MKA Physics -1

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PROBLEMS IN PHYSICS MKA1.

Two particles are simultaneously thrown from roofs of two high buildings, as shown in figure. Their velocities of projection are 2ms-1 and 14 ms-1 respectively. Horizontal and vertical separation between points A and B is 22m and 9 m respectively. Calculate minimum separation between the particles in the process of their motion.

Ans:

6.00 m

MKA 2. A ball of mass m is thrown at an angle of 45 0 to the horizontal from top of a 65 m high tower AB as shown in figure. Another identical ball is thrown with velocity 20 ms-1 horizontally towards AB from top of a 30 m high tower CD one second after the projection of first ball. Both the balls move in same vertical plane. If they collide in mid air (i) Calculate distance A.C. (ii) During collision the two balls get stuck together, calculate the distance between A and the point on the ground, at which the combined ball strikes. Given g = 10 ms-2. Ans:

(i) 40 m

(ii) 15 m

MKA 3. Two inclined planes OA and OB having 3 inclination (with horizontal) 300 and 600 respectively, intersect each other at O as shown in figure. A particle is projected from point P with velocity u = 10 ms -1 along a direction perpendicular to plane OA. If the particle strikes plane OB perpendicularly at Q, calculate (i) velocity with which particle strikes the plane OB, (ii) time of flight, (iii) vertical height h of P from O, (iv) maximum height from O, attained by the particle, and (v) Distance PQ.. Ans:

(i) 10 ms-1 (iii) 5 m (v) 20 m

(ii) 2 sec. (iv) 16.25

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MKA 4. A particle is moving along a vertical 3 circle of radius R = 20 m with a constant speed v = 31.4 ms-1 as shown in figure. Straight line ABC is horizontal and passes through the centre of the circle. A shell is fired from point A at the instant when particle is at C. If distance AB is 20m and shell collides with the particle at B, calculate (i) smallest possible value of the angle  of projection, (ii) Corresponding velocity u of projection. ( = 3.14 and g = 10 ms-2) Ans:

(i)  =  1  (2n  1)2  30o  tan-1  3  (ii) 20 ms-1

MKA 5. A particle is projected from point O on 5 the ground with velocity u = 5 ms -1 at angle =tan-1 (0.5). It strikes at a point C on a fixed smooth plane AB having inclination of 37 0 with horizontal. If the particle does not rebound, calculate (i) (ii)

Ans:

co – ordinates of point C in reference to co-ordinate system shown in figure. maximum height from the ground to which the particle rises. (g = 10 ms-2) (i) (5 m, 1.25 m) (ii) 4.45 m

MKA 6. Two identical shells are fired from a point on the ground with same muzzle velocity at angles of elevation  = 450 and  = tan-1 3 towards top of a cliff, 20 m away from point of firing. If both the shells reach the top simultaneously, calculate (i) muzzle velocity, (ii) height of the cliff, and (iii) time interval between two firings. If just before striking the top of cliff the two shells get stuck together, considering elastic collision of combined body with the top, calculate (iv) maximum height reached by the combined body. (i) 20 ms-1 (iii) (above the ground Ans:

(iv)

12

(ii) 10 m m ( 10  2)sec  1.72 sec

MKA 7. A shell of mass m = 700 gm is fired from ground with a velocity 40 ms -1. At highest point of its trajectory, it collides inelastically with a ball of mass M = 1.3 kg, suspended by a flexible thread of length 1.40 m. If thread deviates through an angle of 120 0, calculate (i) angle of projection of shell, (ii) maximum height of combined body from ground, and (iii) distance between point of suspension of ball and point of projection of shell. Ans:

(i) 60

(ii) 62.3625 m

(iii) 92.57 m

MKA 8. A circle of radius R = 2 m is marked on upper surface of a horizontal board, initially at rest. A particle starts from rest along the circle with a tangential acceleration a = 0.25 ms -2. At the FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

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same instant board accelerates upwards with acceleration b = 2.5 ms -2. If the co-efficient of friction between board and particle is  = 0.1, what distance with the particle travel on the board without sliding? (g = 10 ms-2) Ans:

2 6m

MKA 9. Two small particles A  155  C and B having   18   masses m = 0.5 kg  each and charge q1 = and q = ( + 100 C) respectively, are connected at the ends of a non– conducting, flexible and inextensible string of length r = 0.5 m. Particle A is fixed and B is whirled along a vertical circle with centre at A. If a vertically upward electric field of strength E = 1.1  105 NC-1 exists in the space, calculate minimum velocity of particle B, required at highest point so that it may just complete the circle. (g =10 ms-2) Ans:

61ms-1

MKA 10. A small sphere of mass m = 0.5 kg carrying a positive charge q = 110 C is connected with a light, flexible and inextensible string of length r = 60 cm and whirled in a vertical circle. If a vertically upwards electric field of strength E = 10 5 NC-1 exists in the space, calculate minimum velocity of sphere required at highest point so that it may just complete the circle. ( g = 10 ms-2) Ans:

6 ms-1

MKA 11. A small sphere of mass m = 0.6 kg carrying positive charge q = 80 C is connected with a light, flexible and inextensible string of length r = 30 cm and whirled in a vertical circle. If a horizontally rightward electric field of strength E = 10 5 NC-1 exists in the space calculate minimum velocity of sphere required at highest point so that it may just complete the circle. ( g = 10 ms-2) Ans:

3 ms-1

MKA 12. A particle of mass m = 0.1 kg and having positive charge q = 75 C is suspended from a point by a thread of length l =10 cm. In the space a uniform horizontal electric field E=10 4 NC-1 exists. The particle is drawn aside so that thread becomes vertical and then it is projected orizontally with velocity v such that the particle starts to move along a circle with the same constant speed v. Calculate radius of the circle and speed v. ( g = 10ms-2) Ans:

6 cm, 0.75 ms-1

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MKA 13. Two blocks A and B of mass 1 kg and 2 kg respectively are connected by a string, passing over a light frictionless pulley. Both the blocks are resting on a horizontal floor and the pulley is held such that string remains just taut. At moment t = 0, a force F = 20 t Newton starts acting on the pulley along vertically upward direction, as shown in figure. Calculate (i) velocity of A when B loses contact with the floor, (ii) height raised by the pulley upto that instant, and (iii) work done by the force F upto that instant. ( g = 10ms -2) Ans:

(i) 5 ms-1

(ii) 5/6 m

(iii) 175/6 joule

MKA 14. A uniform flexible chain of length 1.50 m rests on a fixed smooth sphere of radius R = 2/ 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 figure. Calculate acceleration of chain when the thread is burnt. ( g = 10ms-2) Ans: 4 .g  7.58ms 2 3

MKA 15. In the arrangement shown in figure 15. Pulley D and E are small and frictionless. They do not rotate but threads slip over them without friction and their masses being 4 kg and 11.25kg respectively while masses of blocks A, B and C are 2 m, m and m’ respectively. When the system is released from rest, downward accelerations of blocks B and C relative to A are found to be 5 ms-2 and 3 ms-2 respectively. Calculate (i) accelerations of blocks B and C, relative to the ground, and (ii) mass of each block. ( g = 10ms-2) Ans:

(i) 3 ms-2, 1 ms-2 (Both downwards)

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

MKA 16. In the arrangement shown in figure. mass of block A, B and C is 7.5 kg, 6 kg and 1 kg respectively. the pulley is solid circular disc of mass 0.5 kg, radius 20 cm and thickness 1 cm. Thread between block A and pulley is horizontal and that between pulley and block C is vertical. The pulley is free to rotate about axis O without friction and thread does not slip over its curved surface. Neglecting friction between blocks B and C and that between blocks and the floor, calculate resultant acceleration of block C when the system is released. ( g = 10ms-2) Ans:

5 ms2

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MKA 17.In the arrangement shown in figure, pulley are small, light and frictionless, threads are inextensible and mass of blocks A, B and C is m 1 = 5 kg, m2 = 4 kg and m3 = 2.5 kg respectively. co – efficient of friction for both the planes is  = 0.50. Calculate acceleration of each block when system is released from rest. ( g = 10ms-2) Ans:

a1 = 4 ms-2, a2 = 0, a3 = 2 ms-2

MKA 18. In the arrangement shown in figure mass of blocks A, B and C is 18.5 kg, 8 kg and 1.5 kg 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 (i) acceleration of block C, and (ii) energy lost due to friction during first 0.2 sec. (g = 10 ms-2) Ans:

(i) (ii)

0.19

10 ms 2

joule

MKA 19. A block resting over a horizontal floor has a symmetric track ABC, as shown in figure. Mass of the block is M = 3.12 kg. Length AB = Length BC = 1 m. A block of mass m = 2 kg 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. ( g = 10 ms-2) Ans:

2 sec.

