit 5 under Newton's 2nd and 3rd laws of motion.

3 - 4 Friction


Example 3.1

A body of mass 5 kg rests on a horizontal surface and the coefficients of static and kinetic friction between the two surfaces are 0.33 and 0.25 respectively. a) What horizontal force is required to start the body moving? b) What is the kinetic friction force? c) If the force is applied at 60 to the horizontal, what is the force required to start the

body moving? (a) (c) Example 3.2 A 100 kg block rests on a 30 kg plate as shown. The force P is just enough to slide the plate from under the block. Given that the coefficient of static friction, s= 0.2,

a) draw separate free body diagrams for the block and the plate; b) determine the force P required.

(a)

s = 0.2

P 30 kg plate

100 kg block s = 0.2

cable

5 kg

5 kg

100 kg block

30 kg plate

Friction 3 - 5


20 kg

M kg

Worked Example ( involving two connected masses )

Given that between the table and the 20 kg block shown, k = 0.2 and s = 0.3, find the

smallest value of M needed to just start motion. Solution Let P be the tension in the cable. For 20 kg block: Fy = 0, Rn - 20 x 9.81 = 0 Rn = 196.2 N Fs = sRn

Fx = 0, P - Fs = 0

P = sRn

= 0.3 (196.2) = 58.86 N For the mass M:

Fy = 0, (M x g) - P = 0 M = 58.86/9.81 = 6 kg

cable smooth pulley

table

FBDs

P

Rn

W = 20(9.81) N

Fs

motion impending

M

Mg

P

3 - 6 Friction


TUTORIAL (Friction on horizontal plane) 3.1 If a horizontal force of 50 N was required to start a 10 kg block moving on a

horizontal surface, what is the value of the coefficient of static friction? (0.51) 3.2 Determine the mass of a block if a horizontal force of 15 N is required to start it

moving on a horizontal surface. The coefficient of static friction is 0.4. (3.823 kg) 3.3 Each of the two blocks in the Fig. Q3.3 has a mass of 10 kg and the coefficients of

friction are as stated. With a free body diagram showing all the forces acting on the lower block, determine the force P required to pull one block from under the other. (See Example 3.2)

(88.3 N )

= 0.3

P

= 0.3

Fig. Q3.3

cable

Friction 3 - 7


3.4 In an automatic material handling operation, metal boxes of mass 1.5 kg each are

pushed one at a time from the bottom of a stack six blocks high as shown in Fig. Q3.4. If the coefficient of static friction is 0.25, determine the horizontal force (P) required. (40.5 N)

3.5 A 20 kg block rests on a horizontal surface and is attached to a mass of 3.5 kg as

shown in Fig. Q3.5. The pulley is frictionless. The coefficient of static friction between the block and the horizontal surface is 0.2.

a) Prove that the 3.5 kg mass is not heavy enough to start the block moving. b) What is the additional mass that would be needed to start the motion?

(0.5 kg)

20 kg

3.5 kg

cable smooth pulley

Fig. Q3.5

P

Fig. Q3.4

3 - 8 Friction


3.6 A load of 10 tonne is pulled along a horizontal surface at uniform speed by a force P

as shown in Fig. Q3.6. If the coefficient of kinetic friction is 0.2:

a) Sketch a free body diagram of the block b) Find the normal reaction between the block & the surface (87.9 kN) c) Find the kinetic friction force (17.6 kN)

horizontal axis 30°

P

10 tonne

Fig. Q3.6

Friction 3 - 9


3.5 Angle of Friction s Consider a plane inclined at angle to the horizontal (Fig. 3.5). If a body is left resting on the inclined plane, the FBD would be:

When the angle of inclination is increased to s, such that the body is at the point of impending motion (Fig. 3.6), the friction force reaches the limiting value of Fs and thus

where s is known as the Angle of Friction Example 3.3 (Angle of Friction) A body of mass 2 kg rests on a board which is gradually tilted until, at an angle of 25 to the horizontal, the body is at impending motion down the board. a) Determine the coefficient of static friction, s between the body and the board. b) How will the body move if the board is tilted beyond 25o?

