ASSIGNMENT I

SECTION B

1. An almirah bureau with a mass of 45 kg, including drawers and clothing, rest on the floor.(a) If the coefficient of static friction between the almirah and the floor is 0.45, what is themagnitude of the minimum horizontal force that a person must apply to start the almirahmoving ? (b) If the drawers and clothing, with 17 kg mass, are removed before the almirah

is pushed, what is the new minimum magnitude? (Take )sm8.9g 2=

2. A baseball player with mass m = 79 kg, sliding into second base, is retarded by a frictional

force of magnitude 470 N. What is the coefficient of kinetic friction kµ between the playerand the ground?

3. A person pushed a crate of mass 55kg horizontally with a force of 220N to move it across alevel floor. The coefficient of kinetic friction is 0.35. (a) What is the magnitude of the

frictional force? (b) What is the magnitude of the crate’s acceleration? (Take )sm8.9g 2=

4. A 110 g hockey puck sent sliding over ice is stopped in 15 m by the frictional force on itfrom the ice. (a) If its initial speed is 6.0 m/s, what is the magnitude of the frictional force? (b) What is the coefficient of friction between the puck and the ice?

5. A 12 N horizontal force Fr

pushes a block weighing 5.0 N against a vertical wall (Fig.). Thecoefficient of static friction between the wall and the block is 0.60, and the coefficient ofkinetic friction is 0.40. Assume that the block is not moving initially.(a) Will the block move? (b) In unit-vector notation, what is the force on the block from thewall.

6. A 2.5 kg block is initially at rest on a horizontal surface. A 6.0 N horizontal force and a

vertical force Pr

are applied to the block as shown in Fig. The coefficients of friction for the

block and surface are 40.0s =µ and 25.0k =µ . Determine the magnitude and direction of

the frictional force acting on the block if the magnitude of Pr

is (a) 8.0 N, (b) 10 N and (c) 12

N.(Take )sm10g 2=

7. A 68 kg crate is dragged across a floor by pulling on a rope attached to the crate and inclined15° above the horizontal. (a) If the coefficient of static friction is 0.50, what minimum

force magnitude is required from the rope to start the crate moving? (b) If 35.0k =µ , what

is the magnitude of the initial acceleration of the crate ? (Take )sm8.9g 2=

8. In Fig. 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 sµ between A and the table is 0.20.

(b) Block C suddenly is lifted off A. What is the acceleration of block A if kµ between Aand the table is 0.15?

9. Block B in Fig. weighs 711 N. The coefficient of static friction between block and table is0.25; assume that the cord between B and the knot is horizontal. Find the maximumweight of block A for which the system will be stationary.

10. Consider the situation shown in figure. Calculate (a) the acceleration of the 1.0kg blocks,(b) the tension in the string connecting the 1.0kg blocks and (c) the tension in the stringattached to 0.50kg.

11. If the tension in the string in figure is 16N and the acceleration of each block is 0.52sm ,find the friction coefficients at the two contacts with the blocks.

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

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

14. The friction coefficient between the two blocks shown in figure is µ but the floor is smooth.(a) What maximum horizontal force F can be applied without disturbing the equilibrium ofthe system ? (b) Suppose the horizontal force applied is double of that found in part (a),find the accelerations of the two masses.

15. Body A in Fig. weighs 102 N, and body B weighs 32N. The coefficients of friction between

A and the incline are 56.0s =µ and 25.0k =µ . Angle θ is 40°. Find the acceleration of A if(a) A is initially at rest, (b) A is initially moving up the incline, and (c) A is initially movingdown the incline.

16. Two blocks, of weights, 3.6N and 7.2N, are connected by a massless string and slide down a 30°inclined plane. The coefficient of kinetic friction between the lighter block and the plane is 0.10;that between the heavier block and the plane is 0.20. Assuming that the lighter block leads, find(a) the magnitude of the acceleration of the blocks and (b) the tension in the string. (c) Describethe motion if, instead, the heavier block leads.

