WORK, POWER & ENERGY

A-1. A rigid body of mass m is moving in a circle of radius r with a constant speed v. The force on the

body is 2mv

r and is directed towards the centre. What is the work done by this force in moving thebody over half the circumference of the circle.

a)2

2mv

r b) zero c)2

2mvr d)

2

2r

mv

A-2. If the unit of force and length each be increased by four times, then the unit of energy in increasedbya) 16 times b) 8 times c) 2 times d) 4 times

A-3. A man pushes wall and fails to displace it. He doesa) negative work b) positive but not maximum workc) no work at all d) maximum work

A-4. A rigid body moves a distance of 10 m along a straight line under the action of a force of 5 N. If thework done by this force on the body is 25 joules, the angle which the force makes with the directionof motion of the body isa) 0° b) 30° c) 60° d) 90°

A-5. A rigid body mass m kg is lifted uniformly by a man to a height of one metre in 30 sec. Another manlifts the same mass uniformly to the same height in 60 sec. The work done on the body againstgravitation by them are in ratioa) 1 : 2 b) 1 : 1 c) 2 : 1 d) 4 : 1

A-6. The work done in slowly pulling up a block of wood weighing 2 kN for a length of 10 m on a smoothplane inclined at an angle of 15° with the horizontal by a force parallel to the incline isa) 4.36 kJ b) 5.17 kJ c) 8.91 kJ d) 9.82 kJ

A-7. A 50 kg man with 20 kg load on his head climbs up 20 steps of 0.25 m height each. The work done bythe man on the block during climbing isa) 5 J b) 350 J c) 1000 J d) 3540 J

A-8. A particle moves from position 1ˆˆ ˆr 3i 2j 6k

to position 2ˆˆ ˆr 14i 13j 9k

under the action offorce ˆˆ ˆ4i j 3kN. The work done by this force will bea) 100 J b) 50 J c) 200 J d) 75 J

A-9. A ball is released from the top of a tower. The ratio of work done by force of gravity in first, secondand third second of the motion of the ball isa) 1 : 2 : 3 b) 1 : 4 : 9 c) 1 : 3 : 5 d) 1 : 5 : 3

A-10. A block of mass m is suspended by a light thread from an elevator. The elevator isaccelerating upward with uniform acceleration a. The work done by tension on theblock during t seconds is (u = 0)

a) 2m (g a)at2 b) 2m (g a)at2 c) 2m gat2 d) 0

SECTION AWORK DONE BY CONSTANT FORCE

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B-1. A particle moves under the effect of a forceF = Cx from x = 0 to x = x1. The work done in the process is

a) 21Cx b) 2

11 Cx2 c) Cx1 d) zero

B-2. Two springs have their force constant as k1 and k2(k1 > k2). When they are stretched by the sameforce up to equilibriuma) no work is done by this force in case of both the springsb) equal work is done by this force in case of both the springsc) more work is done by this force in case of second springd) more work is done by this force in case of first spring

B-3. A position dependent force F = 7 – 2x + 3x2 newton acts on a small body of mass 2 kg and displacesit from x = 0 to x = 5m. The work done in joules isa) 70 b) 270 c) 35 d) 135

B-4. Two equal masses are attached to the two ends of a spring of spring constant k. The masses arepulled out symmetrically to stretch the spring by a length x over its natural length. The work doneby the spring on each mass during the above pulling is

a) 21 kx2 b) 21 kx2 c) 21 kx4 d) 21 kx4

B-5. A rigid body is acted upon by a horizontal force which is inversely proportional to the distancecovered ‘s’. The work done by this force will be proportional toa) s b) s2 c) s d) none of these

B-6. The work done by the frictional force on a surface in drawing a circle of radius r on the surface bya pencil of negligible mass with a normal pressing force N (coefficient of friction k) isa) 2

k4 r N b) 2k2 r N c) k2 r N d) zero

B-7. A force acting on a particle varies with the displacement x as F = ax – bx2.Where a = 1 N/m and b = 1 N/m2. The work done by this force for the first one meter (F is newtons,x is in meters) is

a) 1 J6 b) 2 J6 c) 3 J6 d) none of these

C-1. The kinetic energy of a body of mass 2 kg and momentum of 2 Ns isa) 1 J b) 2 J c) 3 J d) 4 J

C-2. A particle of mass m at rest is acted upon by a force F for a time t. Its kinetic energy after aninterval t is

a)2 2F tm b)

2 2F t2m c)

