FRICTIONIntroductionIf we slide or try to slide a body over a surface, the motion is resisted by a bonding between thebody and the surface. This resistance is represented by a single force and is called friction force.The force of friction is parallel to the surface and opposite to the direction of intended motion.Types of Friction(1) Static friction : The opposing force that comes into play when one bodytends to move over the surface of another, but the actual motion has yet notstarted is called static friction.(i) If applied force is P and the body remains at rest then static friction F = P.(ii) If a body is at rest and no pulling force is acting on it, force of friction onit is zero.(iii) Static friction is a self-adjusting force because it changes itself inaccordance with the applied force and is always equal to net external force.(2) Limiting friction : If the applied force is increased, the force of static friction also increases. Ifthe applied force exceeds a certain (maximum) value, the body starts moving. This maximum valueof static friction upto which body does not move is called limiting friction.(i) The magnitude of limiting friction between any two bodies in contact is directly proportional tothe normal reaction between them.

RFR sls orFisfrictionstaticmaximumofforceBut.FfrictionstaticofForce staticmaxs

(ii) Direction of the force of limiting friction is always opposite to the direction in which one body isat the verge of moving over the other(iii) Coefficient of static friction :

(a) s is called coefficient of static friction and is defined as the ratio of force of limiting friction and

normal reactionR

Fs

(b) Dimension : ][ 000 TLM

(c) Unit : It has no unit.(d) Value of depends on material and nature of surfaces in contact that means whether dry or

wet ; rough or smooth polished or non-polished.(e) Value of does not depend upon apparent area of contact.

(f) Hence the force of limiting friction is a parameter which decides whether the object will slide overor not. It is not always equal to force of friction.(3) Kinetic or dynamic friction : If the applied force is increased further and sets the body inmotion, the friction opposing the motion is called kinetic friction.(i) Kinetic friction depends upon the normal reaction.

RFk or RF kk wherek is called the coefficient of kinetic friction

(ii) Value ofk depends upon the nature of surface in contact.

(iii) Kinetic friction is always lesser than limiting frictionlk FF

sk i.e. coefficient of kinetic

friction is always less than coefficient of static friction. Thus we require more force to start amotion than to maintain it against friction. This is because once the motion starts actually ; inertiaof rest has been overcome. Also when motion has actually started, irregularities of one surfacehave little time to get locked again into the irregularities of the other surface.(iv) Kinetic friction does not depend upon the velocity of the body (provided velocity should not betoo large).(v) Types of kinetic friction(a) Sliding friction : The opposing force that comes into play when one body is actually sliding overthe surface of the other body is called sliding friction. e.g. A flat block is moving over a horizontaltable.(b) Rolling friction : When objects such as a wheel (disc or ring), sphere or a cylinder rolls over asurface, the force of friction that comes into play is called rolling friction.*Rolling friction is directly proportional to the normal reaction (R) and inversely proportional to theradius (r) of the rolling cylinder or wheel.

r

RF rrolling

P

R

F

mg

r is called coefficient of rolling friction. It would have the dimensions of length and would be

measured in metre.* Rolling friction is often quite small as compared to the sliding friction. That is why heavy loadsare transported by placing them on carts with wheels.* In rolling the surfaces at contact do not rub each other.* The velocity of point of contact with respect to the surface remains zero all the times although thecentre of the wheel moves forward.Graph Between Applied Force and Force of Friction(1) Part OA of the curve represents static friction )( sF . Its value

increases linearly with the applied force(2) At point A the static friction is maximum. This representlimiting friction )( lF .

(3) Beyond A, the force of friction is seen to decrease slightly. Theportion BC of the curve represents the kinetic friction )( kF .

(4) As the portion BC of the curve is parallel to x-axis thereforekinetic friction does not change with the applied force, it remainsconstant, whatever be the applied force.Angle of FrictionAngle of friction may be defined as the angle which the resultant of limitingfriction and normal reaction makes with the normal reaction.By definition angle is called the angle of friction

R

Fltan

tan = s [As we knows

l

R

F ]

or )(tan 1L

Hence coefficient of static friction is equal to tangent of the angle of friction.Resultant Force Exerted by Surface on Block

In the above figure resultant force 22 RFS 22 )()( mgmgS

12 mgS

when there is no friction )0( S will be minimum

i.e. S = mg

Hence the range of S can be given by, 12 mgSmg

Angle of ReposeAngle of repose is defined as the angle of the inclined plane withhorizontal such that a body placed on it is just begins to slide.

By definition, is called the angle of repose.In limiting condition sinmgF and cosmgR

So tanR

F

tantan sR

F [As we know tan sR

F ]

Thus the coefficient of limiting friction is equal to the tangent of angle of repose.As well as i.e. angle of repose = angle of friction.Calculation of Required Force in Different SituationIf W = weight of the body, = angle of friction, tan coefficient of

frictionThen we can calculate required force for different situation in the followingmanner :

(1) Minimum pulling force P at an angle from the horizontalBy resolving P in horizontal and vertical direction (as shown in figure)For the condition of equilibrium

cosPF and sinPWR

ACB

Fl Fk

Forc

eo

ffr

icti

on

Applied forceO

Fs

P

R

F

mg

S

R

mg cos

F

mg sin

mg

P

R

P sin

P cosF

W

By substituting these value in RF )sin(cos PWP

)sin(cos

sincos

PWP [As tan ]

)(cos

sin

WP

(2) Minimum pushing force P at an angle from the horizontalBy Resolving P in horizontal and vertical direction (as shown in the figure)For the condition of equilibrium

cosPF and sinPWR By substituting these value in RF

)sin(cos PWP

)sin(cos

sincos

PWP [As tan ]