17. The coefficients of friction between the tires of a car and the road are 6.0s =µ and 5.0k =µ .(a) If the resultant force on the car is the force of static friction exerted by the road, what isthe maximum acceleration of the car ? (b) What is the least distance in which the car can

stop if it is initially traveling at 30 m/s?(Take )sm8.9g 2=

18. A slide forms an angle of α = 30 degrees with the horizon. A stone is thrown upward alongit and covers a distance of S = 16 metres in 2 seconds , after which it slides down. Whatis the coefficient of friction between the slide and the stone?

19. A 50-kg box must be moved across a level floor. The coefficient of static friction betweenthe box and the floor is 0.6. One method is to push down on the box at an angle θ with thehorizontal. Another method is to pull up on the box at an angle θ with the horizontal. (a)Explain why one method is better than the other. (b) Calculate the force necessary to movethe box by each method if θ =30° and compare these results with that for θ = 0°.

20. A block is on an incline whose angle can be varied. The angle is gradually increased from0°. At 30°, the block starts to slide down the incline. It slides 3m in 2s. Calculate thecoefficients of static and kinetic friction between the block and incline.

21. A rope so lies on the table that part of it hangs over . The rope begins to slide when thelength of the hanging part is 25 percent of the whole length. What is the coefficient offriction between the rope and the table ?

22. The coefficient of static friction between the flat bed of the truck and the crate it carries is0.30 (see Fig). Determine the minimum distance S in which the truck can stop from a speedof 70km/h with constant deceleration if the crate is not to slip forward.

23. If the truck in the previous problem comes to a stop from an initial speed of 70km/h in adistance of 50m with uniform deceleration, determine whether or not the crate strikes thewall at the forward end of the flat bed. If the crate does strike, calculate its speed relative tothe truck as the impact occurs. Use the friction coefficient

sµ = 0.3 and 25.0k =µ

24. Determine the tension in the cable shown in Fig which will give the 50kg block a steadyacceleration of 2m/s2 up the incline.

25. In the arrangement shown in Fig. , calculate the acceleration of each body and the tension T inthe cable. The coefficient of kinetic friction is 0.2.

B

26. In the arrangement shown in figure, the floor is frictionless, while the friction between m and

the vertical face of M (=10 m) is 5.0ks =µ=µ . The mass M is initially held at rest so that m isat a height of 2.45 m from the base of M. If M is now released, calculate (a) the accelerationof m and M, and (b) the time taken by m to reach the base.

ASSIGNMENT-II

1. Find the reading of the spring balance shown in the figure. The elevator is going up with anacceleration of g/10, the pulley and the string are light and the pulley is smooth.

2. Let kg2m,kg1m 21 == and kg3m3 = in figure. Find the accelerations of 21 m,m and 3m .

The string from the upper pulley to 1m is 20cm when the system is released from rest.

How long will it take before 1m strikes the pulley ?

3. Find the acceleration of the block of mass M in the situation of figure. The coefficient of

friction between the two blocks is 1µ and that between the bigger block and the ground is

2µ .

4. A block of mass 2kg is pushed against a rough vertical wall with a force of 40N, coefficientof static friction being 0.5. Another horizontal force of 15N, is applied on the block in adirection parallel to the wall. Will the block move ? If yes, in which direction ? If no, findthe frictional force exerted by the wall on the block.

5. A bead of mass m is fitted onto a rod of length of 2L, and can move on it without friction. At theinitial moment the bead is in the middle of the rod . The rod moves translation in a horizontal planewith an acceleration a in a direction forming an angle α with rod (Fig ). Find the acceleration ofthe bead relative to the rod, the reaction force exerted by the bead on the rod, and the time whenthe bead leaves the rod.

6. Solve the previous problem, assuming that the moving bead is acted upon by a frictionforce. The coefficient of friction between the bead and the rod is k. Disregard the force ofgravity.

7. A table with a mass of 15 kg can move without friction over a level floor. A block of mass10 kg is placed on the table, and a string passed over two pulleys fastened to the table isattached to it (Fig).The coefficient of friction between the table and the block is 0.6. Whatacceleration will the table move with if a constant force of 80 N applied to the free end ofthe rope ? Consider two cases: ( a ) the force is directed horizontally, (b) the force is directedvertically upwards.