2 2F t3m d) Ft

2mC-3. The graph between the resistive force F acting a body and the distance covered

by the body is 25 kg and initial velocity is 2 m/s. When the distance covered bythe body is 4m, its kinetic energy would bea) 50 J b) 40 J c) 20 J d) 10 J

SECTION BWORK DONE BY VARIABLE FORCE

SECTION CWORK ENERGY THEOREM

Work, Power & Energy

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C-4. A particle of mass 0.1 kg is subjected to a force which varies with distanceas shown in figure. If it starts its journey from rest at x = 0, its velocityat x = 12 m isa) 0 m/s b) 20 2 m / s

c) 20 3 m / s d) 40 m/s

C-5. A particle is dropped from a height h. A constant horizontal velocity is given to the particle. Takingg to be constant every where, kinetic energy E of the particle with respect to time t is correctlyshown in

a) b) c) d)

C-6. A body starts from rest with uniform acceleration and acquires a velocity V in time T. Theinstantaneous kinetic energy of the body after any time t is proportional toa) (V/T)t b) (V2/T)t2 c) (V2/T2)t d) (V2/T2)t2

C-7. If v, p and E denote the velocity, momentum and kinetic energy of the particle, thena) p = dE/dv b) p = dE/dt c) p = dv/dt d) none of these

C-8. A heavy stone is thrown from a cliff of height h with a speed v. The stone will hit the ground withmaximum speed if it is throwna) vertically downward b) vertically upwardc) horizontally d) the speed does not depend on the initial direction

C-9. The work done by all the forces (external and internal) on a system equals the change ina) total energy b) kinetic energy c) potential energy d) none of these

C-10. A body moving at 2 m/s can be stopped over a distance x. If its kinetic energy is doubled, how longwill it go before coming to rest, if the retarding force remains unchanged ?a) x b) 2x c) 4x d) 8x

C-11. A rod of length 1 m and mass 0.5 kg hinged at one end, is initially hanging vertical. The other end is nowraised slowly until it makes an angle 60° with the vertical. The required work is (use g = 10 m/s)

a) 5 J2 b) 5 J4 c) 17 J8 d) 5 3 J4C-12. A retarding force is applied to stop a train. The train stops after 80 m. If the speed is doubled, then

the distance travelled when same retarding force is applied isa) the same b) doubled c) halved d) four times

C-13. A particle moves in a straight line with retardation proportional to its displacement. Its loss ofkinetic energy for any displacement x is proportional toa) x2 b) ex c) x d) logex

C-14. A block weighing 10 N travels down a smooth curved track AB joinedto a rough horizontal surface (figure). The rough surface has a frictioncoefficient of 0.20 with the block. If the block starts slipping on thetrack from a point 1.0 m above the horizontal surface, the distanceit will move on the rough surface isa) 5.0 m b) 10.0 mc) 15.0 m d) 20.0 m

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C-15. A small mass slides down an inclined plane of inclination with the horizontal. The co-efficient offriction is m = 0x where x is the distance through which the mass slides down and 0 a constant.Then the distance covered by the mass before it stops is

a)0

2 tan b)

0

4 tan c)

0

1 tan2 d)

0

1 tan

D-1. A toy car of mass 5 kg starts from rest and moves up a ramp under the influenced of force F(F isapplied in the direction of velocity) plotted against displacement x. The maximum height attainedis given by

a) ymax = 20 m b) ymax = 15 m c) ymax = 11 m d) ymax = 5 mD-2. The negative of the work done by the conservative internal forces on a system equals the change in

a) total energy b) kinetic energy c) potential energy d) none of theseD-3. ........ of a two particle system depends only on the separation between the two particles. The most

appropriate choice for the blank space in the above sentence isa) kinetic energy b) total mechanical energyc) potential energy d) total energy

D-4. A body is dropped from a certain height. When it loses U amount of its energy it acquires a velocity‘v’. The mass of the body isa) 2U/v2 b) 2v/U2 c) 2v/U d) U2/2v

D-5. A stone projected vertically up with a velocity u reaches a maximum height h. When it is at aheight of 3h/4 from the ground, the ratio of KE and PE at that point is (consider PE = 0 at the pointof projectory)a) 1 : 1 b) 1 : 2 c) 1 : 3 d) 3 : 1

D-6. A bob hangs from a rigid support by an inextensible string of length l. If it is displaced through adistance l (from the lowest position) keeping the string straight and then released. The speed ofthe bob at the lowest position isa) gl b) 3gl

c) 2gl d) 5gl

D-7. Figure shows a particle sliding on a frictionless track which terminates in a straight horizontalsection. If the particle starts slipping from the point A, how far away from the track will theparticle hit the ground ?a) at a horizontal distance of 1 m from the end of the track.b) at a horizontal of 2 m from the end of the track.c) at a horizontal distance of 3 m from the end of the track.d) insufficient information.