)(cos

sin

WP

(3) Minimum pulling force P to move the body up on an inclinedplaneBy Resolving P in the direction of the plane and perpendicular to theplane (as shown in the figure)

For the condition of equilibrium cossin WPR

sincos PWR and cossin PWF

sincos WPF

By substituting these values in RF and solving we get

)(cos

)(sin

WP

(4) Minimum force to move a body in downward direction along the surface of inclined plane

By Resolving P in the direction of the plane and perpendicular to the plane (as shown in the figure)For the condition of equilibrium

cossin WPR sincos PWR and sincos WPF By substituting these values in RF and solving we get

)(cos

)sin(

WP

P

R

P sin

P cosF

W

F

R + P sin

P cos

+

W cos

W

P

P

R + P sin

W cos

F + W sin

P cos

W

(5) Minimum force to avoid sliding of a body down on an inclined plane

By Resolving P in the direction of the plane and perpendicular to the plane (as shown in the figure)For the condition of equilibrium

cossin WPR

sincos PWR and sincos WFP

cossin PWF By substituting these values in RF and solving we get

)(cos

)(sin

WP

(6) Minimum force for motion along horizontal surface and its direction

Let the force P be applied at an angle with the horizontal.By resolving P in horizontal and vertical direction (as shown in figure)For vertical equilibrium

mgPR sin

sinPmgR …(i)

and for horizontal motionFP cos

i.e. RP cos …(ii)

Substituting value of R from (i) in (ii))sin(cos PmgP

sincos

mgP …(iii)

For the force P to be minimum )sin(cos must be maximum i.e.

0]sin[cos d

d

0cossin

tan

or frictionofangle )(tan 1

i.e. For minimum value of P its angle from the horizontal should be equal to angle of friction

As tan so from the figure,21

sin

and21

1cos

By substituting these value in equation (iii)

2

2

2 11

1

mg

P21

mg 2

min

1

mgP

P

P

F + P cosR + P sin

W sin W cosW

R + P sin

P cosF

mg

1

21

Acceleration of a Block Against Friction(1) Acceleration of a block on horizontal surfaceWhen body is moving under application of force P, then kinetic frictionopposes its motion.Let a is the net acceleration of the bodyFrom the figure

kFPma

m

FPa k

(2) Acceleration of a block sliding down over a rough inclined planeWhen angle of inclined plane is more than angle of repose, the body placed on the inclined planeslides down with an acceleration a.From the figure Fmgma sin

Rmgma sin

cossin mgmgma

Acceleration ]cos[sin ga

Note : For frictionless inclined plane 0 singa .

(3) Retardation of a block sliding up over a rough inclined planeWhen angle of inclined plane is less than angle of repose, then forthe upward motion

Fmgma sin

cossin mgmgma

Retardation ]cos[sin ga

Note : For frictionless inclined plane 0 singa

Motion of Two Bodies one Resting on the OtherWhen a body A of mass m is resting on a body B of mass M then two conditions are possible(1) A force F is applied to the upper body, (2) A force F is applied to the lower body

We will discuss above two cases one by one in the following manner:(1) A force F is applied to the upper body, then following four situations are possible(i) When there is no friction(a) The body A will move on body B with acceleration (F/m).

mFaA /

(b) The body B will remain at rest

0Ba

(c) If L is the length of B as shown in figure, A will fall from B after time t

F

mL

a

Lt

22

F/mata and

2

1sAs 2

(ii) If friction is present between A and B only and applied force is less than limiting friction(F < Fl)(F = Applied force on the upper body, Fl = limiting friction between A and B, Fk = Kinetic frictionbetween A and B)(a) The body A will not slide on body B till

lFF i.e. mgF s

(b) Combined system (m + M) will move together with common accelerationmM

Faa BA

R

PFk

mg

ma

F

mg sin mg cosmg

R

ma

mg sin + F mg cos

mg

Rma

Fm A

M BL

(iii) If friction is present between A and B only and applied force is greater than limitingfriction (F > Fl)In this condition the two bodies will move in the same direction (i.e. of applied force) but withdifferent acceleration. Here force of kinetic friction mgk will oppose the motion of A while cause the

motion of B.

Note : As both the bodies are moving in the same direction.Acceleration of body A relative to B will be

mM

MmmgMFaaa k

BA

)(

So, A will fall from B after time

)(

22

MmmgMF

MLm

a

Lt

k

(iv) If there is friction between B and floor(where gmMFl )( = limiting friction between B and floor, Fk = kinetic friction between A and B)

B will move only iflk FF and then

Blk aMFF

However if B does not move then static friction will work (not limiting friction) between body B andthe floor i.e. friction force = applied force (= Fk) not

lF .

(2) A force F is applied to the lower body, then following four situations are possible(i) When there is no friction(a) B will move with acceleration (F/M) while A will remain at rest (relative to ground) as there is nopulling force on A.

M

FaB

and 0Aa

(b) As relative to B, A will move backwards with acceleration (F/M) and so will fall from it in time t.