8. A steel ball is suspended from the accelerating frame by the two chords A and B (Fig).Determine the acceleration a of the frame which will cause the tension in A to be twice thatof B.

9. In the arrangement shown in Fig. the masses m of the bar and M of the wedge , as well asthe wedge angle α are known. The masses of the pulley and the thread are negligible. Thefriction is absent. Find the acceleration of the wedge.

10. For the system at rest shown in figure, determine the accelerations of all the loads immediately afterthe lower thread keeping the system in equilibrium has been cut. Assume that the threads areweightless and inextensible, the springs are weightless, the mass of the pulley is negligibly small,and there is no friction at the point of suspension.

C

A

Figure Q.No.10 Figure Q.No. 11

11. Find the acceleration of rod A and wedge B in the arrangement shown in figure above if theratio of the mass of the wedge to that of the rod equals η , and the friction between allcontact surfaces is negligible.

12. In the system shown in figure, M =13.4 kg, m1 = 1 kg, m

2 = 2 kg, and m

3 = 3.6 kg. The

coefficient of static friction betweem m1 and m

2 is 0.75 and that between m

2 and M is 0.6.

All other surfaces are frictionless.(a) What minimum horizontal force F must be applied to M so that m

2 does not slip on M?

(b) What is the maximum value of F for which m1 does not slip on m

2?

(c) If F=100 N, does m1 slip with respect to m

2 or does m

2 slip with respect to M?

13. On a smooth fixed inclined plane of angle of inclinationθ = 60o, is placed a smooth wedgeof mass M = 8 kg with its upper face horizontal (see figure). On this horizontal face of M,another rectangular block of mass m = 4kg is placed, quite away from the inclined plane.Find the acceleration of m, when the system of (M+m) is released.

14. Three blocks P, Q and R are arranged as shown in the figure. Block Q lies on frictionlesssurface whereas coefficient of friction between blocks P and Q is 0.25. The string connect-ing blocks P and R is massless and inextensible, passing over a frictionless massless pulley.The system starts from rest.(i) Find the acceleration of P and Q.(ii) After what time, only one fifth of the block P remains on block Q.

15. A lift of total mass M kg is raised by cables from rest to rest through a height h. The greatesttension the cables can safely bear is nM kg wt. Find the shortest interval of time in whichthe ascent can be made.

16. In the pulley system shown in figure, the movable pulleys A,B,C are of mass 1 kg each . Dand E are fixed pulleys. The strings are vertical and inextensible. Find the tension in thestring and the acceleration of the frictionless pulleys.

Figure Q. 16 Figure Q. No.17

17. Two weights with mass m1 and mass m

2 are connected by a string passed over a pulley. The

surfaces on which they rest form angles α and β with the horizontal (Fig ). The righthandweight is h metres below the lefthand weight. Both weights will be at the same height in tseconds after the motion begins. The coefficient of friction between the weights and thesurfaces is k. Determine the relation between the masses of the weights.

18. In the arrangement (shown in figure), the mass of the ball is n = 1.8 times that of rod 2. Thelength of the latter is l = 1 m. The masses of the pulleys and threads, as well as the friction,are negligible. The ball is set at the same level as the lower end of the rod and then released.How soon will the ball be opposite to the other end of the rod ?

C

2

1 l

19. Two bars 1 and 2 are placed on an inclined plane forming an angle α with the horizontal (seefigure). The masses of the bars are equal to m

1 and m

2, and the coefficients of friction

between the plane and the bars are equal to k1 and k

2, respectively, with k

1 > k

2. Find :

(a) the force of interaction of the bars in the process of motion,(b) the minimum value of α at which the bars start sliding down.

20. In the figure, the distance BQ = 3 m, BP = 14 m at time t = 0. The system of blocks isreleased from rest at time t = 0. The string connecting B and C is suddenly cut at timet = 2 s. Calculate(a) the time t when the block B hits the pulley Q and(b) the velocities of B and A at this instant. The coefficient of friction between B and the

horizontal surface is µ µs k= = 0 25. . Take g = 9.8 m/s2.