D-8. Two springs A and B (kA = 2kB) are stretched by applying forces of equal magnitudes at the fourends. If the energy stored in A is E, that in B isa) E/2 b) 2E c) E d) E/4

SECTION DMECHANICAL ENERGY CONSERVATION

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D-9. When a spring is stretched by 2 cm, it stores 100 J of energy. If it is stretched further by 2 cm, thestored energy will be increased bya) 100 J b) 200 J c) 300 J d) 400 J

D-10. A block of mass m is attached to two unstretched springs of spring constants k1 and k2 as shown infigure. The block is displaced towards right through a distance x and is released. Find the speed ofthe block as it passes through the mean position shown.

a) 1 2k k xm b) 1 2

1 2

k k xm(k k )

c)2 21 2

2 21 2

k k xm(k k )

d)3 21 2

2 21 2

k k xm(k k )

D-11. A spring when stretched by 2 mm its potential energy becomes 4 J. If it is stretched by 10 mm, itspotential energy is equal toa) 4 J b) 54 J c) 415 J d) 100J

D-12. A spring of spring constant k placed horizontally surface is compressed against a block of mass mplaced on the surface so as to store maximum energy in the spring. If the coefficient of frictionbetween the block and the surface is , the potential energy stored the spring is

a)2 2 2m g

k b)

2 2 22 m gk

c)2 2 2m g2k

d)2 23 mgk

D-13. A wedge of mass M fitted with a spring of stiffness ‘k’ is kept on a smooth horizontal surface. A rodof mass m is kept on the wedge as shown in the figure. System is in equilibrium. Assuming that allsurfaces are smooth, the

a)2 2mg tan2k

b)2 2m g tan

2k

c)2 2 2m g tan

2k d)

2 2 2m g tank

D-14. A spring with spring constant k when stretched through 1 cm, the potential energy is U. If isstretched by 4 cm. The potential energy will bea) 4 U b) 8 U c) 16 U d) 2 U

D-15. The spring extends by x on loading, then energy stored by the spring is (if T is the tension inspring and k is spring constant).

a)2T

2k b)2

2T2k c)

2

22kT d)

22Tk

D-16. A body of mass m dropped from a certain height strikes a light vertical fixed spring of stiffness k.The height of its fall before touching the spring if the maximum compression of the spring is equal

to 3mgk is

a) 3mg2k b) 2mg

k c) 3mg4k d) mg

4kD-17. A running man has half the kinetic energy of that of a boy of half of his mass. The man speeds up by

1 m/s so as to have same kinetic energy as that of the body. The original speed of the man will be

a) 2 m/s b) 2 1 m/s c) 1 m/s

2 1 d)1 m/s2

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E-1. A car of mass ‘m’ is driven with acceleration ‘a’ along straight level road against a constant externalresistive force ‘R’. When the velocity of the car is ‘V’, the rate at which the engine of the car isdoing work will bea) RV b) maV c) (R + ma)V d) (ma – R)V

E-2. The average power required to lift a 100 kg mass through a height of 50 metres in approximately50 metres in approximately 50 seconds would bea) 50 J/s b) 5000 J/s c) 100 J/s d) 980 J/s

E-3. A block of mass m is moving with a constant acceleration ‘a’ on a rough horizontal plane. If thecoefficient of friction between the block and plane is . The power delivered by the external agentat a time t from the beginning is equal toa) ma2t b) mgat c) m(a + g)gt d) m(a + g)at

E-4. A particle moves with a velocity ˆˆ ˆv (5i 3j 6k) m/s under the influence of a constant force

ˆˆ ˆF (10i 10j 20k)N.

The instantaneous power applied to the particle isa) 200 J/s b) 40 J/s c) 140 J d) 170 J/s

E-5. An electric motor creates a tension of 4500 N in hoisting cable and reels it at the rate of 2 m/s.What is the power of electric motor ?a) g W b) 9 KW c) 225 W d) 9000 H.P.