F

ML

a

Lt

22

(ii) If friction is present between A and B only and F < Fl

(where F = Pseudo force on body A and Fl = limiting friction between body A and B)

Ak amFF

i.e. m

FFa k

A

m

mgFa k

A)(

Free body diagram of A

Bk aMF

i.e. M

Fa k

B

M

mga k

B

Free body diagram of B

FK

MaB

B

Fl

F

mA

M BL

FA

Fk

maA

FK

MaB

B

(a) Both the body will move together with common accelerationmM

Fa

(b) Pseudo force on the body A,

Mm

mFmaF

and mgF sl

(c)lFF mg

Mm

mFs

gMmF s )(

So both bodies will move together with accelerationMm

Faa BA

if gMmF s ][

(iii) If friction is present between A and B only and F > Fl(where Fl = s mg = limiting friction between body A and B)Both the body will move with different acceleration. Here force of kinetic friction mgk will oppose

the motion of B while will cause the motion of A.

mgma kA

i.e. ga kA

Free body diagram of A

Bk MaFF

i.e. M

mgFa k

B][

Free body diagram of B

Note : As both the bodies are moving in the same directionAcceleration of body A relative to B will be

M

MmgFaaa k

BA

)(

Negative sign implies that relative to B, A will move backwards and will fall it after time

)(

22

MmgF

ML

a

Lt

k

(iv) If there is friction between B and floor and F > Fl :(where Fl = s(m+M)g = limiting friction between body B and surface)The system will move only if ''

lFF then replacing F by lFF . The entire case (iii) will be valid.

However if 1FF the system will not move and friction between B and floor will be F while between

A and B is zero.Motion of an Insect in the Rough BowlThe insect crawl up the bowl, up to a certain height h only till the component of its weight alongthe bowl is balanced by limiting frictional force.

Let m = mass of the insect, r = radius of the bowl, = coefficient of frictionfor limiting condition at point A

cosmgR ......(i) and sinmgFl ......(ii)

Dividing (ii) by (i)

A

Fk

maA

FK

MaB

BF

Fl

mg sin

mgmg cos

A

R

rO

y

h

R

Fltan RFl As

y

yr 22

or21

r

y

So

21

11

ryrh ,

21

11

rh

Minimum Mass Hung from the String to Just Start the Motion(1) When a mass m1 placed on a rough horizontal plane Another mass

2m hung from the string

connected by frictionless pulley, the tension (T) produced in string will try to start the motion ofmass

1m .

At limiting conditionlFT

Rgm 2 gmgm 12

12 mm this is the minimum value of

2m to start the motion.

Note: In the above condition Coefficient of friction

1

2

m

m

(2) When a mass m1 placed on a rough inclined plane Another mass2m hung from the string

connected by frictionless pulley, the tension (T) produced in string will try to start the motion ofmass

1m .

At limiting conditionFor gmTm 22 …(i)

For FgmTm sin11

RgmT sin1

cossin 11 gmgmT …(ii)

From equation (i) and (ii) ]cos[sin12 mm

this is the minimum value of2m to start the motion

Note : In the above condition Coefficient of friction

tan

cos1

2

m

m

Maximum Length of Hung ChainA uniform chain of length l is placed on the table in such a manner that its 'l part is hanging over

the edge of table without sliding. Since the chain have uniform linear density therefore the ratio ofmass and ratio of length for any part of the chain will be equal.

We knowtabletheonlyingmass

tablethefromhangingmass

1

2

m

m

For this case we can rewrite above expression in the following manner

tabletheonlyinglength

tablethefromhanginglength [As chain have uniform linear density]

ll

l

by solving)1(

ll

m2

m1

T

R

m1g

Fl

T

m2g

m2

RT

m1g sin + F m1g cos

T

m2g

m1

m1g

l

( l – l )

Coefficient of Friction Between a Body and WedgeA body slides on a smooth wedge of angle and its time of descent is t.

If the same wedge made rough then time taken by it to come down becomes n times more (i.e. nt)The length of path in both the cases are same.

For smooth wedge, 2

2

1attuS

2)sin(2

1tgS …(i)

andAs 0[ u ]singa

For rough wedge, 2

2

1attuS

2)()cos(sin2

1ntgS …(ii)

)]cos(sin0[ gau andAs

From equation (i) and (ii)

2)sin(2

1tg = 2)()cos(sin

2

1ntg

2)cos(sinsin n

2

11tan

n

Stopping of Block Due to Friction(1) On horizontal road(i) Distance travelled before coming to rest : A block of mass m is moving initially with velocity uon a rough surface and due to friction, it comes to rest after covering a distance S.

Retarding force RmaF mgma ga

From aSuv 222 Sgu 20 2

],0 gav [As

g

uS

2

2

orgm

PS

2

2

2

[As momentum P = mu](ii) Time taken to come to restFrom equation tauv tgu 0

],0[ gav As

g

ut

(2) On inclined road : When block starts with velocity u its kinetic energy will be converted intopotential energy and some part of it goes against friction and after travelling distance S it comes torest i.e. v = 0.We know that retardation ]cos[sin ga

By substituting the value of v and a in the following equation

Sauv 222

Sgu ]cos[sin20 2

S

Smooth wedge

S

Rough wedge

v = 0S

u

u

v = 0

S

)cos(sin2

2

g

uS

Stopping of Two Blocks Due to FrictionWhen two masses compressed towards each other and suddenly released then energy acquired byeach block will be dissipated against friction and finally block comes to resti.e., F × S = E[Where F = Friction, S = Distance covered by block, E = Initial kinetic energy of the block]

m

PSF

2

2

[Where P = momentum of block]

m

PSmg

2

2

[As F = mg]

gm

PS

2

2

2

In the given condition P and are same for both the blocks.

So,2

1

mS ;

2

1

2

2

1

m

m

S

S

Sticking of a Block With Accelerated CartWhen a cart moves with some acceleration toward right then a pseudo force (ma) acts on blocktoward left.This force (ma) is action force by a block on cart.