21. A plank of mass M is placed on a rough inclined plane and a man of mass m walks down theplank as shown in the figure. Find the accelerations of the man so that the plank does notslip on the incline, if the coefficient of friction between the plank and the incline is µ.

Q.No.21 Q.No. 22

22. Two blocks of masses 3 kg and 5 kg hang over a pulley, as shown in the figure. The 5 kg block

is initially held 4 m above the floor and then released. What is the maximum height reached

by the 3 kg block.

ASSIGNMENT LEVEL-III

1. A cylinder of mass m rests on a supporting carriage as shown in Fig. If β = 45 degrees and

θ = 30 degrees , calculate the maximum acceleration a which the carriage may be given upthe incline so that the cylinder does not lose contact at B.

β

β

A

B

θ

2. A bar of mass m is pulled by means of a thread up an inclined plane forming an angle αwith the horizontal (Fig). The coefficient of friction is equal to k. Find the angle β whichthe thread must form with the inclined plane for the tension in the thread to be minimum.

Also, find the tension in the thread for this angle β .

F

β

3. At the moment t = 0 the force F = at is applied to a small body of mass m resting on a smoothhorizontal plane (a is a constant ) . The permanent direction of this force forms an angle αwith the horizontal (figure). Find :(a) the velocity of the body at the moment of its breaking off the plane;(b) the distance traversed by the body up to this moment.

F

4. Two balls are placed as shown in figure on a “weightless” support formed by two smoothinclined planes each of which forms an angle α with the horizontal. The support can slidewithout friction along a horizontal plane. The upper ball of mass m

1 is released. Determine

the condition under which the lower ball of mass m2 starts “climbing” up the support.

2m1m

5. A 20 kg block is originally at rest on a horizontal surface for which the coefficient of frictionis 0.6. A horizontal force F is applied such that it varies with time as shown in the figure

11(a) &11(b). Determine the speed of the block in 10 s. (Take g = 2s/m10 )

200

)(NF

0 5 10)(st

F

Fig. 11(a) Fig. 11(b)

6. A flexible rope l = 10m long, weighing 0.5 kg per meter, passes over a small frictionlesspulley. It is released from rest with 4 m of the rope hanging from one side and 6 m from theother side of the pulley. Calculate the acceleration of rope when the smaller end of twolength x metres is hanging on one side. What is the velocity of the rope when the smallerend of the rope reaches the pulley.

7. The figure shows one end of a string being pulled down by a constant velocity v. Assumethat the pulleys are massless and smooth, find the tension in the string AB. x is theinstantaneous position of m as shown.

8. A trolley A has a simple pendulum suspended from a stand fixed to its deck. A block B is in contact

with its vertical wall. The trolley is accelerated to the right such that the block just does not fall

down. Calculate the inclination θ of the pendulum thread to the vertical. The coefficient of friction

between the block and trolley is µ = 0.5.

9. Show that the tension on two sides of the string wrapped over a rough pulley are related as

T T e T T2 1 2 1= >µθ ( )

where µ is the coefficient of friction and θ is the angle subtended by the string as shown infigure.

10. A fixed pulley carries a weightless thread with masses m1 and m

2 at its ends. There is friction

between the thread and the pulley. It is such that the thread starts slipping when the ratio

m

m1

2

2= . Find

(a) the friction coefficient

(b) the acceleration of the masses when m

m1

2

3=

11. A block of mass m is projected on a larger block of mass 10 m with a velocity v as shown.The larger block is initially at rest and has a length l. The coefficient of friction between thetwo blocks is µ

2 while that between the lower block and the ground is µ

1 . Given that µ

2 >

11µ1,

(a) Find the minimum value of v such that the mass m falls off the block of mass 10 m.(b) If v has this minimum value, find the time taken by block m to do so.

12. A chain of mass M and length 2l hangs in equilibrium over a smooth pulley, as shown in thefigure. An insect of mass m sits at one end of it and begins to crawl up with uniform velocityu

rel with respect to the chain. Find the velocity with which the chain leaves the pulley.