E-6. A man M1 of mass 80 kg runs up a staircase in 15 s. Another man M2 also of mass 80 kg runs up thestair case in 20 s. The ratio of the power developed by them (P1/P2) will bea) 1 b) 4/3 c) 16/9 d) 9/16

CONSERVATIVE AND NON CONSERVATIVE FORCES AND EQUILIBRIUM

F-1. The potential energy of a particle in a field is 2a bU ,rr

where a and b are constant. The value ofr in terms of a and b where force on the particle is zero will bea) a

b b) ba c) 2a

b d) 2ba

F-2. Potential energy v/s displacement curve of one demesnial conservative field is shown. Force at Aand B is respectivelya) positive, positive b) positive, negativec) negative, positive d) negative, negative

F-3. The potential energy of a particle u varies with distance x as shown inthe graph. The force acting on the particle is zero ata) C b) Bc) B and C d) A and D

F-4. The diagrams represent the potential energy U of a function of the inter-atomic distance r. Whichdiagram corresponds to stable molecules found in nature.

a) b) c) d)

SECTION EPOWER

SECTION F

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F-5. For the path PQR in a conservative force field (figure), the amount of work done in carrying a bodyfrom P to Q and from Q to R are 5J and 2J respectively. The workdown in carrying the body fromP to R will bea) 7 J b) 3 Jc) 21J d) zero

F-6. The potential energy for a force field F

is given by U(x, y). The force acting on the particle of mass

m at 0, 4

is

a) 1 b) 2 c)12 d) 0

F-7. A particle is taken from point A to point B under the influence of a force field. Now it is taken backfrom B to A and it is observed that the work done in taking the particle from A to B is not equal tothe work done in taking it from B to A. If Wnc and Wc is the work done by non-conservative forcesand conservative forces present in the system respectively, K is the change in kinetic energy,thena) ncW U k b) cW U c) nc cW W k d) All above correct

RANK DECIDING QUESTIONS (RDQs)G-1. A force ˆ ˆF K(yi xj)

where K is a positive constant, acts on a particle moving in the x-y plane.

Starting from the origin, the particle is taken along the positive x-axis to the point (a, 0) and thenparallel to the y-axis to the point (a, a). The total work done by the force F

on the particle is

a) –2Ka2 b) 2Ka2 c) –Ka2 d) Ka2

G-2. A pump motor is used to deliver water at a certain rate from a given pipe. To obtain “n” timeswater from the same pipe in the same time, the amount by which the power of the motor should beincreased isa) n2 b) n3 c) n4 d) n1/2

G-3. A wind-powered generator converts wind energy into electrical energy. Assume that the generatorconverts a fixed fraction of the wind energy intercepted by its blades into electrical energy. Forwind speed v, the electrical power output will be proportional toa) v b) v2 c) v3 d) v4

G-4. A particle, which is constrained to move along the x-axis, is subjected to a force in the same directionwhich varies with the distance x of the particle from the origin as F(x) = –kx + ax3. Here k and a arepositive constants. For x > 0, the functional form of the potential energy U(x) of the particle is

a) b) c) d)

G-5. An ideal spring with spring-constant k is hung from the ceiling and a block of mass M is attachedto its lower end. The mass is released with the spring initially unstreched. Then the maximumextension in the spring isa) 4 Mg/k b) 2 Mg/k c) Mg/k d) Mg/2k

SECTION G

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G-6. A particle moves under the influence of a force F = kx in one dimensions (k is a positive constantand x is the distance of the particle from the origin). Assume that the potential energy of theparticle at the origin is zero, the schematic diagram of the potential energy U as a function of x isgiven by

a) b) c) d)

G-7. Statement-1A block of mass m starts moving on a rough horizontal surface with a velocity v. It stops due tofriction between the block and the surface after moving through a certain distance. The surface isnow tilted to an angle of 30° with the horizontal and the same block is made to go up on the surfacewith the same initial velocity v. The decrease in the mechanical energy in the second situation issmaller than that in the first situation.BecauseStatement-2The coefficient of friction between the block and the surface decreases with the increase in theangle of inclination.a) statement-1 is true, statemetn-2 is true; statement-2 is a correct explanation for statement-1b) statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1c) statement-1 is true, statement-2 is falsed) statement-1 is false, statement-2 is true