MKA 20. In the arrangement shown in figure 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, FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

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flexible thread passes over the pulley. Two blocks having mass m 1 = 1.3 kg and m2 = 1.5 kg are connected at the ends of the thread. m 1 is on smooth horizontal surface and m 2 rests on inclined surface of the wedge. Base length of wedge is 2m and inclination is 37 0. m2 is initially near the top edge of the wedge. If the whole system is released from rest, calculate (i) velocity of wedge when m2 reaches its bottom, (ii) velocity of m2 at that instant and tension in the thread during motion of m 2. All the surface are smooth. ( g = 10 ms-2) Ans:

(i) 2 ms-1 13 ms1, 3.9 newton (ii)

MKA 21. A small, light pulley is attached with a block C of mass 4 kg is placed on the top horizontal surface of C. Another lock 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. Ans: Vertical acceleration of A = 6.25 ms -2 (Downward) 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

MKA 22. A board is fixed to the floor of 2 an elevator such that the board forms angle  = 370 with horizontal floor of the elevator accelerating upwards. A block is placed on point A of the board as shown in figure. When motion with velocity v1 = 4 ms1 is given to the block, it comes to rest after moving a distance l = 1.6m relative 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  between the board and the block. (g = 10 ms2) Ans: 2.5ms-2, 0.25

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MKA 23. A block B of mass 10 kg is resting over a smooth horizontal plane. Distance of B from the wall is 40 cm and it is held at rest by an inextensible thread BD. Another thread is connected to left face of B and a block A of mass 2 kg is suspended as shown in figure. Block C of mass 2 kg is resting against the vertical wall. Blocks B and C are hinged at the ends of a light rigid rod. Assuming friction to 20 cm is absent, calculate acceleration of each block when thread BD is burnt. (g = 10 ms2) Ans:

Acceleration of A = 3 ms-2() Acceleration of B = 3 ms-2( ) Acceleration of C = 4 ms-2()

MKA 24. Three identical blocks A, B and C, each of mass m = 7 kg are connected with each other by light and inextensible strings, as shown in figure. Strings pass over light and frictionless pulleys fixed to the edges of trolly of mass M = 21 kg. If coefficient of friction between blocks and trolly surfaces is  = 4/7, calculate maximum possible value of angle  so that block B remains stationary relative to the trolly. Calculate also, the force F to be applied horizontally on the trolley. Calculate also, the force F to be applied horizontally on the trolley. Ans:

37, 315 newton

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MKA 25. Two blocks of mass m1 and m2 are attached at the ends of an ideal spring of force constant K and natural length l0. The system rests on a smooth horizontal plane. Blocks having mass m1 and m2 are pulled apart by applying force F1 and F2 respectively as shown in figure. Calculate maximum elongation of the spring. Ans:

2  m1F2  m2F1   K  m1  m2 

MKA 26.A uniform solid sphere of radius R = 44 cm is cut into two parts by a plane. Distance of the plane from centre of the sphere is a = 26.4 cm as shown in figure. Calculate distance of centre of mass of heavier part from centre O. Ans:

132 cm 35

MKA 27. A vehicle of mass m starts moving S along a horizontal circle of radius R such that its speed varies with distances s covered by the vehicle as v = K, where K is a constant. Calculate i) tangential and normal force on vehicle as function of s, ii) distance s in terms of time t, and iii) work done by the resultant force in first t seconds after the beginning of motion. Ans:

(i)

(ii)

(iii)

2 1 11 2 2mK s K, t42t 2 mKmK 2 84 R

MKA 28. A particle of mass m moves along a horizontal circle of radius R such that normal acceleration of particle varies with time as an = Kt2, where K is a constant. Calculate (i) tangential force on particle at time t, (ii) total force on particle at time t. (iii) power developed by total force at time t, and (iv) average power developed by total force over first t second. Ans: (i) (ii) m m k(RKR  Kt 4 ) (iii) (iv) 1mKRT MKA 29. Two identical blocks A and B, each of 2 mKRt mass m = 2 kg are connected to the ends of and ideal spring having force constant K = 1000 Nm1. System of these blocks and spring is placed on a rough force. Coefficient of friction between blocks and floor is  = 0.5. Block B is pressed towards left so that spring gets compressed. (i) Calculate initial minimum compression x -0 of spring such that block A leaves contact with the wall when system is released. (ii) If initial compression of spring is x = 2 x 0, calculate velocity of spring is x = 2 x 0, calculate velocity of centre of mass of the system when block A just leaves contact with the wall. (g = 10 ms–2) Ans: 1 1.05 ms1 2

(i) 3cm (ii) = 0.51 ms-1

MKA 30. An ice cube of size a = 2-0 cm is floating in a tank (base area A = 50 cm x 50 cm) partially filled with water. Density of water is 1 = 1000 kg m–3 and that of ice is 2 -= 900 kgm–3. Calculate increase in gravitational potential energy when ice melts completely. Ans:

-0.72 J FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

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MKA 31. A cubical block of wood (density 1 = 500 kg m –3) has side a = 30 cm. It is floating in a rectangular tank partially filled with water (density 2 = 1000 kg m–3 and having base area A = 45 cm x 60 cm. Calculate work done to press the block so that it is just. Immersed in water. Ans:

6.75 J

MKA 32. A block of mass m is held at rest on a smooth horizontal floor. A light frictionless, small pulley is fixed at a height of 6 m from the floor. A light inextensible string of length 16 m, connected with A passes over the pulley and another identical block B is hung from the string. Initial height of B is 5 m from the floor as shown in figure. When the system is released from rest, B starts to move vertically downwards and A sides on the floor towards right. (i) If at an instant string makes an angle  with horizontal, calculate relation between velocity u of A and v of B, (ii) Calculate v when B strikes the floor. (g = 10 ms–2) v

40 41

ms1

Ans:

(i) u =v sec  (ii)

MKA 33. A string with one end fixed on a rigid wall, passing over a fixed frictionless pulley at a distance of 2 m from the wall, has a point mass M of 2 kg attached to it at a distance of 1 m from the wall. A mass m of 0.5 kg is attached to the free end. The system is initially held at rest so that the string is horizontal between wall and pulley and vertical beyond the pulley as shown in Figure What will be the speed with which the point mass M will hit the wall when the system is released? (g = 10 ms–2) Ans: 5x

5 5 ms1  3.39ms 1 6

MKA 34.Two identical buggies each of mass 150 kg move one after the other without friction with same velocity 4 ms–1 . A man of mass m rides the rear buggy. At a certain moment the man jumps into the front buggy with a velocity v relative to his buggy. As a result of this process rear buggy stops. If the sum of kinetic energies of man and front buggy just after collision differs from that just before collision by 2700 joule, calculate values of m and v. Ans:

50 kg, 16 ms-1

MKA 35. Two balls of mass m1 = 100 gm and 3 m2 = 300 gm are suspended from point A by l length l = 32/35 m. Ball of mass m1 is drawn two equal inextensible threads, each of 2 aside and held at the same level as A but at a distance from A, as shown in figure. When ball m1 is released, it collides elastically with the stationary ball of mass m2. Calculate

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(i) velocity u1 with which the ball of mass m1 collides with the other ball, and (ii) maximum rise of centre of mass of the ball of mass m2. (g = 10 ms–2) Ans:

(i) 4 ms-1

(ii) 0.20 m

MKA 36. Two identical blocks A and B, each of mass m = 1.5 kg and carrying positive charge q = 30 Care kept stationary on a smooth horizontal floor. When the blocks are released, due to electrostatic repulsion. A moves towards left while B towards right. After moving 10 cm, A comes into contact with a mass less spring of force constant K = 6750 Nm –1 while after moving 10 cm, B collides inelastically with a rigid wall as shown in Figure. Calculate. (i) velocity of A when it comes in contact with the spring, and (ii) maximum compression of the spring. Ans:

(i) 6 ms-1

(ii) 10 cm

MKA 37. Two small blocks A and B of masses, m1 = 0.5 kg and m2 = 1 kg respectively, each carrying positive charge of q = 40 Care kept stationary on a smooth horizontal floor. When the blocks are released, due to electrostatic repulsion, A moves towards left while B towards right. After moving 40 cm, A comes in contact with a mass less spring of force constant K = 7600 Nm –1 while after moving 20 cm, B collides inelastically with a rigid wall as shown in figure. Calculate (i) velocity of A when it comes in contact with the spring and (ii) maximum compression of the spring. Ans:

(i) 12 ms-1

(ii) 101 cm

MKA 38. Two identical blocks A and B, each of mass m = 600 gm, each carrying positive charge of q = 40 Care kept stationary on a smooth horizontal floor, When the blocks are released, due to electrostatic repulsion, A moves towards left while B towards right. After moving 40 cm, A comes in contact with a mass less spring of force constant K = 6400 Nm–1, while after moving 20cm, B collides inelastically with a rigid wall as shown in figure. Calculate. (i) velocity of A when it comes in contact with the spring and (ii) maximum compression of the spring. Ans:

(i) 10 ms-1

(ii) 101 cm

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MKA 39. A wooden block of mass 700 gm is suspended by a light rigid rod of length 1 m form A. The rod is free to rotate in a vertical plane through A, without friction. A bullet of mass 10 gm is fired from point O on the ground with velocity 100 ms –1 at angel of elevation . At highest point of its trajectory, it strikes the wooden block. At that instant block was moving in vertical circle with velocity 7 ms –1and inclination of rod with vertical was 37 as shown in Fig. The bullet gets embedded into the block and the combined body just completes vertical circle. Calculate (i) velocity of the combined body just after collision. (ii) velocity of bullet just before collision, and (iii) co–ordinates of A, in reference to co–ordinate system as shown in figure. (g = 10 ms–2) Ans:

(i) 6 ms-1 (iii) (480.6 m, 180.8 m)

(ii) 80 ms-1

MKA 40. A bullet of mass m, moving horizontally with velocity v0 = 3 ms– 1 strikes elastically with a body of mass M = 2 m suspended by two identical threads of length l = 1 m each as shown in Fig. Calculate. (i) maximum deflection angle  with vertical of thread, and (ii) period of small oscillations of body M. (g = 10 ms–2) Ans: (i) 37 (ii) 2 sec 10

MKA 41. A body of mass M = 2 m rests on a smooth horizontal plane. A small block of mass m rests over it at left end A as shown in Fig. 41. A sharp impulse is applied on the block, due to which it starts moving to the right with velocity v0 = 6 ms –1. At highest point of its trajectory, the block collides with a particle of the same mass m moving vertically downwards with velocity v =2 ms–1 and gets stuck with it. If the combined body lands at the end point A of body of mass M, calculate length l. (Neglect Friction). (g = 10 ms–2) FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

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Ans:

40 cm

MKA 42. In the arrangement shown in Fig. 42, ball and block have the same mass m = 1 kg each,  = 600 and length l = 2.50 m. Co–efficient of friction between block and floor is 0.5. When the ball is released from the position shown in the figure, it collides with the block and the block stops after moving a distance 2.50 m. Find coefficient of restitution for collision between the ball and the block. (g = 10 ms–2) Ans:

1

MKA 43. A block B of mass m = 0.5 kg is attached with upper end of a vertical spring of force costant K = 1000 Nm –1 as shown in Fig. 43. Another identical block A fall from a height h = 49.5 cm on the block B and gets stuck with it. The combined body starts to perform vertical oscillations.

Ans:

Calculate amplitude of these oscillations. 5 cm

(g = 10 ms–2)

MKA 44.Three identical balls each of mass m = 0.5 kg are connected with each other as shown in Fig. 43 and rest over a smooth horizontal table. At moment t = 0, ball B is imparted a velocity v 0 = 9 ms–1. Calculate velocity of A when it collides with bass C. Ans:

6 ms-1

MKA 45. Two small particles, each of mass m carrying positive charge q each are attached to the ends of a non– conducting light thread of length 2 l. A third particle of mass 2 m is attached at mid–point of the thread. The whole system is placed on a smooth horizontal floor and the particle of mass 2 m is given a velocity v as shown in figure. Calculate minimum distance between the two charged particles during the process of FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

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motion. Ans:

2q2l 40mv 2l  q2 MKA 46. A pan of mass m = 1.5 kg and a block of mass M = 3 kg are connected with each other by a flexible, light and inextensible string, passing over a small, light and frictionless pulley. Initially the block is resting over a horizontal floor as shown in figure. At t = 0, an inelastic ball of mass m 0 = 0.5kg collides with the pan with velocity v0 = 16 ms–1 (vertically downwards). Calculate (i) maximum height, upto which the block rises, (ii) the time t at which block strikes wit1h the floor, iii) If the block comes to rest just after striking the floor, calculate velocity of pan at t = 2 second. (g = 10 ms–2) Ans: (i) 0.64 m (ii) 1.60 sec (iii) 0.48 ms-1 (downward)

MKA 47. Two identical blocks A and B each of mass 2 kg are hanging stationary by a light inextensible flexible string, passing over a light and frictionless pulley, as shown in Fig. 47. A shell C, of mass 1 kg moving vertically upwards with velocity 9 ms –1 collides with block B and gets stuck to it. Calculate (i) (ii) (iii) Ans:

time after which block B starts moving downwards, maximum height reached by B, and loss of mechanical energy up to that instant.

(i) 0.9 second (iii) 32.4 joule

(ii) 0.81 m

MKA 48. A light flexible thread passes over a small, frictionless pulley. Two blocks of mass m = 1 kg and M = 3 kg are attached with the thread as shown in Fig. Heavier block rests on a slab. A shell of mass 1 kg, moving upwards with velocity 10 ms –1, collides with the hanging block at time t = 0. Calculate. (i) maximum height ascended by M when it is jerked into motion, and (ii) time t at that instant : (a) If shell gets stuck the hanging block. (b) If shell collides with the hanging block elastically. (g = 10 ms–2) Ans:

a. (i) 1 m b. (i) 0.625 m

a. (ii) 2 sec b. (ii) 2.50 sec.

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PHY-PP-14

MKA 49. In the arrangement shown in Fig. pulleys are light and frictionless and thread are flexible and inextensible. Mass of each of the blocks A and B is m = 0.5 kg. Initially B is resting over a slab and A is hanging. A shell of equal mass m = 0.5 and moving vertically upwards with velocity v0 = 12 ms–1 strikes the block A and gets embedded into at t = 0. Calculate. (i) maximum height ascended by B when it is jerked into motion and time t at that instant, and (ii) time t when A strikes the slab. Initial height of block A from the slab is h = 10 cm. (g = 10 ms–2) Ans:

(i) 1m, 1.65 second

(ii) 1.25 second

MKA 50. A ball of mass m = 1 kg is hung vertically by a thread of length l = 1.50 m. Upper end of the thread is attached to the ceiling of a trolley of mass M = 4 kg. initially. Trolley is stationary and it is free to move along horizontal rails without friction. A shell of mass m = 1 kg, moving horizontally with velocity v0 = 6 ms–1, collides with the ball and gets stuck with it. As a result, thread starts to deflect towards right. Calculate its maximum deflection with the vertical. (g = 10 ms–2) Ans:

37

MKA 51. A small steel ball A is suspended by an inextensible thread of length l = 1.5 m from O. Another identical ball is thrown vertically downwards such that its surface remains just in contact with thread during downward motion and collide elastically with the suspended ball. If the suspended ball just. Completes vertical circle after collision. Calculate the velocity of the falling ball just before collision and its distance from O after t = 0.1 second after the collision. (g = 10 ms–2) Ans:

12.5 ms-1, 1.302 m

MKA 52. A block A of mass m = 5 kg is attached with a spring having force constant k = 2000 Nm–1. The other end of the spring is fixed to a rough plane, inclined at 37 with horizontal and having coefficient of friction  = 0.25 Block A is gently placed on the plane such that the spring has no tension. Then block A is released slowly.

