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 the magnitude of the minimum horizontal force that a person must apply to start the almirah moving ? (b) If the drawers and clothing, with 17 kg mass, are removed before the almirah is pushed, what is the new minimum magnitude? (Take g = 9.8 m s 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 µ k between the player and the ground?
3.
A person pushed a crate of mass 55kg horizontally with a force of 220N to move it across a level 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 g = 9.8 m s 2 )
4.
A 110 g hockey puck sent sliding over ice is stopped in 15 m by the frictional force on it from 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 F pushes a block weighing 5.0 N against a vertical wall (Fig.). The coefficient of static friction between the wall and the block is 0.60, and the coefficient of kinetic 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 the wall.
6.
A 2.5 kg block is initially at rest on a horizontal surface. A 6.0 N horizontal force and a r vertical force P are applied to the block as shown in Fig. The coefficients of friction for the block and surface are µ s = 0.40 and µ k = 0.25 . Determine the magnitude and direction of
r
r
the frictional force acting on the block if the magnitude of P is (a) 8.0 N, (b) 10 N and (c) 12 N.(Take g = 10 m s 2 )
7.
A 68 kg crate is dragged across a floor by pulling on a rope attached to the crate and inclined 15° 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 µ k = 0.35 , what is the magnitude of the initial acceleration of the crate ? (Take g = 9.8 m s 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 µs between A and the table is 0.20. (b) Block C suddenly is lifted off A. What is the acceleration of block A if µ k between A and the table is 0.15?
9.
Block B in Fig. weighs 711 N. The coefficient of static friction between block and table is 0.25; assume that the cord between B and the knot is horizontal. Find the maximum weight of block A for which the system will be stationary.
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 string attached to 0.50kg.
11.
If the tension in the string in figure is 16N and the acceleration of each block is 0.5 m s 2 , 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 the tensions in the two strings.
13.
The friction coefficient between the board and the floor shown in figure is µ . Find the maximum force that the man can exert on the rope so that the board does not slip on the floor.
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 of the 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 µ s = 0.56 and µ k = 0.25 . 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 moving down 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) Describe the motion if, instead, the heavier block leads.
17.
The coefficients of friction between the tires of a car and the road are µs = 0.6 and µ k = 0.5 . (a) If the resultant force on the car is the force of static friction exerted by the road, what is the 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 g = 9.8 m s 2 )
18.
A slide forms an angle of α = 30 degrees with the horizon. A stone is thrown upward along it and covers a distance of S = 16 metres in 2 seconds , after which it slides down. What is 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 between the box and the floor is 0.6. One method is to push down on the box at an angle θ with the horizontal. 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 move the 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 from 0°. At 30°, the block starts to slide down the incline. It slides 3m in 2s. Calculate the coefficients 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 the length of the hanging part is 25 percent of the whole length. What is the coefficient of friction between the rope and the table ?
22.
The coefficient of static friction between the flat bed of the truck and the crate it carries is 0.30 (see Fig). Determine the minimum distance S in which the truck can stop from a speed of 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 a distance of 50m with uniform deceleration, determine whether or not the crate strikes the wall at the forward end of the flat bed. If the crate does strike, calculate its speed relative to the truck as the impact occurs. Use the friction coefficient µ s = 0.3 and µ k = 0.25
24.
Determine the tension in the cable shown in Fig which will give the 50kg block a steady acceleration of 2m/s2 up the incline.
25. In the arrangement shown in Fig. , calculate the acceleration of each body and the tension T in the 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 µ s = µ k = 0.5 . The mass M is initially held at rest so that m is at a height of 2.45 m from the base of M. If M is now released, calculate (a) the acceleration of 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 an acceleration of g/10, the pulley and the string are light and the pulley is smooth.
2.
Let m1 = 1kg, m 2 = 2kg and m3 = 3kg in figure. Find the accelerations of m1, m2 and m 3 . The string from the upper pulley to m1 is 20cm when the system is released from rest. How long will it take before m1 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 µ1 and that between the bigger block and the ground is
µ2 .
4.
A block of mass 2kg is pushed against a rough vertical wall with a force of 40N, coefficient of static friction being 0.5. Another horizontal force of 15N, is applied on the block in a direction parallel to the wall. Will the block move ? If yes, in which direction ? If no, find the 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 the initial moment the bead is in the middle of the rod . The rod moves translation in a horizontal plane with an acceleration a in a direction forming an angle α with rod (Fig ). Find the acceleration of the bead relative to the rod, the reaction force exerted by the bead on the rod, and the time when the bead leaves the rod.