Now block will remain static w.r.t. cart. If friction force mgR

mgma ]As[ maR

ga

ga min

This is the minimum acceleration of the cart so that block does not fall.and the minimum force to hold the block together

minmin )( amMF

gmMF )(min

m1

A

m1

B

m2 m2

S1 S2

M

a

F

CART

m

F

mg

m Rma

Understanding Concept:-1. Determine the maximum acceleration of the train in which a box lying on its floor will remainstationary, given that the co-efficient of static friction between the box and the train’s floor is 0.15.2. See Fig. A mass of 4 kg rests on a horizontal plane. The plane is gradually inclined until at anangle θ = 15° with the horizontal, the mass just begins to slide. What is the coefficient of static friction between the block and the surface?

3. What is the acceleration of the block and trolley system shown in a Fig., if the coefficient ofkinetic friction between the trolley and the surface is 0.04? What is the tension in the string? (Takeg = 10 m s-2). Neglect the mass of the string.

4. Two bodies A and B of masses 5 kg and 10 kg in contact with each other rest on a table againsta rigid wall (Fig.). The coefficient of friction between the bodies and the table is 0.15. A force of 200N is applied horizontally to A. What are (a) the reaction of the partition (b) the action-reactionforces between A and B ? What happens when the wall is removed? Does the answer to (b) change,when the bodies are in motion? Ignore the difference between μs and μk.

5. A block of mass 15 kg is placed on a long trolley. The coefficient of static friction between theblock and the trolley is 0.18. The trolley accelerates from rest with 0.5 m s-2 for 20 s and thenmoves with uniform velocity. Discuss the motion of the block as viewed by (a) a stationary observeron the ground, (b) an observer moving with the trolley.6. The rear side of a truck is open and a box of 40 kg mass isplaced 5 m away from the open end as shown in Fig. Thecoefficient of friction between the box and the surface below it is0.15. On a straight road, the truck starts from rest and accelerateswith 2 m s-2. At what distance from the starting point does the boxfall off the truck? (Ignore the size of the box).7. A brick slides on a horizontal surface. Which of the following will increase the magnitude of thefrictional force on it?A. Putting a second brick on topB. Decreasing the surface area of contactC. Increasing the surface area of contactD. Decreasing the mass of the brickE. None of the above8. The coefficient of kinetic friction:A. is in the direction of the frictional forceB. is in the direction of the normal forceC. is the ratio of force to area

D. can have units of newtonsE. is none of the above9. When the brakes of an automobile are applied, the road exerts the greatest retarding force:A. while the wheels are slidingB. just before the wheels start to slideC. when the automobile is going fastestD. when the acceleration is leastE. at the instant when the speed begins to change10. A forward horizontal force of 12N is used to pull a 240-N crate at constant velocity across ahorizontal floor. The coefficient of friction is:A. 0.5 B. 0.05 C. 2 D. 0.2 E. 2011. The speed of a 4.0-N hockey puck, sliding across a level ice surface, decreases at the rate of0.61m/s2. The coefficient of kinetic friction between the puck and ice is:A. 0.062 B. 0.41 C. 0.62 D. 1.2 E. 9.812. A crate rests on a horizontal surface and awoman pulls on it with a 10-N force. No matter whatthe orientation of the force, the crate does not move.Rank the situations shown below according to themagnitude of the frictional force of the surface on thecrate, least to greatest.A. 1, 2, 3 B. 2, 1, 3 C. 2, 3, 1 D. 1, 3, 2 E. 3, 2, 113. A crate with a weight of 50N rests on a horizontal surface. A personpulls horizontally on it with a force of 10N and it does not move. To start itmoving, a second person pulls vertically upward on the crate. If thecoefficient of static friction is 0.4, what is the smallest vertical force forwhich the crate moves?A. 4N B. 10N C. 14N D. 25N E. 35N14. A 40-N crate rests on a rough horizontal floor. A 12-N horizontal force is then applied to it. If

the coefficients of friction are s = 0.5 and k = 0.4, the magnitude of the frictional force on thecrate is:A. 8N B. 12N C. 16N D. 20N E. 40N15. A 24-N horizontal force is applied to a 40-N block initially at rest on a rough horizontal surface.If the coefficients of friction are s = 0.5 and k = 0.4, the magnitude of the frictional force on theblock is:A. 8N B. 12N C. 16N D. 20N E. 400N16. A horizontal shove of at least 200N is required to start moving a 800-N crate initially at rest ona horizontal floor. The coefficient of static friction is:A. 0.25 B. 0.125 C. 0.50 D. 4.00 E. none of these17. A force F (larger than the largest possible force of static friction) is applied to the left to anobject moving to the right on a horizontal surface. Then:A. the object must be moving at constant speedB. F and the friction force act in opposite directionsC. the object must be slowing downD. the object must be speeding upE. the object must come to rest and remain at rest

18. A bureau rests on a rough horizontal surface (s = 0.50, k = 0.40). A constant horizontalforce, just sufficient to start the bureau in motion, is then applied. The acceleration of the bureauis:A. 0 B. 0.98m/s2 C. 3.3m/s2 D. 4.5m/s2 E. 8.9m/s2

19. A car is traveling at 15m/s on a horizontal road. The brakes are applied and the car skids to astop in 4.0 s. The coefficient of kinetic friction between the tires and road is:A. 0.38 B. 0.69 C. 0.76 D. 0.92 E. 1.1120. A boy pulls a wooden box along a rough horizontal floor atconstant speed by means of a force P as shown. In thediagram f is the magnitude of the force of friction, N is themagnitude of the normal force, and Fg is the magnitude of theforce of gravity. Which of the following must be true?A. P = f and N = Fg B. P = f and N >Fg C. P >f and N <FgD. P >f and N = Fg E. none of these