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PHY-PP-15

(i) Calculate elongation of the spring when equilibrium is achieved. Now an inextensible thread is connected with block A and passed below pulley C and over pulley D, as shown in figure,. Other end of the thread is connected with another block B of mass 3 kg. Block B is resting over a table and thread is loose.If the table collapses suddenly and B falls freely through 80/9 cm the thread becomes taut, calculate (ii) combined speed of blocks at that instant, and (iii) maximum elongation of spring in the process of motion. (g = 10 ms–2) Ans:

(i) 1 cm (iii) 6 m

(ii) 0.5 ms-1

MKA 53. A right angled wedge ABC of mass M = 4 kg and base angel  = 53 is resting over a smooth horizontal plane. A shell of mass m = 0.5 kg moving horizontally with velocity v0 = 40 ms–1, collides with the wedge, just above point A. As a consequence, wedge starts to move towards left with velocity v -= 5 ms–1. Calculate (i) heat generated during collision. (ii) maximum height reached by the shell, and (iii) distance of point A of wedge from the shell when shell strikes the plane. (g = 10 ms–2) Ans:

(i) 125 joule (iii) 30 m

(ii) 45 m

MKA 54. A uniform chain A’ B’ of length 2l having mass  per unit length is hanging from ceiling of an elevator by two light, inextensible threads AA’ and BB’ of equal length as shown in figure. Distance AB is very small. At a certain instant, elevator starts ascending FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

PHY-PP-16

Ans:

with constant acceleration a. Two seconds after the beginning of motion, thread BB’ is burnt. Assuring that instant to be t = O, calculate tension in thread AA’ at time t. 3 l(g a) (ga)2 t 2 4

MKA 55. A turn table is free to rotate about a  fixed vertical axis and has a smooth groove made on its upper surface along a radius. The table is rotated about the axis with constant angular velocity and a particle of mass m is gently placed in the groove at distance a from the axis of rotation. Calculate magnitude of resultant velocity of the particle as a function of its distance x from axis of rotation. Calculate also, torque required to keep the angular velocity  constant. Ans:

 2x 2  a2 ,2m2 x x 2  a 2

MKA 56. A light inextensible string  is passed through a hole made in a smooth horizontal table top. Two masses m 1 = 3 kg and m2 = 6.2 kg are connected at the ends of the strings as shown in Fig. 56. Initially, m2 is held at rest and m1 is rotated along a horizontal circle of radius r0 = 20 cm with angular velocity 0 = 18 rad sec–1. Calculate. (i) acceleration of m2 when it is released from rest, and velocity of m1 when radius of its circular path becomes 30 cm. Ans:

(i) (ii)

1 331 6.76ms ms 2 23

MKA 57. A uniform solid sphere of mass 1 kg and radius 10 cm is kept stationary on a rough inclined plane by fixing a highly dense particle at B. Inclination of plane is 37 with horizontal and AB is the diameter of the sphere which is parallel to the plane, as shown in Fig. 57 Calculate. (i) mass of the particle fixed at B, and (ii) minimum required coefficient of friction between sphere and plane to keep sphere in equilibrium. Ans:

(i) 3 kg

(ii) 0.75

MKA 58. A ball of radius R = 20 cm has mass m = 0.75 kg and moment of inertia (about its diameter) I = 0.0125 kg m2. The ball rolls without sliding over a rough horizontal floor with velocity v 0 = 10 ms–1 towards a smooth vertical wall. If coefficient of restitution between the wall and the ball is e = 0.7 calculate velocity v of the ball long after the collision. (g = 10 ms–2) Ans:

2 ms-1

MKA 59. AB is a horizontal diameter of a ball of mass m = 0.4 kg and radius R = 0.10 m. At time t = 0, a sharp impulse is applied a B at angle of 450 with the horizontal, as shown in figure. So that the ball immediately starts to move with velocity v0 = 10 ms–1. (i) Calculate the impulse. If coefficient of kinetic friction between the floor and the ball is  = 0.1, calculate, (ii) velocity of ball when it stops sliding. (iii) time t at that instant. (iv) horizontal distance traveled by the ball upto that instant, (v) angular displacement of the ball about horizontal diameter perpendicular to AB, upto that instant, and (vi) energy lost due to friction. FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

PHY-PP-17

2 kgms 1 Ans: (i) 4 (ii) Zero (iii) 10 second (iv) 50 m(Leftward) (v) 1250 radians (clockwise) (vi) 70 joule MKA 60. A solid ball of diameter d = 11 cm is rotating about its one of the horizontal diameters with angular velocity 0 = 120 rad/sec. It is released from a height so that it falls h = 1.8 m freely and then collides with the horizontal floor. Co–efficient restitution is e = 5/6 and co–efficient of friction between the ball and the ground is  = 0.2. Calculate fraction of energy lost during collision and the distance between the points where the ball strikes the floor for the first and second time. ( g = 10 ms–2) Ans:

0.432, 2.2 m

MKA 61. A steel ball of radius R = 20 cm and mass m = 2 kg is rotating about a horizontal diameter with angular velocity 0 = 50 rad/sec. This rotating ball is dropped on to a rough horizontal floor and falls freely through a height h = 1.25 m. The coefficient of restitution is e = 1.0 and coefficient of friction between the ball and the floor is  = 0.3. Calculate (i) distance between points of first and second impact of the ball with the floor, and (ii) loss of energy due to friction. Ans:

(i) 3 m

(ii) 38.5 joule

MKA 62. A uniform rod of length l and mass M is suspended on two vertical inextensible strings as shown in figure. Calculate tension T in left string at the instant, when right string snaps. Ans:

T

Mg 4

MKA 63. A triangular prism of mass M = 1.12 kg having base angle 370 is placed on a smooth horizontal floor. A solid cylinder of radius R = 20 cm and mass m = 4 kg is placed over the inclined surface of the prism. If sufficient friction exists between the cylinder surface and the prism, so that cylinder does not slip, calculate also, force of friction existing between the cylinder and the prism. ( g = 10 ms–2) Ans:

3.75 ms2, 12 newton Angular acceleration of cylinder = 30 radian/sec 2 (clockwise)

MKA 64. A solid metallic cylinder of mass m = 1 kg and radius R = 20 cm is free to roll (without sliding) over the inclined surface of a wooden wedge of mass M = 0.28 kg. Surface of wedge in inclined at 37 0 with the horizontal and the wedge lies on a smooth horizontal floor. When the system is released from rest, calculate (i) acceleration of the wedge, (ii) angular acceleration of the cylinder, and FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

PHY-PP-18

(iii) force of interaction between cylinder and the wedge. (g = 10 ms–2) (i) 3.75 ms-2 (ii) 30 rad sec-2 (iii) Normal reaction = 5.75 N Friction = 3.00 N Interaction force = 5.752  32  6.49N Ans:

MKA 65. A uniform rod of mass m = 30 kg and length l = 0.80 m is free to rotate about a horizontal axis O passing through its centre. A particle P of mass M = 11.2 kg falls vertically through a height h = 36/245 m and collides elastically with the rod at a distance l/4 from O. At the instant of collision the rod was stationary and was at angle  = 370 with horizontal as shown in figure. Calculate (i) angular velocity of the rod just after collision, and (ii) velocity (direction and magnitude of particle P after collision. (g = 10 ms–2) Ans:

(i) 3 rad/sec

(ii) 9/7 ms-1 (horizontally rightward)

MKA 66. A homogeneous rod AB of length L and mass M is hinged at the centre O in such a way that it can rotate freely in the vertical plane. The rod is initially in horizontal position. An insect S of the same mass M falls vertically with speed V on point C, midway between the points O and B. Immediately after falling, the insect starts to move towards B such that the rod rotates with a constant angular velocity . (i) calculate angular velocity  in terms of V and L,

Ans:

(ii) if insect reaches the end B when the rod has turned through an angle of 90 0, calculate v in terms of L. (i) (ii) 7 12V 2gL 12 7L

MKA 67. A square frame is formed by four rods, each of length l = 60 cm. Mass of two rods AB and BC is m = 25/18 kg each while that of rods AD and CD is 2m each. The frame is free to rotate about a fixed horizontal axis passing through its geometric centre O shown in figure. A spring is placed on the rod AB at a distance a = 15 cm from B. The spring is held vertical and a block is placed on upper end of the spring so that rod AB is horizontal.