Note: Friction F = W sin Rn = W cos

Note: Fs = W sinsRn = W coss

Wcoss

Rn

s s

Fig. 3.6

Fs

Wsins

Friction F

Weight W

Normal reaction Rn

Fig. 3.5

No motion

ss

s

n

s tancosW

sinW

R

F

ss tan

3 - 10 Friction


TUTORIAL 3.7 If the normal and frictional forces between two surfaces which are about to slip are

100 N and 35 N respectively, determine the:

a) coefficient of friction. b) angle of friction. (0.35, 19.3) 3.8 A body of 2 kg rests on a board which is gradually tilted until, at an angle of 27 to

the horizontal, the body is at the point of slipping down the plane (board). Determine

a) the coefficient of static friction, s. (0.51) b) the normal reaction and friction when the body is at the point of slipping

down. (17.5 N , 8.91 N)

Friction 3 - 11


3.9 A truck of mass 6 tonnes is resting on the incline shown in Fig. Q3.9. a) Sketch a free body diagram of the truck. b) If the truck is stationary on the incline and = 10, what is the magnitude of the friction force exerted on it by the road? (10.22 kN)

c) If the coefficient of static friction between the truck’s tyres and the road is 0.6, what is the largest value of for which the truck can remain stationary?

(30.96o)

Fig. Q3.9

3 - 12 Friction


3.6 Further Examples of Friction on Inclined Plane

We can illustrate with several worked examples of motion on an inclined plane. The motion can be impending, or at constant velocity, up or down the incline. Example 3.4 (Impending motion up and down incline, and < s )

A mass of 100 kg rests on a rough plane inclined at 20o to the horizontal. If s = 0.5, find the

force applied parallel to the plane required to start the mass moving: a) up the plane; b) down the plane. a) b)

20º

20º

Friction 3 - 13


Example 3.5 ( Motion with Constant Speed ) A 1 tonne load is pulled at constant speed up a track inclined at 30o to the horizontal by a force, P, inclined at 20o above the track. Calculate the value of P if the coefficient of kinetic friction is 0.15. TUTORIAL (Friction on inclined plane) 3.10 A load of 20 kg resting on an inclined plane is at impending motion when the plane is

tilted to an angle of 25 to the horizontal. What is the coefficient of static friction?

If the angle of the plane is increased to 35what force applied parallel to the plane will be necessary to keep the load from slipping down?

(0.4663, 37.6 N )

P

mg

Rn

Fk

3 - 14 Friction


3.11 The coefficient of static friction between the block and incline is 0.4 (Fig. Q3.11).

Determine the force P which will cause impending motion of the block: a) up the incline. b) down the incline. (452 N , 139 N )

Fig. Q3.11

37°

P

50 kg

Friction 3 - 15


3.12 The crate shown in Fig. Q3.12 has a mass of 350 kg and is subjected to a towing force

P acting at 20 to the horizontal. If s = 0.5, determine: a) magnitude of P to just start the crate moving downwards. b) normal reaction. c) magnitude of the frictional force. (981 N , 2891 N , 1446 N )

20°

10°

P

s = 0.5

Fig. Q3.12

3 - 16 Friction


3.13 A block of mass 4.5 kg rests on an incline of 20. If the coefficient of friction for the

contact surfaces is 0.6, determine the force required to move the block up the plane if the force is applied:

a) horizontally. b) parallel to the inclined plane. c) at an angle of 15 above the inclined plane. (54.4 N , 40 N , 35.7 N )

Friction 3 - 17


3.14 The coefficient of static friction between the block and the surface of the inclined

plane shown in Fig. Q3.14 is 0.5. Determine: a) the least value of M and, b) the greatest value of M for the system to be at rest. (7.39 kg , 13.4 kg )

******************

600 900

12 kg

Fig 7 23

M

Fig. Q3.14

smooth pulley

cable

unit 3 mm9400 ver1.1(2014)

Engineering