21. A boy pulls a wooden box along a roughhorizontal floor at constant speed by means of a forceP as shown. In the diagram f is the magnitude of theforce of friction, N is the magnitude of the normalforce, and Fg is the magnitude of the force of gravity.Which of the following must be true?A. P = f and N = Fg B. P = f and N >FgC. P >f and N <Fg D. P >f and N = FgE. none of these22. A 400-N block is dragged along a horizontal surfaceby an applied force F as shown. The coefficient of kinetic

friction is k = 0.4 and the block moves at constantvelocity. The magnitude of F is:A. 100N B. 150NC. 200N D. 290NE. 400 N23. A block of mass m is pulled at constant velocity along arough horizontal floor by an applied force T as shown. Themagnitude of the frictional force is:

A. T cos B. T sin C. zero D. mg

E. mg cos 24. A block of mass m is pulled along a rough horizontal floorby an applied force T as shown. The vertical component of theforce exerted on the block by the floor is:

A. mg B. mg - T cosC. mg + T cos D. mg - T sin E. mg + T sin 25. A 12-kg crate rests on a horizontal surface and a boy pullson it with a force that is 300 below the horizontal. If thecoefficient of static friction is 0.40, the minimum magnitude force he needs to start the cratemoving is:A. 44N B. 47N C. 54N D. 56N E. 71N26. A crate resting on a rough horizontal floor is to be moved horizontally. The coefficient of staticfriction is 0.40. To start the crate moving with the weakest possible applied force, in what directionshould the force be applied?A. Horizontal B. 240below the horizontal C. 220 above the horizontalD. 240 above the horizontal E. 660 below the horizontal27. A 50-N force is applied to a crate on a horizontal rough floor, causing it to move horizontally. Ifthe coefficient of kinetic friction is 0.50, in what direction should the force be applied to obtain thegreatest acceleration?A. Horizontal B. 600above the horizontal C. 300 above the horizontalD. 270above the horizontal E. 300 below the horizontal28. A professor holds an eraser against a vertical chalkboard by pushing horizontally on it. Hepushes with a force that is much greater than is required to hold the eraser. The force of frictionexerted by the board on the eraser increases if he:A. pushes with slightly greater forceB. pushes with slightly less forceC. stops pushingD. pushes so his force is slightly downward but has the same magnitudeE. pushes so his force is slightly upward but has the same magnitude29. A horizontal force of 12N pushes a 0.5-kg book against a vertical wall. The book is initially at

rest. If the coefficients of friction are s = 0.6 and k = 0.8 which of the following is true?A. The magnitude of the frictional force is 4.9NB. The magnitude of the frictional force is 7.2NC. The normal force is 4.9ND. The book will start moving and accelerateE. If started moving downward, the book will decelerate

30. A horizontal force of 5.0N pushes a 0.50-kg book against a vertical wall. The book is initially at

rest. If the coefficients of friction are s = 0.6 and k = 0.80, the magnitude of the frictional force is:A. 0 B. 4.9N C. 3.0N D. 5.0N E. 4.0N31. A horizontal force of 12N pushes a 0.50-kg book against a vertical wall. The book is initially atrest. Ifs = 0.6 and k = 0.80, the acceleration of the book in m/s2 is:A. 0 B. 9.4m/s2 C. 9.8m/s2 D. 14.4m/s2 E. 19.2m/s2

32. A horizontal force of 5.0N pushes a 0.50-kg block against a vertical wall. The block is initially

at rest. If s = 0.60 and k = 0.80, the acceleration of the block in m/s2 is:A. 0 B. 1.8 C. 6.0 D. 8.0 E. 9.833. A heavy wooden block is dragged by a force F along arough steel plate, as shown below for two possible situations.The magnitude of F is the same for the two situations. Themagnitude of the frictional force in (ii), as compared with thatin (i) is:A. the same B. greater C. lessD. less for some angles and greater for othersE. can be less or greater, depending on the magnitude of the applied force.34. A block is first placed on its long side and then onits short side on the same inclined plane, as shown.The block slides down the plane on its short side butremains at rest on its long side. A possible explanationis:A. the short side is smootherB. the frictional force is less because the contact area islessC. the center of gravity is higher in the second caseD. the normal force is less in the second caseE. the force of gravity is more nearly down the plane in the second case35. A box rests on a rough board 10 meters long. When one end of the board is slowly raised to aheight of 6 meters above the other end, the box begins to slide. The coefficient of static friction is:A. 0.8 B. 0.25 C. 0.4 D. 0.6 E. 0.7536. A block is placed on a rough wooden plane. It is found that when the plane is tilted 30 to thehorizontal, the block will slide down at constant speed. The coefficient of kinetic friction of theblock with the plane is:A. 0.500 B. 0.577 C. 1.73 D. 0.866 E. 4.9037. A crate is sliding down an incline that is 350 above the horizontal. If the coefficient of kineticfriction is 0.40, the acceleration of the crate is:A. 0 B. 2.4m/s2 C. 5.8m/s2 D. 8.8m/s2 E.10.3m/s2