G-8. A block (B) is attached to two unstretched springs S1 and S2 with spring constants k and 4k,respectively (see figure I). The other ends are attached to identical supports M1 and M2 not attachedto the walls. The springs and supports have negligible mass. There is no friction anywhere. Theblock B is displaced towards wall 1 by a small distance x(figure II) and released. The block returnsand moves a maximum distance y towards wall 2. Displacements x and y are measured with respectto the equilibrium position of the block B. The ratio y/x is

a) 4 b) 2 c) 1/2 d) 1/4G-9. A light inextensible string that goes over a smooth fixed pulley as shown in

the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10m/s2, find the work done (in joules) by the string on the block of mass 0.36 kgduring the first second after the system is released from rest.a) 16 J b) 8 Jc) 24 J d) 4 J

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G-10. A block of mass 50 kg is projected horizontally on a rough horizontal floor. The coefficient of frictionbetween the block and the floor is 0.1. The block strikes a light spring of stiffnessk = 100 N/m with a velocity 2 m/s. The maximum compression of the spring isa) 1 m b) 2 m c) 3 m d) 4 m

G-11. A car of mass m starts moving so that its velocity varies according to the law v s, where is aconstant and s is the distance covered. The total work performed by all the forces which are actingon the car during the first t seconds after the beginning of motion isa) m4t2/8 b) m2t4/8 c) m4t2/4 d) m2t4/4

G-12. A block of mass m is placed in side a smooth hollow cylinder of radius Rwhose axis is kept horizontally. Initially system was at rest. Now cylinder isgiven constant acceleration 2g in the horizontal direction by external agent.The maximum angular displacement of the block with the vertical isa) 2 tan–12 b) tan–12 c) tan–11 d) tan–1(1/2)

G-13. A block of mass 250 g is kept on a vertical spring of spring constant 100 N/m fixed from below. Thespring is now compressed to have length 10 cm shorter than its natural length and the system isreleased from this position. How high does the block rise ? Take g = 10 m/s2.a) 20 cm b) 30 cm c) 40 cm d) 50 cm

G-14. In a projectile motion, KE varies with time as in graph

a) b) c) d)

G-15. If W1, W2 and W3 represent the work done in moving a particle from A to B

along three different paths 1, 2, 3 respectively (as shown) in the gravitationalfield of a point mass m, find the correct relation between W1, W2 and W3

a) W1 > W2 > W3 b) W1 = W2 = W3

c) W1 < W2 < W3 d) W2 > W1 > W3

G-16. An open knife edge of mass ‘m’ is dropped from a height ‘h’ on a woodenfloor. If the blade penetrates upto depth ‘d’ into the wood, the averageresistance offered by the wood to the knife edge is

a) mg b) hmg 1 d

c) hmg 1 d

d)2hmg 1 d

G-17. A body is moved along straight line by a machine delivering constant power. The distance movedby the body in time t is proportional to (u = 0)a)t1/2 b) t3/4 c) t3/2 d) t2

G-18. The potential energy of a system is represented in the first figure,the force acting on the system will be represented by

a) b) c) d)

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G-19. Which of the following graphs is correct between kinetic energy (E), potential energy (U) andheight (h) from the ground of the particle (h << RE and U = 0 at h = 0)

a) b) c) d)

G-20. The graph between E and 1/p is (E = kinetic energy and p = momentum)

a) b) c) d)

G-21. The force acting on a body moving along x-axis varies with the position of the particle as shown inthe figure. The body is in stable equilibrium at

a) x = x1 b) x = x2 c) both x1 and x2 d) neither x1 nor x2

G-22. The graph between kinetic energy E and velocity v is

a) b) c) d)

G-23. A 10 kg is pulled in the vertical plane along a frictionless surface in the form of an arc of a circle ofradius 10 m. The applied force is os 200 N as shown in the figure. If the block started from rest atA, the velocity at B would be

a) 3 m/s b) 10 3 m/s c) 100 3 m/s d) 3 10 m/s

CENTRE OF MASS & COLLISION

A-1. The centre of mass of a bodya) lies always at the geometrical centre b) lies always inside the bodyc) lies always outside the body d) lies within or outside the body

A-2. The centre of mass of the shaded portions of the disc is : (The mass is uniformly distributed in theshaded portion) :

a) R20 to the left of A b) R

12 to the left of AA

c) R20 to the right of AA d) R

12 to the right of AA

A-3. A thin uniform wire is bent to form the two equal sides AB and AC of triangle ABC, whereAB = AC = 5 cm. The third side BC, of length 6 cm, is made from uniform wire of twice the densityof the first. The distance of centre of mass from A is

a) 3411 cm b) 11

34 cm c) 349 cm d) 11

45 cm

A-4. A semicircular portion of radius ‘r’ is cut from uniform rectangular plate as shown in figure. Thedistance of centre of mass ‘C’ of remaining plate, from point ‘O’ is

a)2r

(3 ) b)3r

2(4 )

c)2r

(4 ) d)