13. A uniform flexible chain of length 1.50 m rests on a fixed smooth sphere of radius mR π/2=such that one end A of chain is at top of the sphere while the other end B is hanging freely.Chain is held stationary by a horizontal thread PA as shown in fig.Calculate acceleration of chain when the thread is burnt.

PA

R

B

14. In the arrangement shown in fig., pulleys D and E are small and frictionless, their massesbeing 4kg and 11.25kg respectively while masses of blocks A,B and C are 2m ,m and 'mrespectively. When the system B is released from rest, downward accelerations of blocks B

and C relative to A are found to be 25 −ms land 23 −ms respectively. Calculate

i) accelerations of blocks B and C, relative to the ground andii) mass of each block

D

E

A B C

m2 'm

Figure Q.No.15

Figure Q.No.14

15. A chain AB of length l is located in a smooth horizontal tube so that its fraction of length h hangsfreely and touches the surface of the table with its end B. At a certain moment, the end A of thechain is set free. With what velocity will this end of the chain slip out of the tube ?

16. In the arrangement shown in fig., pulleys are small, light and frictionless, threads are

inextensible and mass of blocks A,B and C are kgmkgm 4,5 21 == and kgm 5.23 = respec-

tively. Co-efficient of friction for both the planes is 50.0=µ . Calculate acceleration of eachblock when system is released from rest.

°= 37θ

A

B

C

17. In the arrangement shown in fig. mass of blocks A , B and C are 18.5 kg, 8kg and 1.5kgrespectively. Bottom surface of A is smooth, while co-efficient of friction between B andfloor is 0.2 and that between blocks A and C is 1/3. System is released from rest at 0=t andpulleys are light and frictionless. Calculate

AC

B

i) acceleration of block C, andii) energy lost due to friction during first 0.2 sec.

18. A block resting over a horizontal floor has a symmetric track ABC, as shown in fig. Mass ofthe block is kgM 12.3= . Length =AB Length BC = 1m. A block of mass m =2kg is put onthe track at A and the system is released from rest. Neglecting friction and impact at B,calculate period of horizontal oscillations performed by the block of mass M.

°37 °37

A C

B M

19. In the arrangement shown in fig., a wedge of mass kgm 45.33 = is placed on a smoothhorizontal surface. A small and light pulley is connected on its top edge, as shown. A light,

flexible thread passes over the pulley. Two blocks having mass kgm 3.11 = and kgm 5.12 =

are connected at the ends of the thread. Mass 1m is on smooth horizontal surface and 2m

rests on inclined surface of the wedge. Base length of wedge is 2m and inclination is 2.37 m° isinitially near the top edge of the wedge.If the whole system is released from rest, calculate

1m

2m3m

°37

i) velocity of wedge when 2m reaches its bottom

ii) velocity of 2m at that instant and tension in the thread during motion of 2m .All the surfaces are smooth.

20. A small, light pulley is attached with a block C of mass 4kg as shown in fig. A block B ofmass 1.5 kg is placed on the top horizontal surface of C. Another block A of mass 2 kg is

hanging from a string, attached with B and passing over the pulley. Taking 210 −= msg and

neglecting friction, calculate acceleration of each block when the system is released fromrest. If initial height of lower surface of block A is 12.5 cm from bottom of a hole cut in C,calculate kinetic energy of each block and loss of potential energy of A when it hits thebottom of the hole.

cm5.12 C

B

A

21. A board is fixed to the floor of an elevator such that the board forms angle °= 37θ withhorizontal floor of the elevator accelerating upwards.A block is placed on point A of the board as shown if fig. When motion with velocity

11 24 −= msv is given to the block, it comes to rest after moving a distance ml 6.1= rela-

tive to the board. Its velocity was 12 4 −= msv down the board when it returns to point A.

Calculate acceleration a of elevator and co-efficient of friction µ between the board and theblock.