6.
Solve the previous problem, assuming that the moving bead is acted upon by a friction force. The coefficient of friction between the bead and the rod is k. Disregard the force of gravity.
7.
A table with a mass of 15 kg can move without friction over a level floor. A block of mass 10 kg is placed on the table, and a string passed over two pulleys fastened to the table is attached to it (Fig).The coefficient of friction between the table and the block is 0.6. What acceleration will the table move with if a constant force of 80 N applied to the free end of the rope ? Consider two cases: ( a ) the force is directed horizontally, (b) the force is directed vertically 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 that of B.
9.
In the arrangement shown in Fig. the masses m of the bar and M of the wedge , as well as the wedge angle α are known. The masses of the pulley and the thread are negligible. The friction 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 after the lower thread keeping the system in equilibrium has been cut. Assume that the threads are weightless and inextensible, the springs are weightless, the mass of the pulley is negligibly small, and there is no friction at the point of suspension.
A C
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 the ratio of the mass of the wedge to that of the rod equals η , and the friction between all contact surfaces is negligible.
12.
In the system shown in figure, M =13.4 kg, m1 = 1 kg, m2 = 2 kg, and m3 = 3.6 kg. The coefficient of static friction betweem m1 and m2 is 0.75 and that between m2 and M is 0.6. All other surfaces are frictionless. (a) What minimum horizontal force F must be applied to M so that m2 does not slip on M? (b) What is the maximum value of F for which m1 does not slip on m2? (c) If F=100 N, does m1 slip with respect to m2 or does m2 slip with respect to M?
13.
On a smooth fixed inclined plane of angle of inclination θ = 60o, is placed a smooth wedge of 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 frictionless surface whereas coefficient of friction between blocks P and Q is 0.25. The string connecting 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 greatest tension the cables can safely bear is nM kg wt. Find the shortest interval of time in which the ascent can be made.
16.
In the pulley system shown in figure, the movable pulleys A,B,C are of mass 1 kg each . D and E are fixed pulleys. The strings are vertical and inextensible. Find the tension in the string and the acceleration of the frictionless pulleys.
Figure Q. 16
Figure Q. No.17
17.
Two weights with mass m1 and mass m2 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 righthand weight is h metres below the lefthand weight. Both weights will be at the same height in t seconds after the motion begins. The coefficient of friction between the weights and the surfaces 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. The length 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
19.
l
Two bars 1 and 2 are placed on an inclined plane forming an angle α with the horizontal (see figure). The masses of the bars are equal to m1 and m2, and the coefficients of friction between the plane and the bars are equal to k1 and k2, respectively, with k1 > k2. Find :
(a) (b)
the force of interaction of the bars in the process of motion, 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 is released from rest at time t = 0. The string connecting B and C is suddenly cut at time t = 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 s = µ 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 the plank as shown in the figure. Find the accelerations of the man so that the plank does not slip on the incline, if the coefficient of friction between the plank and the incline is µ.
Q.No.21
22.
Q.No. 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 up the incline so that the cylinder does not lose contact at B. β B
β
A
θ
2.
A bar of mass m is pulled by means of a thread up an inclined plane forming an angle α with the horizontal (Fig). The coefficient of friction is equal to k. Find the angle β which the 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 smooth horizontal plane (a is a constant ) . The permanent direction of this force forms an angle α with 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 smooth inclined planes each of which forms an angle α with the horizontal. The support can slide without friction along a horizontal plane. The upper ball of mass m1 is released. Determine the condition under which the lower ball of mass m2 starts “climbing” up the support.
m1 m2
5.
A 20 kg block is originally at rest on a horizontal surface for which the coefficient of friction is 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 = 10m / s 2 ) F (N )
200
0
F
5
Fig. 11(a)
10
t (s )
Fig. 11(b)
6.
A flexible rope l = 10m long, weighing 0.5 kg per meter, passes over a small frictionless pulley. It is released from rest with 4 m of the rope hanging from one side and 6 m from the other side of the pulley. Calculate the acceleration of rope when the smaller end of two length x metres is hanging on one side. What is the velocity of the rope when the smaller end of the rope reaches the pulley.