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PHY-PP-19

(i) Calculate mass M of the block, (ii) If the spring is initially compressed by connecting a thread between its ends and energy stored in it is 76.5 joule, calculate velocity with which block bounces up when the thread is burnt. Calculate also maximum angular velocity of frame during its rotational motion assuming that the block does not collide with the frame in subsequent motion. ( g = 10 ms–2) Ans:

(ii) 7.2 ms-1

(i) 25/9 kg (iii) 3.53 rad/sec

MKA 68. A heavy plank of mass 102.5 kg is placed over two cylindrical rollers of radii R = 10 cm and r = 5 cm. Mass of rollers is 40 kg and 20 kg respectively. Plank is pulled towards right by applying a horizontal force F = 25 N as shown in figure. During first second of motion the plank gets displaced by 10 cm. If plank remains horizontal and slipping does not 1take place, calculate magnitude and direction of force of friction acting between (i) plank an bigger roller, (ii) plank and smaller roller, (iii) bigger roller and floor, and (iv) smaller roller and floor. Ans:

(i) 3N (iii) 1.00 N

(g = 10 ms–2) (ii) 1.50 N (iv) 0.50 N

MKA 69. A semi  circular of radius R = 62.5 cm is cut in a block. Mass of block, having track, is M = 1 kg and rests over a smooth horizontal floor. A cylinder of radius r = 10 cm and mass m = 0.5 kg is hanging by a thread such that axes of cylinder and track are in same level and surface of cylinder is in contact with the track as shown in figure. When the thread is burnt, cylinder starts to move down the track. Sufficient friction exists between surface of cylinder and track, so that cylinder does not slip. Calculate velocity of axis of cylinder bottom of the track. Ans:

when it reaches (g = 10 ms–2)

2 ms-1

MKA 70. A trolley initially at rest with a solid cylinder placed on its bed such that cylinder axis makes angle  with direction of motion of trolley as shown in figure, starts to move forward with constant acceleration a. If initial distance of mid point of cylinder axis from rear edge of trolley bed is d, calculate the distance s which the trolley goes before the cylinder rolls off the edge of its horizontal bed. Assume dimensions of cylinder to be very small in comparison to other dimensions. Neglect slipping. Calculate also, frictional force acting on the cylinder. Ans: 3 1 dcos ec 2 , ma sin2   9 cos2  2 3

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PHY-PP-20

MKA 71. A uniform circular disc of mass M and radius R is free to rotate about a vertical axis O passing through its rim. An insect of mass m is at point A such that line OA is the diameter of the disc as shown in figure. The insect describes a complete circle relative to disc and returns to the starting point A. Calculate the angle moved by the disc relative to the ground. Ans:

 3M    1  3M  8m 

MKA 72. A unifo1rm rod AB of mass m = 2 kg and length l = 100 cm is placed on a sharp support O such that AO = a = 40 cm and OB = b = 60 cm. A spring of force constant K = 600 Nm1 is attached to end B as shown in figure. To keep the rod horizontal, its end A is tied with a thread such that the spring is elongated by y = 1 cm. Calculate reaction of support O on the rod when the thread is burnt. (g 10 ms–2) Ans:

20 newton

MKA 73. In the system shown in figure, blocks A 26 3 and B have mass m1 = 2 kg and m2 = kg respectively. Pulley having moment of 7 inertia I = 0.11 kg m2 can rotate with out friction 10 of pulley are a 10 cm and b = 15 cm about a fixed axis. Inner and outer radii respectively. B is hanging with the thread wrapped around the pulley, while A lies on a rough inclined plane. Coefficient of friction being  =. Calculate (i) tension in each thread, and (ii) acceleration of each block. ( g = 10 ms–2) Ans:

(i) Tension in thread connected with A is 17 N (ii) Tension in thread connected with B is 26 N (iii) Acceleration of A = 2 ms-2 (up the plane) (iv) Acceleration of B = 3 ms-2 (vertically downward)

MKA 74. In the arrangement shown in figure, mass of blocks A and B is m1 = 0.5 kg and m2 = 10 kg, respectively and mass of spool is M = 8 kg. Inner and outer radii of the spool are a = 10 cm and b = 15 cm respectively. Its moment of inertia about its own axis is I0 = 0.10 kg m2. If friction be sufficient of prevent sliding, calculate acceleration of blocks A and B. ( g = 10 ms–2) Ans:

3 ms-2 (upward), 0.5 ms-2 (downward)

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PHY-PP-21

MKA 75. A pulley of radius b = 20 cm is fixed 33 with a shaft of radius a = 10 cm. Moment of inertia of shaftpulley system is I = kg 800 m2 and the system is free to rotate about axis O of the shaft without friction. A block B of mass m2 = 8 kg is resting over and ideal spring of force constant. K = 2048 Nm 1. Lower end of the spring is fixed to the floor and the spring is vertical. Thread connected between shaft and block B is just taut. Another thread is connected between pulley and block A of mass m 1 = 4 kg. Initially this thread is loose. When block A is released, first it falls freely through a height h = 405/1024 m, then the thread becomes taut and block B is jerked into motion. Calculate (i) initial compression of the spring, (ii) velocity of block B when it is jerked into motion, (iii) loss of energy during that jerk, and (iv) maximum elongation of spring (from its natural length) in the process of motion. (g = 10 ms–2) Ans:

(i) 125/32 cm (ii) 80 cm sec-1 (iii) joule or 6.82 joule (iv) 13

873 128 m  10.16cm

128 MKA 76. A wheel of radius R = 10 cm and moment of inertia I = 0.05 kgm2 is rotating about a fixed horizontal axis O with angular velocity 0 = 10 rad/sec. A uniform rigid rod of mass m = 3 kg and length l = 50 cm is hinged at one end A such that it can rotate about end A in a vertical plane. End B of the rod is tied with a thread as shown in figure such that the rod is horizontal and is just in contact with the surface of rotating wheel. Horizontal distance between axis of rotation. O of cylinder and A is equal to a = 30 cm. If the wheel stops rotating after one second after the thread has burnt, calculate coefficient of friction  between the rod and the surface of the wheel. (g = 10 ms–2) Ans:

0.2

MKA 77. In the arrangement shown in figure, ABC is a straight, light and rigid rod of length 90 cm. End A is pivoted so that the rod can rotate freely about it, in vertical plane. A pulley, having internal and external radii R = 7.5 cm and r = 5 cm is fixed to a shaft of radius 5 cm. The pulleyshaft system can rotate about a fixed horizontal axis O. B is point of contact of the pulley and the rod. From free end C of the rod a mass m2 = 2 kg is suspended by a thread. Another thread is wound over the shaft and a block of mass m1 = 4 kg is suspended from it. If coefficient of friction between the rod and the pulley surface is  = 0.4 and moment of inertia of pulleyshaft system about axis O is I = 0.045 kg m2, calculate acceleration of block m1, when the system is released. ( g = 10 ms–2) Ans:

1 ms-2

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PHY-PP-22

MKA 78. A uniform rod of length l = 75 cm is hinged t one of its end and is free to rotate in vertical plane. It is released from rest when the rod is horizontal. When rod becomes vertical, it is broken at mid point and lower part now moves freely. Calculate distance of the centre of lower part from hinge, when it again becomes vertical for the first time. ( g = 10 ms–2) Ans: 2.52 m MKA 79. A man can jump over b = 4 m wide trench on earth. If mean density of an imaginary planet is twice that of the earth, calculate its maximum possible radius so that he may escape from it by jumping. Given radius of earth, R e = 6.4  106 m. Ans: 6.4 km MKA 80. A thin uniform rod of length 2 a has mass  per unit length. Calculate magnitude of gravitational field strength an potential as a function of distance r from centre of the rod along the straight line (i) perpendicular to the rod and passing through the centre, (ii) coinciding with the rod’s axis (at points lying outside the rod). 2 2 2 Ans: (i) (ii) 2Ga2Ga ,G log a r ar  a    2 ,  22G log  e  2 (r 2  a )  r  ar  r r  a MKA 81. A particle of mass m is placed on centre of curvature of a fixed, uniform  semicircular ring R and mass M as shown in figure. Calculate (i) interaction force between the ring and the particle, and (ii) work required to displace the particle from centre of curvature to infinity. Ans:

(i)

2GMm GMm RR2

(ii)