2r3(4 )

A-5. A uniform solid cone of height 40 cm is shown in figure.The distance of centre of mass of the cone frompoint B (centre of the base) isa) 20 cm b) 10/3 cmc) 20/3 cm d) 10 cm

A-6. The centre of mass of a system of particles is at the origin. It follows thata) the number of particles to the right of the origin is equal to the number of particle to the leftb) the total mass of the particles to the right of the origin is same as the total mass to the left of the originc) the number of particles on X-axis may be equal to the number of particles on Y-axisd) if there is a particle on the positive X-axis, there must be at least one particle on the negative X-axis.

A-7*. A body has its centre of mass at the origin. The x-coordinates of the particlesa) may be all positiveb) may be all negativec) may be all non-negatived) may be positive for some particles and negative in other particles

SECTION ACALCULATION OF CENTRE OF MASS

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A-8*. In which of the following cases the centre of mass of a rod is certainly not at its centre ?a) the density continuously increases from left to rightb) the density continuously decreases from left to rightc) the density decreases from left to right upto the centre and then increasesd) the density increases from left to right upto the centre and then decreases

A-9. A body of mass 1 kg moving in the x-direction, suddenly explodes into two fragments of mass 1/8 kgand 7/8 kg. An instant later, the smaller fragment is 0.14 m above the x-axis. The position of theheavier fragment isa) 0.02 m above x-axis b) 0.02 m below x-axis c) 0.14 m below x-axis d) 0.14 m above x-axis

SECTION BMOTION OF CENTRE OF MASS

B-1. Two particles of mass 1 kg and 0.5 kg are moving in the same direction with speed of 2 m/s and 6 m/srespectively on a smooth horizontal surface. The speed of centre of mass of the system is

a) 103 m/s b) 10

7 m/s c) 112 m/s d) 12

3 m/s

B-2. Two particles of equal mass have initial velocities 1 1ˆ ˆ2ims and 2jms . First particle has a constant

acceleration 2ˆ ˆi j ms while the acceleration of the second particle is always zero. The centre ofmass of the two particles moves ina) circle b) parabola c) ellipse d) straight line

B-3. Two particle having mass ratio n : 1 are interconnected by a light inextensible string that passesover a smooth pulley. If the system is released, then the acceleration of the centre of mass of thesystem is

a) (n – 1)2 g b)2n 1 gn 1

c)

2n 1 gn 1

d) n 1 gn 1

B-4. A bomb travelling in a parabolic path under the effect of gravity, explodes in mid air. The centre of

mass of fragments willa) move vertically upwards and then downwardsb) move vertically downwardsc) move in irregular pathd) move in the parabolic path which the unexploded bomb would have ravelled

B-5. If a ball is thrown upwards from the surface of eartha) the earth remains stationary while the ball moves upwardsb) the ball remains stationary while the earth moves downwardsc) the ball and earth both movies towards each otherd) the ball and earth both move away from each other

B-6. Internal forces can changea) the linear momentum but not the kinetic energy of the systemb) the kinetic energy but not the linear momentum of the systemc) linear momentum as well as kinetic energy of the systemd) neither the linear momentum nor the kinetic energy of the system

B-7*. If the external forces acting on a system have zero resultant, the centre of massa) must not move b) must not acceleratec) may move d) may accelerate

Centre of Mass & Collision

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B-8. Two balls are thrown in air. The acceleration of the centre of mass of the two balls while in air(neglect air resistance)a) depends on the direction of the motion of the ballsb) depends on the masses of the two ballsc) depends on the speeds of the two ballsd) is equal to g

B-9. Two blocks of masses 10 kg and 30 kg are placed along a vertical line. The first block is raisedthrough a height of 7 cm. By what distance should the second mass be moved to raise the centre ofmass by 1 cm ?a) 1 cm downward b) 2 cm downward c) 3 cm downward d) 4 cm downward

B-10. Mr. Verma (50 kg) and Mr. Mathur (60 kg) are sitting at the two extremes of a 4 m long boat (40 kg)standing still in water. To discuss a mechanics problem, they comes to the middle of the boat.Neglecting friction with water, how far does the boat move in the water during the process ?

a) 20 cm3 b) 40 cm3 c) 80 cm3 d) 40 cm

SECTION CCONSERVATION OF LINEAR MOMENTUM

C-1. Two particles A and B initially at rest move towards each other under a mutual force of attraction.The speed of centre of mass at the instant when the speed of A is v and the speed of B is 2v isa) v b) zero c) 2 v d) 3v/2