7.
The figure shows one end of a string being pulled down by a constant velocity v. Assume that the pulleys are massless and smooth, find the tension in the string AB. x is the instantaneous 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 T2 = T1e µθ
(T2 > T1 )
where µ is the coefficient of friction and θ is the angle subtended by the string as shown in figure.
10.
A fixed pulley carries a weightless thread with masses m1 and m2 at its ends. There is friction between the thread and the pulley. It is such that the thread starts slipping when the ratio m1 = 2. Find m2
11.
(a)
the friction coefficient
(b)
m1 the acceleration of the masses when m = 3 2
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 the two blocks is µ2 while that between the lower block and the ground is µ1 . Given that µ2 > 11µ1,
(a) (b)
Find the minimum value of v such that the mass m falls off the block of mass 10 m. 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 the figure. An insect of mass m sits at one end of it and begins to crawl up with uniform velocity urel 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 R = 2 / π m 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. P
A R
B
14.
In the arrangement shown in fig., pulleys D and E are small and frictionless, their masses being 4kg and 11.25kg respectively while masses of blocks A,B and C are 2m ,m and m ' respectively. When the system B is released from rest, downward accelerations of blocks B and C relative to A are found to be 5 ms −2 land 3 ms −2 respectively. Calculate i) accelerations of blocks B and C, relative to the ground and ii) mass of each block
D
E
Figure Q.No.15 A 2m
B
C m'
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 hangs freely and touches the surface of the table with its end B. At a certain moment, the end A of the chain 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 m1 = 5kg , m2 = 4kg and m3 = 2.5 kg respectively. Co-efficient of friction for both the planes is µ = 0.50 . Calculate acceleration of each block when system is released from rest. A B θ = 37°
C
17.
In the arrangement shown in fig. mass of blocks A , B and C are 18.5 kg, 8kg and 1.5kg respectively. Bottom surface of A is smooth, while co-efficient of friction between B and floor is 0.2 and that between blocks A and C is 1/3. System is released from rest at t = 0 and pulleys are light and frictionless. Calculate
A
i) ii) 18.
B
C
acceleration of block C, and energy lost due to friction during first 0.2 sec.
A block resting over a horizontal floor has a symmetric track ABC, as shown in fig. Mass of the block is M = 3.12kg . Length AB = Length BC = 1m. A block of mass m =2kg is put on the 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. A
C 37° B
19.
37° M
In the arrangement shown in fig., a wedge of mass m3 = 3.45 kg is placed on a smooth horizontal 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 m1 = 1.3 kg and m2 = 1.5 kg are connected at the ends of the thread. Mass m1 is on smooth horizontal surface and m2
rests on inclined surface of the wedge. Base length of wedge is 2m and inclination is 37°. m2 is initially near the top edge of the wedge. If the whole system is released from rest, calculate m1 m2
m3
37°
20.
i)
velocity of wedge when m2 reaches its bottom
ii)
velocity of m2 at that instant and tension in the thread during motion of m2 . All the surfaces are smooth.
A small, light pulley is attached with a block C of mass 4kg as shown in fig. A block B of mass 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 g = 10 ms −2 and neglecting friction, calculate acceleration of each block when the system is released from rest. 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 the bottom of the hole. B
A 12.5 cm
21.
C
A board is fixed to the floor of an elevator such that the board forms angle θ = 37° with horizontal floor of the elevator accelerating upwards. A block is placed on point A of the board as shown if fig. When motion with velocity v1 = 4 2 ms −1 is given to the block, it comes to rest after moving a distance l = 1.6 m rela-
tive to the board. Its velocity was v2 = 4 ms −1 down the board when it returns to point A. Calculate acceleration a of elevator and co-efficient of friction µ between the board and the block.
Figure Q.No.21 37°
ASSIGNMENT I SECTION B 1. 2. 3. 4.
(a) 198.5; (b) 123.5 N 0.61 (a) 188.6 N; (b) 0.57 m/s2 (a) 0.13N; (b) 0.12
5.
no; (b) (−12ˆi + 5ˆj) N
6. 7. 8. 9.
(a) − 6ˆiN (b) − 3.75ˆiN (a) 304.2 N; (b) 1.3 m/s2 (a) 66N; (b) 2.3 m/s2 102.6 N
(c) − 3.25ˆiN
10.