38. A 5.0-kg crate is resting on a horizontal plank. The coefficient of static friction is 0.50 and thecoefficient of kinetic friction is 0.40. After one end of the plank is raised so the plank makes anangle of 250 with the horizontal, the force of friction is:A. 0 B. 18N C. 21N D. 22N E. 44N39. A 5.0-kg crate is resting on a horizontal plank. The coefficient of static friction is 0.50 and thecoefficient of kinetic friction is 0.40. After one end of the plank is raised so the plank makes anangle of 300 with the horizontal, the force of friction is:A. 0 B. 18N C. 21N D. 22N E. 44N40. A 5.0-kg crate is on an incline that makes an angle of 300 with the horizontal. If the coefficientof static friction is 0.50, the minimum force that can be applied parallel to the plane to hold thecrate at rest is:A. 0 B. 3.3N C. 30N D. 46N E. 55N41. A 5.0-kg crate is on an incline that makes an angle of 300 with the horizontal. If the coefficientof static friction is 0.5, the maximum force that can be applied parallel to the plane without movingthe crate is:A. 0 B. 3.3N C. 30N D. 46N E. 55N42. Block A, with mass mA, is initially at rest on a horizontal floor. Block B, with mass mB, isinitially at rest on the horizontal top surface of A. The coefficient of static friction between the twoblocks is s. Block A is pulled with a horizontal force. It begins to slide out from under B if theforce is greater than:A. mAg B. mBg C. s mA g D. s mB g E. s (mA +mB) g

43. The system shown remains at rest. Each block weighs 20 N.The force of friction on the upper block is:A. 4N B. 8NC. 12N D. 16NE. 20N

44. Block A, with a mass of 50 kg, rests on a horizontal table top. Thecoefficient of static friction is 0.40. A horizontal string is attached to Aand passes over a massless frictionless pulley as shown. The smallestmass mB of block B, attached to the dangling end, that will start Amoving when it is attached to the other end of the string is:A. 20 kg B. 30 kgC. 40 kg D. 50 kgE. 70 kg

45. Block A, with a mass of 10 kg, rests on a 350 incline. The coefficientof static friction is 0.40. An attached string is parallel to the incline andpasses over a massless, frictionless pulley at the top. The largest massmB of block B, attached to the dangling end, for which A begins toslide down the incline is:A. 2.5kg B. 3.5kgC. 5.9kg D. 9.0kgE. 10.5kg46. Block A, with a mass of 10 kg, rests on a 350 incline. The coefficientof static friction is 0.40. An attached string is parallel to the incline andpasses over a massless, frictionless pulley at the top. The largest massmB, attached to the dangling end, for which A remains at rest is:A. 2.5kg B. 3.5kgC. 5.9kg D. 9.0kgE. 10.5kg47. Block A, with a mass of 10 kg, rests on a 300 incline. The coefficient ofkinetic friction is 0.20. The attached string is parallel to the incline andpasses over a massless, frictionless pulley at the top. Block B, with a massof 8.0 kg, is attached to the dangling end of the string. The acceleration of Bis:A. 0.69m/s2, up the planeB. 0.69m/s2, down the planeC. 2.6m/s2, up the planeD. 2.6m/s2, down the planeE. 048. Block A, with a mass of 10 kg, rests on a 300 incline. The coefficientof kinetic friction is 0.20. The attached string is parallel to the inclineand passes over a massless, frictionless pulley at the top. Block B, witha mass of 3.0 kg, is attached to the dangling end of the string. Theacceleration of B is:A. 0.20m/s2, upB. 0.20m/s2, downC. 2.8m/s2, upD. 2.8m/s2, downE. 049. A 1000-kg airplane moves in straight flight at constant speed. The force of air friction is 1800N. The net force on the plane is:A. zero B. 11800N C. 1800N D. 9800N E. none of these50. Why do raindrops fall with constant speed during the later stages of their descent?A. The gravitational force is the same for all dropsB. Air resistance just balances the force of gravityC. The drops all fall from the same heightD. The force of gravity is negligible for objects as small as raindrops

E. Gravity cannot increase the speed of a falling object to more than 9.8m/s51. A ball is thrown downward from the edge of a cliff with an initial speed that is three times theterminal speed. Initially its acceleration isA. upward and greater than gB. upward and less than gC. downward and greater than gD. downward and less than gE. downward and equal to g52. A ball is thrown upward into the air with a speed that is greater than terminal speed. On theway up it slows down and, after its speed equals the terminal speed but before it gets to the top ofits trajectory:A. its speed is constantB. it continues to slow downC. it speeds upD. its motion becomes jerkyE. none of the above53. A ball is thrown upward into the air with a speed that is greater than terminal speed. It landsat the place where it was thrown. During its flight the force of air resistance is the greatest:A. just after it is thrownB. halfway upC. at the top of its trajectoryD. halfway downE. just before it lands.54. In a situation the contact force by a rough horizontal surface on a body placed on it hasconstant magnitude. If the angle between this force and the vertical is decreased , the force and thevertical is decreased, the friction force between the surface and the body will(a) increse (b*) decrease (c) remain the same (d) may increase or decrease55. While walking on ice, one should take small steps to avoid slipping . This is because smallersteps to avoid slipping. This is because smaller steps ensure(a) larger friction (b*) smaller friction (c) larger normal force (d) smaller normal force

56. A body of mass M is kept on a rough horizontal surface (friction coefficient = ). A person istrying to pull the body by applying a horizontal force but the body is not moving. The force by thesurface on A is F where