C-2. If the KE of a body becomes four times its initial value, then the new momentum will be more thatits initial momentum bya) 50% b) 100% c) 125% d) 150%

C-3. A particle of mass 4m which is at rest explodes into three fragments. Two of the fragments each ofmass m are found to move with a speed ‘v’ each in mutually perpendicular directions. The energyreleased in the process of explosion isa) 2/3 mv2 b) 3/2 mv2 c) 4/3 mv2 d) 3/4 mv2

C-4. A 500 kg boat has an initial speed of 10 ms–1 as it passes under a bridge. At that instant a 50 kg manjumps straight down into the boat from the bridge. The speed of the boat after the man and boatattain a common speed is

a) 1100 ms11 b) 110 ms11

c) 150 ms11 d) 15 ms11

C-5. A man of mass ‘m’ climbs on a rope of length L suspends below a balloon of mass M. The balloon isstationary with respect to ground. If the man begins to climb up the rope at a speed vrel (relative torope). In what direction and with what speed (relative to ground) will the balloon move ?

a) downwards, relmvm M

b) upwards, relmvm M

c) downwards, relmvM d) upwards, rel(M m)v

M

C-6. A shell is fired from a cannon with a velocity V at an angle with the horizontal direction. At thehighest point in its path, it explodes into two pieces of equal masses. One of the pieces retraces itspath to the cannon. The speed of the other piece immediately after the explosion isa) 3V cos b) 2V cos c) 3/2 V cos d) V cos

Centre of Mass & Collision

XIth Class

[ 14 ]

SECTION DSPRING - MASS SYSTEM

D-1. Two blocks of masses m and M are moving with speeds v1 and v2 (v1 > v2) in the same direction onthe frictionless surface respectively, M being ahead of m. An ideal spring of force constant k isattached to the backside of M (as shown). The maximum compression of the spring when the blockcollides is

a) 1mv k b) 2

Mv k

c) 1 2mM(v v ) (M m)K

d) none of above is correct

D-2. Two masses m1 and m2 are connected by a spring of spring constant k and are placed on a frictionlesshorizontal surface. Initially the spring is stretched through a distance x0 when the system is releasedfrom rest. Find the distance moved by the two masses before they again come to rest.

a) 2 0 1 0

1 2 1 2

2m x 2m x,m m m m b) 2 0 1 0

1 2 1 2

2m x 2m x,m m m m c) 2 0 1 0

1 2 1 2

m x m x,m m m m d) 2 0 1 0

1 2 1 2

2m x 2m x,m m m m

SECTION EIMPULSE

E-1. A ball of mass 50 gm is dropped from a height h = 10 m. It rebounds losing 75 percent of its kineticenergy. If it remains in contact with the ground for t = 0.01 sec., the impulse of the impact force isa) 1.3 N-s b) 1.05 N-s c) 1300 N-s d) 105 N-s

E-2. The area of F-t curve is A, where ‘F’ is the force on the mass due to the other.If one of the colliding bodies of mass M is at rest initially, its speed just afterthe collision is

a) A/M b) M/A c) AM d) 2AM

E-3. A bullet of mass m moving vertically upwards with a velocity ‘u’ hits the hanging block of mass ‘m’and gets embedded in it. The height through which block rises after the collision, assume sufficientspace above blocka) u2/2g b) u2/g c) u2/8g d) u2/4g

E-4. Velocity of a particle of mass 2 kg varies with time t according to the equation ˆ ˆv 2ti 4j m/s . Here

is in seconds. Find the impulse imparted to the particle in the time interval from t = 0 to t = 2 s.

a) ˆ10 i m/s b) ˆ15 i m/s c) ˆ8 i m/s d) ˆ4 i m/s

E-5. A ball of mass 50 g moving at a speed of 2.0 m/s strikes a plane surface at an angle of incidence 60°.The ball is reflected by the plane at equal angle of reflection with the same speed. Calculate themagnitude of the change in momentum of the ball.a) 0.5 kg-m/s b) 0.1 kg-m/s c) 1 kg-m/s d) 2 kg-m/s

Centre of Mass & Collision

XIth Class

[ 15 ]

SECTION FCOLLISION

F-1*. In an elastic collision in absence of external force, which of the following is/are correcta) the linear momentum is conservedb) the potential energy is conserved in collisionc) the final kinetic energy is less than the initial kinetic energyd) the final kinetic energy is equal to the initial kinetic energy