(a) 0.4 m s 2 (b) 2.4 N
(c) 4.8 N
11. 12.
µ1 = 0.75, µ 2 = 0.06 96 N in the left string and 68N in the right
13.
µ ( M + m)g 1+ µ
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
2µmg in opposite directions M+m (a) 0; (b) 3.9 m/s2 down the incline; (c) 1.0 m/s2 down the incline (a) 3.5 m/s2; (b) 0.21 N; (c) blocks move independently (a) 5.89 m/s2 (b) 76.5 m
(a) 2 µmg
(b)
µ k = 0.37
(b) pushing at 30°, 520 N; pulling at 30°, 252 N; pushing or pulling at 0°’ 294 N µ s = 0.577; µ k = 0.401 µ =1/3 S = 64.3 m The crate will strike the wall with a velocity of 2.82 m/s2 P = 227 N aA = 1.45 m/s2, aB = 0.725 m/s2, T = 105.4 N.
(a)
g g towards right and downwards (b) 8 4
2s
Assignment Level-II 1.
4.4 kg
2.
19 17 21 g ( up), (down ), g (down ),0.25s 29 29 29
3.
[2m − µ 2 (M + m)]g M + m[5 + 2(µ1 − µ 2 )]
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
it will move at an angle of 53° with the 15N force N = ma cos α, w = a (cos α − k sin α), τ = (2L / a (cos α − k sin α))1 / 2 (a) a = 314 cm/s2 (b) a = - 131 cm/ s2 a = g/(27)1/2 w = mg sin α /( M +2m (1-cos α ))
a1 = a 2 = a 3 = 0; a 4 =
(m3 + m 4 − m1 − m 2 ) .g m4
wA=g/(1 + η cot2 α ), wB=g/(tan α + η cot α )
8.182 m s-2.
3 g aP = g , aQ = 8 16
(ii) t =
16 3L 5 g
12
15.
2nh t= (n − 1)g
16.
T = 6.5N; aA = aB = –aC =g/3.
17. 18.
m1 : m 2 = (gτ 2 (sin α + sin β)(k cos β + sin β) + 2h )(gτ 2 (sin α + sin β)(sin α − k cos α ) − 2 h )
1.4 sec
19.
(a)
20.
55 sec ; 7 m/s 4
21.
M +m m+m (sin θ + µ cosθ )g (sinθ − µ cosθ )g ≤ a ≤ m m
22.
5m
F=
(K1 − k 2 )m1m2 g cosα m1 + m2
(b)
k m +k m tan α = 1 1 2 2 m1 + m2
Assignment Level-III 1. 2.
a = 0.366 g. tan β = k , T = mg (sin α + k cos α )/(1+k2)1/2
3.
(a) v =
4.
mg 2 cos α
2a sin 2 α m2 < m1 cos2 α .
6.
2x 1 − g ; l
8.
tan −1 2
10.
µ=
11.
(a)
12. 14.
ln 2 (b) π
; (b) s =
m 2g 3 cos α 6a 2 sin 3 α
6.86 m/s
a=
22(µ 2 − µ1 )gl 10
V = 24 m/s
7.
b 2v 2 mg 1 + 4 gx3
9.
-
g 5
(b)
m M + 2m U rel gl + M +m M + m
i) ii)
5.
20l 11g (µ 2 − µ1 ) 2
13.
4 +π .g = 7.58 ms − 2 3π
3ms −2 ,1ms −2 (both downwards)
Mass of A = 18 kg Mass of B = 9 kg Mass of C = 7 kg
15.
l v = 2gh ln h
16.
a1 = 4 ms −2 , a2 = 0, a3 = 2ms −2
17. 18.
i) 2sec
19.
i)
20.
Vertical accleration of A = 6.25 ms −2 (Downward)
10ms −2
2ms −1
ii) ii)
0.19 joule
13 m s ,3.9 N
Horizontal acceleration of A = 1.25 ms −2 (Rightward) Acceleration of B = 5.00 ms −2 (Leftward) Acceleration of C = 1.25 ms −2 (Rightward) KE of A = 1.625 J KE of B = 0.75 J KE of C = 0.125 J Loss of PE = 2.50 J 21.
2.5 ms −2 ,0.25