MKA 82. A system consists of a thin ring of radius R and a very long uniform wire oriented along axis of the ring with one of its ends coinciding with the centre of the ring. If mass of ring be M and mass of wire be  per unit length, calculate interaction force between the ring and the wire. GM R MKA 83. Inside a fixed sphere of radius R and uniform density , there is a spherical cavity of radius R/2 such that surface of the cavity passes through the centre of the sphere as shown in figure. A particle of mass m is released from rest at centre B of the cavity. Calculate velocity with which particle strikes the centre A of the sphere. Neglect earth’s gravity. Ans:

Ans: 2 πGρR2 3

MKA 84. In a vertical cylindrical vessel of base area A = 80 cm 2 water is filled to a height h = 30 cm. If density and Bulk Modulus of water be  = 1000 kg m3 and B = 2  109 Nm2, calculate elastic deformation energy of water in the vessel. ( g = 10 ms –2)

ρ2g2 Ah3 =1.8 x10-6 joule 6B MKA 85. A ring of radius R = 4 m is Ans:

made of a highly dense material. Mass of the ring is m 1 = 5.4  109 kg. Distributed uniformly over its circumference. A highly dense particle of mass m 2 = 6  10 8 kg is placed on the axis of the ring at a distance x0 = 3 m from the centre. Neglecting all other force, S except mutual gravitational interaction of the two, calculate FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

PHY-PP-23

(i) displacement of the ring when particle is closest to it, and (ii) speed of the particle at this instant. Ans: (i) 0.3 m (ii) 18 cm/sec

MKA 86. An artificial satellite of mass m of a planet of mass M, revolves in a circular orbit whose radius in n times the radius R of the planet. In the process of motion the satellite experiences a slight resistance due to cosmic dust. Assuming resistance force on satellite to depend on velocity as F = a. v.2 where a is a constant, calculate how long the satellite will stay in orbit before it falls onto the planet’s surface. Ans:

m R( n  1)

a GM MKA 87. A satellite is revolving round the  5   1 earth in a circular orbit of radius a   with velocity v0. A particle is projected  4  from the satellite in forward direction with relative velocity v = v0. Calculate, during subsequent motion of the particle its minimum and maximum distance from earth’s centre. Ans: 5a a, 3 MKA 88. A solid sphere of mass m = 2 kg and specific gravity s = 0.5 is held stationary relative to a tank filled with water as shown in figure. The tank is accelerating vertically upward with acceleration a = 2 ms2. (i) Calculate tension in the thread connected between the sphere and the bottom of the tank. (ii) If the thread snaps, calculate acceleration of sphere with respect to the tank. (density of water is  = 1000 kg m3) (g = 10 ms–2) Ans: (i) 24 N

(ii) 12 ms-2 (upward)

MKA 89. Length of a horizontal arm of a U  tube is l = 21 cm and ends of both of the vertical arms are open to surroundings of pressure 10500 Nm 2. A liquid of density  = 103 kg m3 is poured into the tube such that liquid just fills horizontal part of the tube. Now, one of the open ends is sealed and the tube is then rotated about a vertical axis passing through the other vertical arm with angular velocity 0 = 10 radian/ sec. If length of each vertical arm be a = 6 cm, calculate the length of air column in the sealed arm. ( g = 10 ms–2) Ans: 5 cm

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PHY-PP-24

MKA 90. A cylindrical tank having crosssectional area A = 0.5 m 2 is filled with two liquids of density 1 = 900 kg m3 and 2 = 600 kg m3, to a height h = 60 cm each as shown in figure. A small hole having area a = 5 cm 2 is made in right vertical wall at a height y = 20 cm from the bottom. Calculate (i) velocity of efflux, (ii)horizontal force F to keep the cylinder in static equilibrium, if it is placed on a smooth horizontal plane, and (iii) minimum and maximum values of F to keep the cylinder in static equilibrium, if coefficient of friction between the cylinder and the plane is  = 0.01 (g = 10 ms–2) Ans: (i) 4 ms-1 (iii) Zero, 52.2 N

(ii) 7.2 N

MKA 91. Curved surface of a vessel has shape of a truncated cone having semi  vertex angle  = 370. Top and bottom radii of the vessel are r1 = 3 cm and r2 = 12 cm respectively and height is h = 12 cm. The vessel is full of water (density = 1000 kg m3) and is placed on a smooth horizontal plane in vacuum. Calculate (i) mass of the liquid in the vessel, (ii) force on the bottom of the vessel, (iii) resultant force on curves walls. A hole having area S = 1.5 cm 2 is made in curved wall near the bottom. Calculate (iv) velocity of efflux, (v) horizontal range of water jet, and (vi) horizontal force required to keep the vessel in static equilibrium. Neglect atmospheric pressure. Ans:

(i) 0.756.  kg (iii) 9.72  N (vertically 24 ms1 upward) (iv) (v) 23.04 cm

(ii) 17.28.  N

(vi) 0.288 N

MKA 92. A cylindrical tank of base area A has a small orifice of area a at the bottom. At time t = 0, a tap starts to supply water into the tank at a constant rate Q m 3 s1. Calculate relation between height h of water in the tank and time t. Ans:

t=

A 

 Q  2gh Q loge    2h   Q   MKA 93. A steel rod of length l1 g= 30 cm gand two identical brass rods of length l2 = 20 cm each, support a light horizontal platform as shown in figure. Cross sectional area of each of the three rods is A = 1 cm2. Calculate stress in each rod when a vertically downward force F = 5000 N is applied on the platform.

  

Given, Young’s modulus of elastically for steel, FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

PHY-PP-25

Ys = 2  1011 Nm2. Young’s modulus of elasticity for brass, Y b = 1  1011 Nm2. Ans:

In steel, s = 2 x 107 Nm-2 In brass, b = 1.5 x 107 Nm-2

MKA 94.Two vertical wires, one of steel having cross-sectional area As =2 x 10-6 m2 other of bronze having cross sectional area Ab = 1x10-6 m2 are suspended from a ceiling as shown in Fig., horizontal distance between the two being r = 55 cm. Each wire is l = 150 cm long. A light but horizontal cross piece connects the lower ends of he wires. Where should a force F = 1100 N be applied on this cross piece, so that it remains horizontal after the force is applied. Given, Young’s modulus of elasticity of steel, Ys = 2 x 1011 Nm-2, Young’s modulus of elasticity of bronze, Yb = 1.5 x 1011 Nm-2. Ans:

x = 15 cm

MKA 95. Distance between centers of two stars is 10 a. Mass of these stars is M and 16 M and their radii are a and 2a respectively. A body of Mass m is fired straight form the surface of larger star directly towards the smaller star. Calculate minimum initial speed of the body so that it can reach the surface of smaller star. Obtain the expression in terms of G, M and a. Ans: 45GM 4a MKA 96. A steel bolt of cross-section area Ab = 5 x 10-5 m2 is passed through a cylindrical tube made of aluminium. Cross-sectional area of the tube material is A t = 10 x 10-5 m2 and its length is l = 50 cm. The bolt is just taut so that there is no stress in the bolt. Calculate stress in bolt and tube when temperature of the assembly is increased through  = 10C. Given : Young’s modulus of steel, Yb = 2 x 1011 Nm2 Young’s modulus pf aluminum, Yt = 1 x 1011 Nm2 Coefficient of linear thermal expansion of steel, b = 1 x 105/C Coefficient of linear thermal expansion of aluminum, t = 2 x 105/C. Ans:

Stress in tube = 5 x 10 6 Nm-2 (compressive) Stress in bolt = 1 x 10 7 Nm-2 (tensile)

MKA 97. One end of an ideal spring is fixed to a wall at origin O and axis of spring is parallel to x-axis. A block of mass m = 1 kg is attached to the free end of the spring and it is performing S.H.M FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

PHY-PP-26

Equation of position of the block in co-ordinate system shown in Fig. 97 is x = 10 + 3 sin (10t), when t is in second and x in cm. (i) Calculate force constant of the spring, Another identical block, moving towards origin with velocity 0.6 ms -1 collides elastically with the block performing S.H.M. at t = 0. Calculate (ii) new amplitude of oscillations. (iii) equation of position of block performing S.H.M., and (iv) percentage increase in oscillation energy. Neglect friction. Ans:

(i) 100 Nm-1 (ii) 6 cm (iii) x = 10+6 sin(10t+) or x = 10–6 sin (10t) cm (iv) 300 %

MKA 98. One end of an ideal spring is fixed to a wall at origin O and axis of spring parallel to x-axis. A block of mass m = 1 kg is attached to free end of the spring and it is performing S.H.M. Equation of position of the block in co-ordinate system shown in figure is x = 10 + 3. sin (10.t), t is in second and x in cm.