(a) F = Mg (b) F = Mg (c*) 21MgFMg (d) 21MgFMg

57. A scooter starting from rest moves with a constant acceleration for a time Δt1, then with aconstant deceleration for the next Δt2 and finally with a constant deceleration for the next Δt3 tocome to rest. A 500N man sitting on the scooter behind the driver manges to stay at rest withrespect to the scooter without touching any other part. The force exerted by the seat on the man is(a) 500 N throughout the journey (b*) less than 500N throughout the journey(c) more than 500N throughout the journey (d) > 500 N for time Δt1 and Δt3 and 500 N for Δt2.58. Consider the situation shown in figure. The wall is smooth but the surface ofA and B in contact are rough. The friction on B due to A in equilibrium(a) is upward (b) is downward(c) is zero (d*) the system cannot remain in equilibrium.59. Suppose all the surfaces in the previous problem are rough. The direction of friction on B dueto A(a*) is upward (b) is downward(c) is zero (d) depends on the masses of A and B.60. Two cars of unequal masses use similar tyres. If they are moving at the same initial speed, theminimum stopping distance(a) is smaller for the heavier car (b) is smaller for the lighter car(c*) is same for both cars (d) depends on the volume of the car.61. In order to stop a car in shortest distance on a horizontal road, one should(a) apply the brakes very hard so that the wheels stop rotating(b*) apply the brakes hard enough to just prevent slipping(c) pump the brakes (press and release)(d) shut the engine off and not apply brakes .

62. A block A kept on an inclined surface just begins to slide if the inclination is 30º. The block isreplaced by another block B and it is found that it just begins to slide if the inclination is 40º.(a) mass of A > mass of B (b) mass of A < mass of B(c) mass of A = mass of B (d*) all the three are possible.63. A boy of mass M is applying a horizontal force to slide a box of mass M’ on a rough horizontal

surface . The coefficient of friction between the shoes of the boy and the floor is and that betweenthe box and the floor is ’. In which of the following cases it is certainly not possible to slide the box?

(a*) <’, M < M’ (b) >’, M < M’(c) <’, M > M’ (d) >’, M > M64. Let F, FN and f denote the magnitudes of the contact force , normal force and the frictionexerted by one surface on the other kept in contact. If none of these is zero,(a*) F > FN (b*) F > f (c) FN > f (d*) FN – f < F < FN + f65. The contact force exerted by a body A on another body B is equal to the normal force betweenthe bodies. We conclude that(a) the surface must be frictionless(b*) the force of friction between the bodies is zero(c) the magnitude of normal force equals that of friction(d*) the bodies may be rough but they don’t slip on each other.66. Mark the correct statements about the friction between two bodies.(a) Static friction is always greater than the kinetic friction.(b*) Coefficient of static friction is always greater than the coefficient of kinetic friction.(c*) Limiting friction is always greater than the kinetic friction.(d*) Limiting friction is never less than static friction.67. A block is placed on a rough floor and a horizontal force F is applied on it. The force of friction fby the floor on the block is measured for different values of F and a graph is plotted between them.(a) The graph is a straight line of slope 45º(b) The graph is straight line parallel to the F-axis.(c*) The graph is a straight line of slope 45º for small F and a straight line parallel to the F-axis forlarge F.(d*) There is a small kink on the graph.68. Consider a vehicle going on a horizontal road towards east. Neglect any force by the air. Thefrictional forces on the vehicle by the road.(a*) is towards east if the vehicle is accelerating(b*) is zero if the vehicle is moving with a uniform velocity(c) must be towards east.(d) must be towards east.

FRICTION{H Series}

1. A body of mass 400 g slides on a rough horizontal surface. If the frictional force is 3.0 N, find (a)the angle made by the contact force on the body with the vertical and (b) the magnitude of thecontact force. Take g = 10m/s2.

2. The coefficient of static friction between a block of mass m and an incline is s = 0.3. (a) Whatcan be the maximum angle of the incline with the horizontal so that the block does not slip onthe plane? (b) If the incline makes an angle /2 with the horizontal, find the frictional force on theblock.3. A horizontal force of 20 N is applied to a block of mass 4 kg resting on a rough horizontal table.If the block does not moue on the table, how much frictional force the table is applying on theblock? What can be said about the coefficient of static friction between the block and the table?Take g = 10m/s2.

4. The coefficient of static friction between the block of 2 kg and the table shown in figure1 is s =0.2. What should be the maximum value of m so that the blocks do not move? Take g = 10m/s2.The string and the pulley are light and smooth.

Fig.1

5. The coefficient of static friction between the two blocks shown in figure 2 is µ and the table issmooth. What maximum horizontal force F can be applied to the block of mass M so that theblocks moue together?

Fig.26. A block slides down an incline of angle 30° with acceleration g/4. Find the kinetic frictioncoefficient.7. A block of mass 2.5 kg is kept on a rough horizontal surface. It is found that the block does notslide if a horizontal force less than 15 N is applied to it. Also it is found that it takes 5 seconds toslide through the first 10 m if a horizontal force of 15 N is applied and the block is gently pushed tostart the motion. Taking g =10 m/s2, calculate the coefficients of static and kinetic friction betweenthe block and the surface.

8. A block placed on a horizontal surface is being pushed by a force F making an angle with thevertical. If the friction coefficient is µ, how much force is needed to get the block just started.

Discuss the situation when tan<.9. Find the maximum value of M/m in the situation shown in figure 3 so that the system remains

at rest. Friction coefficient at both the contacts is Discuss the situation when tan < µ.

Fig.3

10. Consider the situation shown in figure 4. The horizontal surface below the bigger block issmooth. The coefficient of friction between the blocks is µ. Find the minimum and the maximumforce F that can be applied in order to keep the smaller blocks at rest with respect to the biggerblock.

Fig.411. Figure 5 shows two blocks connected by a light string placed on the two inclined parts of atriangular structure. The coefficients of static and kinetic friction are 0.28 and 0.25 respectively ateach of the surfaces. (a) Find the minimum and maximum values of m for which the systemremains at rest. (b) Find the acceleration of either block if m is given the minimum value calculatedin the first part and is gently pushed up the incline for a short while.