Figure Q.No.21

°37

ASSIGNMENT I

SECTION B

1. (a) 198.5; (b) 123.5 N2. 0.613. (a) 188.6 N; (b) 0.57 m/s2

4. (a) 0.13N; (b) 0.12

5. no; (b) N)j5i12( +−

6. (a) Ni6− (b) Ni75.3− (c) Ni25.3−7. (a) 304.2 N; (b) 1.3 m/s2

8. (a) 66N; (b) 2.3 m/s2

9. 102.6 N

10. (a) 0.4 2sm (b) 2.4 N (c) 4.8 N

11. 06.0,75.0 21 =µ=µ12. 96 N in the left string and 68N in the right

13. µ++µ

1

g)mM(

14. (a) 2 mgµ (b) mM

mg2

+µ

in opposite directions

15. (a) 0; (b) 3.9 m/s2 down the incline; (c) 1.0 m/s2 down the incline16. (a) 3.5 m/s2; (b) 0.21 N; (c) blocks move independently17. (a) 5.89 m/s2 (b) 76.5 m18. µ k = 0.3719. (b) pushing at 30°, 520 N; pulling at 30°, 252 N; pushing or pulling at 0°’ 294 N

20. 401.0;577.0 ks =µ=µ21. µ =1/322. S = 64.3 m23. The crate will strike the wall with a velocity of 2.82 m/s2

24. P = 227 N25. a

A = 1.45 m/s2, a

B = 0.725 m/s2, T = 105.4 N.

26. (a) 8

g towards right and

4

g downwards (b) s2

Assignment Level-II

1. 4.4 kg

2. s25.0),down(g29

21),down(

29

17),up(g

29

19

3. )](25[mM

g)]mM(m2[

21

2

µ−µ+++µ−

4. it will move at an angle of 53° with the 15N force5. -6. 2/1))sink(cosa/L2(),sink(cosaw,cosmaN α−α=τα−α=α=7. (a) a = 314 cm/s2 (b) a = - 131 cm/ s2

8. a = g/(27)1/2

9. w = mg sinα /( M +2m (1-cosα ))

10. g.m

)mmmm(a;0aaa

4

21434321

−−+====

11. wA=g/(1 + η cot2 α ), w

B=g/(tanα + η cotα )

12. -13. 8.182 m s-2.

14. g83

aP = , 16g

aQ = (ii) g

L3

5

16t =

15.21

g)1n(

nh2t

−=

16. T = 6.5N; aA = a

B = –a

C =g/3.

17. )h2)cosk)(sinsin(sing)(h2)sincosk)(sin(sing(m:m 2221 −α−αβ+ατ+β+ββ+ατ=

18. 1.4 sec

19. (a) ( )21

2121 cos

mm

gmmkKF

+−= α (b)

21

2211tanmm

mkmk

++=α

20. ;sec4

55 7 m/s

21. ( ) ( )gm

mMag

m

mm θµθθµθ cossincossin +

+≤≤−

+

22. 5 m

Assignment Level-III

1. a = 0.366 g.2. tanβ = k , T = mg (sinα + k cosα )/(1+k2)1/2

3. (a) ;sina2

cosmgv

2

2

αα= (b)

αα=

32

32

sina6

cosgms

4. m2 < m

1 cos2α . 5. V = 24 m/s

6. ;2

1 gl

x

− 6.86 m/s 7.

+

3

22

41

gx

vbmg

8. 2tan 1− 9. -

10. π=µ 2ln

(b)5

ga =

11. (a)( )

10

22 12 glµµ −(b) ( )1211

20

µµ −g

l

12.2

2

++

++

relUmM

mgl

mM

mM13. 258.7.

3

4 −=+msg

ππ

14. i) 22 1,3 −− msms (both downwards)

ii) Mass of A = 18 kgMass of B = 9 kgMass of C = 7 kg

15.

=h

llngh2v

16. 232

21 2,0,4 −− === msaamsa

17. i) 210 −ms ii) joule19.0

18. 2sec

19. i) 12 −ms ii) N9.3,sm13

20. Vertical accleration of 225.6 −= msA (Downward)

Horizontal acceleration of 225.1 −= msA (Rightward)

Acceleration of 200.5 −= msB (Leftward)

Acceleration of 225.1 −= msC (Rightward)

KE of JA 625.1=

KE of JB 75.0=

KE of JC 125.0=Loss of JPE 50.2=

21. 25.0,5.2 2−ms