Ans:

Another block of mass M = 3 kg, moving towards the origin with velocity 30 cm/sec collides with the block performing S.H.M. at t = O and gets stuck to it. Calculate. (i) new amplitude of oscillations, (ii) new equation for position of the combined body, and (iii) loss of energy during collision. Neglect friction. (i) 3 cm (ii) x = 10 + 3. sin (5t + ) or x = 10 – 3 sin (5t) cm (iii) 0.135 joule

MKA 99. One end of an ideal spring is fixed with a wall and the other end is fixed with a block of mass m = 1 kg. Force constant of spring is K = 100 Nm -1 and block is performing S.H.M. with amplitude 3 cm. When the block is at left extreme position, an other block of mass M = 3 kg, moving directly toward with velocity 80/3 cm/sec, collides and gets stuck to it. (i) Calculate angular frequency and amplitude of oscillations of the combined body. (ii) Assuming that the collision takes place at t = 0, and right hand direction to be positive xdirection. Calculate initial phase of oscillations of the combined body. Neglect friction. Ans:

(i) 5 rad /sec, 5 cm (ii) 217 or 217  rad 180

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PHY-PP-27

MKA 100. Two identical blocks A and B of mass m = 3 kg are attached with ends of an ideal spring of force constant K = 2000 Nm -1 and rest over a smooth horizontal floor. Another identical block C moving with velocity V0 = 0.6ms-1as shown in figure strikes of block A and gets stuck to it. Calculate for subsequent motion (i) velocity of centre of mass of the system. (ii) frequency of oscillations of the system, (iii) oscillation energy of the system, and (iv) maximum compression of the spring. Ans: (i) 0.2 ms-1 (ii) 5 10 H 2 (iii) 0.09 joule 3 10mm  (iv)

MKA 101. Two block A and B of masses m 1 = 3 kg and m2 = 6 kg respectively connected with each other by a spring of force constant K = 200 Nm -1 as shown in Fig. 101. Blocks are pulled away from each other by xo = 3 cm and then released. When spring is in its natural length and blocks are moving towards each other, another block of mass m = 3 kg moving with velocity v 0 = 0.4ms-1 (towards right) collides with A and gets stuck to it. Neglecting friction, calculate (i) velocities v1 and v2 of the blocks A and B respectively just before collision and their angular frequency, (ii) velocity of centre of mass of the system, after collision, (iii) amplitude of oscillations of combined body, and (iv) loss of energy during collision. Ans:

(i) 0.2 ms-1, 0.1 ms-1, 10 rad/sec (ii) 0.1 ms-1 (towards right) (iii) 24 cm (iv) 0.03 joule

MKA 102. In the arrangement shown in pulleys are small and light and spring are ideal, K1, K2, K3 and K4 are force constants of the springs. Calculate period of small vertical oscillations of block of mass m. Ans:

 1 1 1 1     K K K K  1 2 3 4

4 m 

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PHY-PP-28

MKA 103. AB and CD are two ideal springs having force constant K1 and K2respectively. Lower ends of these springs are attached to the ground so that the springs remain vertical. A light rod of length 3a is attached with upper ends B and C of springs. A particle of mass m is fixed with the rod at a distance a from end B and in equilibrium, the rod is horizontal. Calculate period of small vertical oscillations of the system. 2 m(K1  4K 2 Ans: 3 K1K 2 MKA 104. Fig shows a particle of mass m = 100 gm, attached with four identical springs, each of length l = 10 cm. Initial tension in each spring if F 0 = 25 newton. Neglecting gravity, calculate period of small oscillations of the particle along a line perpendicular to the plane of the figure. Ans:

0.02 sec

MKA 105. In the arrangement shown in Fig. 105, body B is a solid cylinder radius R = 10 cm with mass M = 4 kg. It can rotate without friction about a fixed horizontal axis O, A block A of mass m = 2 kg suspended by an inextensible thread is wrapped around the cylinder. A horizontal light spring of force constant K = 100 Nm -1 fixed at one end keeps the system in static equilibrium. Calculate (i) initial elongation in the spring, and (ii) period of small vertical oscillations of the block. (g = 10 ms-2) Ans:

(i) 20 cm

(ii) 0.4  second

MKA 106. A solid uniform sphere of radius r rolls without sliding along the inner surface of a fixed spherical shell of radius R and performs small oscillations. Calculate period of these oscillations. Ans: 7(R  r) 2 5g MKA 107. One end of each of two identical springs, natural length 9 cm and force constant K = 45 Nm -1 is attached with a small particle of mass m = 30 gm. Other end of right spring if fixed with a wall and other end of left spring is attached with a fixed block having a positive charge q = 1 C as shown in figure. The particle rests over a smooth horizontal plane and springs are non-deformed. Calculate deformation of springs when a positive charge q = 1 C is given to the particle and equilibrium is attained. Calculate also, frequency of small longitudinal oscillations of the particle. Ans: 1 cm, 30 Hz 

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PHY-PP-29

MKA 108. A non-conducting piston of mass m and area S divides a non-conducting, closed cylinder into two parts as shown in Fig. 108. Piston is connected with left wall of cylinder by a spring of force constant K. Left part is evacuated and right part contains an ideal gas at pressure P. Adiabatic constant of the gas is  and in equilibrium length of each part is l. Calculate angular frequency oscillations of the piston.  PS  Kl Ans:  ml   

of

small

MKA 109. A rectangular tank having base 15 cm x 20 cm is filled with water (density  = 1000 kg m-3) upto 20 cm height. One end of an ideal spring of natural length h 0 = 20 cm and force constant K = 280 Nm-1 is fixed to the bottom of a tank so that spring, remains vertical. This system is in an elevator moving downwards with acceleration a = 2 ms-2. A cubical block of side l = 10 cm and mass m = 2 kg is gently placed over the spring and released gradually, as shown in Fig. 109. (i) Calculate compression of the spring in equilibrium position. (ii) If block is slightly pushed down from equilibrium position and released, calculate frequency of its vertical oscillations. Ans: (i) 4 cm 5 2 sec 1 (ii)  MKA 110. Both the limbs of a U-tube are vertical. One end of a light spring of force constant K = 78 Nm -1is fixed with top of left limb and a piston of mass m = 50 gm is attached with lower end of the spring as shown in Fig. 110. Cross-sectional area of tube is S = 1 cm 2. Water (density  = 1000 kg ms-3) is poured into right limb till elongation of spring reduces to a = 6 mm. (i) Calculate difference h between level of water in right limb and level of lower face of the piston (ii) If mass of whole in the tube is M = 150 gm, calculate angular frequency of small oscillations. (Neglect Atmospheric pressure). Ans:

(i) 32 mm

(ii) 20 rad/sec

MKA 1* A bus is traveling along a straight road with velocity v = 6.4ms-1. A boy is sitting a distance ‘a’ way from line of motion of the bus. He throws a stone with velocity u = 10 ms -1 at the instant when a glass window of the bus is infront of the him. If the stone strikes this glass window at highest point of its trajectory and height of the window above the point of projection is H – 1.8 m, calculate. (i) time of flight of the stone, (ii) distance ‘a’, and (iii) inclination of plane of trajectory of stone with the road. (g = 10 ms-2) Ans:

(i) 0.6 second (iii) 37

(ii) 2.88 m

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PHY-PP-30

MKA 2*. A particle of mass m=1kg is moving 5 along x-axis with constant velocity of magnitude v0=2 ms-1. When it passes through origin, it experience a constant force F= Newton inclined at angle =tan-1 (2) with x-axis so that the particle now moves in negative quadrant of x-y plane. Neglecting gravity, calculate equation to the trajectory of the particle.” Ans:

4x2 + 4xy + y2 + 16y =0

MKA 3*. In the arrangement shown in Fig., pulleys are light, small and smooth. Mass of blocks A, B and C is m 1= 14 kg, m2 = 11 kg and M = 52 kg respectively. The block A can slide freely along a vertical rail, fixed to left vertical face of block C. Assuming all the surfaces to be smooth, calculate magnitude of resultant acceleration of each of the blocks A, B and C. (g = 10 ms-2)