Fig.512. If the tension in the string in figure 6 is 16 N and the acceleration of each block is 0.5 m/s2,find the friction coefficients at the two contacts with the blocks.

Fig.613. The friction coefficient between the table and the block shown in figure 7 is 0.2. Find thetensions in the two strings.

Fig.714. The friction coefficient between a road and the tyre of a vehicle is 4/3. Find the maximumincline the road may have so that once hard brakes are applied and the wheel starts skidding, thevehicle going down at a speed of 36 km/hr is stopped within 5 m.15. Figure 8 shows two blocks in contact sliding down an inclined surface of inclination 30°. Thefriction coefficient between the block of mass 2.0 kg and the incline is 1, and that between the

block of mass 4.0 kg and the incline is 2. Calculate the acceleration of the 2.0 kg block if (a) 1 =0.20 and 2 = 0.30, (b) 1 = 0.30 and 1= 0.20. Take g = 10 m/s2.

Fig.816. Two masses Ml and M2 are connected by a light rod and the system is slipping down a rough

incline of angle with the horizontal. The friction coefficient at both the contacts is . Find theacceleration of the system and the force by the rod on one of the blocks.17. A block of mass M is kept on a rough horizontal surface. The coefficient of static friction

between the block and the surface is . The block is to be pulled by applying a force to it. Whatminimum force is needed to slide the block ? In which direction should this force act?

18. The friction coefficient between the board and the floor shown in figure 9 is . Find themaximum force that the man can exert on the rope so that the board does not slip on the floor.

Fig.919. A 2 kg block is placed over a 4 kg block and both are placed on a smooth horizontal surface.The coefficient of friction between the blocks is 0.20. Find the acceleration of the two blocks if ahorizontal force of 12 N is applied to (a) the upper block, (b) the lower block. Take g=10m/s2.20. Find the accelerations a1, a2, a3 of the three block; shown in figure 10 if a horizontal force of 10N ii applied on (a) 2 kg block, (b) 3 kg block, (c) 7 kg block. Take g=10m/s2.

Fig.10

21. The friction coefficient between the two blocks shown it y figure 11 is but the floor is smooth.(a) What maximum horizontal force F can be applied without disturbing the equilibrium of thesystem? (b) Suppose the horizontal force applied is double of that found in part (a). Find theaccelerations of the two masses.

Fig.1122. Suppose the entire system of the previous question is kept inside an elevator which is comingdown with an acceleration a <g. Repeat parts (a) and (b).23. Consider the situation shown in figure 11. Suppose small electric field E exists in the space inthe vertically upward direction and the upper block carries a positive charge Q on its top surface.The friction coefficient between the two blocks is N but the floor is smooth. What maximumhorizontal force F can be applied without disturbing the equilibrium?24. A block of mass m slips on a rough horizontal table under the action of a horizontal force

applied to it. The coefficient of friction between the block and the table is . The table does notmove on the floor. Find the total frictional force applied by the floor on the legs of the table: Do youneed the friction coefficient between the table and the floor or the mass of the table?25. Find the acceleration of the block of mass M in the 1ituation of figure 12. The coefficient of

friction between the two blocks is and that between the bigger block and the ground is µ2.

Fig.1226. A block of mass 2 kg is pushed against a rough vertical wall with a force of 40 N, coefficient ofstatic friction being 0.5. Another horizontal force of 15 N, is applied on the block in a directionparallel to the wall. Will the block move? If yes, in which direction? If no, find the frictional forceexerted by the wall on the block.27. A person (40 kg) is managing to be at rest between two vertical walls by pressing one wall A byhis hands and feet and the other wall B by his back 13. Assume that the friction coefficientbetween his body and the walls is 0.8 and that limiting friction acts at all the contacts. (a) Showthat the person pushes the two walls with equal force. (b) Find the normal force exerted by eitherwall on the person. Take g=10m/s2.

Fig.13

28. Figure 14 shows a small block of mass m kept at the left end of a larger block of mass M andlength L. The system can slide on a horizontal road. The system is started towards right with aninitial velocity v. The friction coefficient between the road and the bigger block is µ and thatbetween the blocks is 2. Find the time elapsed before the smaller block separates from the biggerblock.

Fig.14

ANSWERS

1. (a) 370 (b) 5.0N

2. (a) tan-1(0.3) (b) mg sin(/2)3. s≥0.5 4. 0.4 kg

5. g(M+m)

6. 1/2 3

7. s = 0.60,k = 0.528. mg/ (sin cos)

9. /( sin cos)

10. g)m2M(-1

1andgm2M

1

1

11. (a) 9/8 kg, 32/9 kg (b) 0.31m/s2

12. 1 = 0.75,2 = 0.0613. 96 N in the left string and 68 N in the right string14. 160

15. 2.7 m/s2, 2.4 m/s2.

16. a = g(sin cos), zero17. mg/( 21 ) at an angle tan-1 with the horizontal.

18. m) g/ (1+ )19. (a) upper block 4 m/s2, lower block 1 m/s2.(b) both blocks 2 m/s2.20. (a) a1 = 3 m/s2, a2 = a3 = 0.4 m/s2

(b) a1 = a2 = a3 = 5/6 m/s2 (c) same as (b)

21. (a) 2mg (b) 2mg/(M + m) in opposite direction.22. (a) 2mg (b) 2m(g-a) /(M + m)23. 2mg-QE)24. mg

25.

)](25[mM

g]mMm2[

21

2

26. it will move at an angle of 530 with the 15 N force.27. (b) 250 N

28. gmM

Ml4