2ms2 , 10 ms2 ,1ms2 Ans:

MKA 4*. A shell of mass m =1 kg is fired from a point O on the ground with velocity u =6 ms -1 at angle  = 60 with the horizontal. At highest point of trajectory, the shell just comes into contact to a horizontal plank of mass M = 2 kg which is resting over a horizontal platform as shown in figure. Coefficient of friction between shell and plank is 2 = 0.5 and that between plank and platform is 1 = 0.1. In the figure, x-axis is horizontal axis through O and is in the line of trajectory of the shell and y-axis is vertical axis through O. Calculate co-ordinates f the point where the shell finally comes to rest and displacement of plank upto that instant. (g = 10 ms-2) Ans:

[(1  0.9 3 )m,1.35m],0.25m

MKA 5*. A particle is projected form ground with velocity u = 10 ms -1 at an angle with horizontal. At highest point of its trajectory, it comes into contact with lowest point of a vertical circular track of radius R = 1 m as shown in figure and it starts to move along inner surface of the track. Height of lowest point of the track from ground is h = 3.10 m. Neglecting friction between particle and the track, calculate maximum height reached by the particle above the ground (g = 10 ms-2) Ans:

4.892 m

MKA 6*. A rod AB of length a = 90 cm can rotate freely in a horizontal plane about a vertical axis OO’, passing through its one end A as shown in figure, A particle is suspended from other and rod by a light, inextensible thread of the length l = 50 cm. The thread is capable of with standing a maximum tension equal to 1.25 times the weight of the particle. If rod starts to FIITJEE (Hyderabad Classes) Limited., 5-9-14/B, Saifabad, (Opp. Secretariat) Hyderabad. 500 004. Phone: 040-55777000 – 03 Fax: 040-55777004 Regd. Off: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 16, Ph : 26515949 , 26865182, 26854102, Fax: 26513942

PHY-PP-31

Ans:

rotate and its angular velocity increases slowly and if height OA is equal to h = 120 cm, calculate. (i) speed of the particle when it strikes the ground, and (ii) distance of the point at which particle strikes the ground from foot O of the axis of rotation. (g = 10 ms-2) –1 (i) 5 ms 120 2 cm (ii)

MKA 7*. A small block of mass m = 1 kg is attached with one end of a spring of force constant K = 110 Nm-1. Other end of the spring is fixed to a rough plane having inclination =sin–1(3/5) with the horizontal and having coefficient of friction  = 0.2. The spring is kept in its natural length by an inextensible thread tied between its ends as shown in figure. If the thread is burnt, calculate elongation of spring when the block attains static equilibrium position. Ans: 64

11

cm

MKA 8*. A stationary light, smooth pulley can rotate without friction about a fixed horizontal axis. A light rope passes over the pulley. One end of the rope supports a ladder with a man and the other end supports a counterweight of mass M, Mass of the man is m. Initially, centre of mass of the counterweight is at a height h from that of the man as shown in Fig. 8*. If the man starts to climb up the ladder slowly, calculate work done by him to reach his centre of mass level with that of the counterweight. Ans:

mMgh (M  m) r

MKA 9*. A steel ball of mass m is suspended by Vv an inextensible thread of length l = 1 m: The 0 point of suspension is at height h = 6 m 0 from the ground. The bass is drawn aside and an impulse is given to it so that it passes through the equilibrium position with velocity= 5ms –1. Another ball of mass m/2 is projected from the ground with velocity u at angle  with horizontal such that its plane of trajectory passes through the point of suspension and is normal to . At highest point of trajectory, second ball collides with the first ball. If at the instant of collision, first ball was passing through its exactly lowest position and first ball, subsequently, completes vertical circle, calculate. (i) maximum possible value of , and (ii) corresponding value of u. (g = 10 ms–2) Ans:

(i) 53

(ii) 12.5 ms–1

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PHY-PP-32

MKA 10*. A ball of mass m = 1 kg is suspended from point ‘O’ of a toy cart of mass M = 3 kg by an inextensible thread of length l = 1.5625 m. The ball is first raised to point A and then the ball is released from rest. Point A is in the same level as O and distance OA is such that the ball falls freely through a height h = 1.25m as shown in figure, then thread becomes taut. Neglecting friction, calculate (i) velocity of cart just after the thread becomes taut, and (ii) loss of energy when thread becomes taut. Ans: (i) (ii) 5350 1

msJ 749

MKA 11*.A wedge of mass M = 3.6 kg and having base angle  = 3.7 is resting over a smooth horizontal surface. A ball of mass m = 1 kg is thrown vertically downwards such that it strikes the wedge with velocity v0 = 11 ms-1 at height h = 48 cm from base of the wedge as shown in figure. Coefficient of restitution between ball and wedge is e = 0.5 Assuming all the surfaces to be smooth, calculate (i) velocity of wedge just after collision, (ii) vertical component of velocity of ball just after collision, (iii) time of flight of ball from the instant of collision with wedge to the instant when ball strikes the floor, and (iv) distance between ball and right edge of the wedge ball strikes the floor. (g = 10 ms-2) Ans:

(i) 2 ms–1 (iii) 0.2 second

(ii) 1.4 ms–1 (downwards) (iv) 1.20 m

MKA 12*. A uniform rod of length 4l and mass m  is free to rotate about a horizontal axis passing through a point distant l from its one end. When the rod is horizontal, its angular velocity is as shown figure. Calculate (i) reaction of axis at this instant, (ii) acceleration of centre of mass of the rod at this instant, (iii) reaction of axis and acceleration of centre of mass of the rod when rod becomes vertical for the first time, and (iv) minimum value of so that centre of  rod can complete circular motion. Ans: (iv)

(i) (ii) (iii)

2

2

 3g  7l2 2 4mg 1    (l)    6g  13 7 7  6g  2 2   4g mg  ml  ,  7   7  l   71 

MKA 13*. A uniform solid cylinder of mass m = 2 kg and radius R = 10 cm is gently placed on a rough having inclination  = 37 with horizontal and having coefficient of friction  = 1/8, such that axis of the cylinder is normal to line of greatest slope of the plane. Calculate after one second. (i) Kinetic energy of the cylinder, and (ii) loss of energy against friction. (g = 10 ms-2) Ans:

(i) 27 J

(ii) 3 J

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PHY-PP-33

MKA 14*. A solid ball of radius R = 10 cm and mass m = 0.8 kg rolls down, from rest, along a rough plane inclined at angle  = 37 with horizontal. Vertical component of displacement of ball is h 1 = 1.75 m as shown in the figure. Base of inclined plane is at height h 2 = 37 cm from the ground. If coefficient of friction and restitution between the ball and ground are  = 0.025 and e = 0.5 respectively, calculate distance, from O, of the point at which the ball strikes the ground. (i) for the first time, and (ii) for the second time, Calculate also, loss of energy against friction during first collision with the ground. What amount of energy is lost during first collision with the ground ? (G = 10 ms-2) Ans:

(i) 46 cm (ii) 212 cm, 88.5 ml, 4.8885 J

MKA 15*. Two identical balls, each of mass m = 1 kg are attached at ends of an ideal spring of natural length l0 = 10 cm and having force constant K = 13750 Nm-1. The system is placed on a smooth horizontal table. A sharp impulse is applied at one of these balls in a direction normal to axis of the spring. If the impulse is horizontal and is equal to J = 6 N-s, calculate (i) energy supplied by the impulse, and (ii) an expression from maximum elongation x 0 of the spring during subsequent motion.

x02  4l0 x0 )  2mKx 0 (l0  2x 0 )2 Ans:

(i) 18 J

(ii) J2(4

v

MKA 16*. An earth satellite is I revolving in a circular orbit of radius ‘a’ with velocity v 0. A gun 0 is in the satellite and is aimed directly towards the earth. A bullet is fired from the gun with muzzle velocity . Neglecting 2 resistance offered by cosmic dust, calculate. (i) minimum and maximum distance of bullet from earth during its subsequent motion, and (ii) period of revolution. Ans:

(i)

(ii)

2a 16a ,2a 33 3v 0

MKA 17*. Two cylindrical tanks having crosssectional area A and 2A are kept on a horizontal floor. First tank is filled with water to a height h while the other is empty . If two tank are connected by a pipe of cross-sectional area a(a