F-2*. A body is moving towards a finite body which is initially at rest collides with it. In the absence ofany external impulsive force, it is possible thata) both the bodies come to restb) both the bodies come to restc) the moving body comes to rest and the stationary body starts movingd) the stationary body remains stationary, the moving body changes its velocity

F-3*. A block moving in air explodes in two parts then just after explosiona) the total momentum must be conservedb) the total kinetic energy of two parts must be same as that of block before explosionc) the total momentum must changed) the total kinetic energy must increase

F-4*. In head on elastic collision of two bodies of equal massesa) the velocities are interchangedb) the speeds are interchangedc) the momenta are interchangedd) the faster body slows down and the slows body speeds up

F-5. A bullet of mass m = 50 gm strikes a sand bag of mass M = 5 kg hanging from a fixed point, with ahorizontal velocity pv . If bullet sticks to the sand bag then the ratio of final & initial kineticenergy of the bullet isa) 10–2 b) 10–3 c) 10–6 d) 10–4

F-6. In the arrangement shown, the pendulum on the left is pulled aside. It isthen released and allowed to collide with other pendulum which is at rest. Aperfectly inelastic collision occurs and the system rises to a height h/4. Theratio of the masses (m/M) of the pendulum isa) 1 b) 2 c) 3 d) 4

F-7. There are hundred identical sliders equally spaced on a frictionless track as shown in the figure.Initially all the sliders are at rest. Slider 1 is pushed with velocity v towards slider 2. In a collisionthe sliders stick together. The final velocity of the set of hundred stucked sliders will be

a) v/99 b) v/100 c) zero d) vF-8. A solid iron ball A of radius r collides head on with another stationary solid iron ball B of radius 2r.

The ratio of their speeds just after the collision (e = 0.5) isa) 3 b) 4 c) 2 d) 1

F-9. A particle of mass m moves with velocity v0 = 20 m/sec towards a wall that ismoving with velocity v = 5 m/sec. If the particle collides with the wall elastically,the speed of the particle just after the collision isa) 30 m/s b) 20 m/s c) 25 m/s d) 22 m/s

Centre of Mass & Collision

XIth Class

[ 16 ]

F-10. Two perfectly elastic balls of same mass m are moving with velocities u1 and u2. They collideelastically n times. The kinetic energy of the system finally is

a) 21

1 m u2 n b) 2 21 2

1 m u u2 n c) 2 21 2

1 m u u2 d) 2 21 2

1 mn u u2

F-11. A super-ball is to bounce elastically back and forth between two rigid walls at a distance d fromeach other. Neglecting gravity and assuming the velocity of super-ball to be v0 horizontally, theaverage force being exerted by the super-ball on one wall is

a)201 mv

2 d b)20mv

d c)202mv

d d)204mv

dF-12. A massive ball moving with speed v collides head-on with a tiny ball at rest having a mass very less

than the mass of the first ball. If the collision is elastic, then immediately after the impact, thesecond ball will move with a speed approximately equal toa) v b) 2v c) v/2 d)

F-13. A sphere of mass m moving with a constant velocity hits another stationary sphere of the samemass. If e is the coefficient of restitution, then ratio of speed of the first sphere to the speed of thesecond sphere after collision will be

a) 1 e1 e

b) 1 e

1 e

c) e 1

e 1

d) e 1

e 1

F-14. A ball of mass ‘m’, moving with uniform speed, collides elastically with another stationary ball.

The incident ball will lose maximum kinetic energy when the mass of the stationary ball isa) m b) 2m c) 4m d) infinity

F-15. Three balls A, B and C are placed on a smooth horizontal surface. Given that mA = mC = 4mB. Ball Bcollides with ball C with an initial velocity v as shown in figure. Find the total number of collisionsbetween the balls. All collisions are elastic.a) 4 b) 2c) 3 d) 5

F-16. A ball of mass m moving at a speed v makes a head on collision with an identical ball at rest. Thekinetic energy of the balls after the collision is 3/4 of the original kinetic energy. Calculate thecoefficient of restitution.

a)12 b) 1

2 c) 34 d) 4

3F-17. A ball of mass 0.50 kg moving at a speed of 5.0 m/s collides with another ball of mass 1.0 kg. After

the collision the balls stick together and remain motionless. What was the velocity of the 1.0 kgblock before the collision ?a) 5 m/s b) 10 m/s c) 2.5 m/s d) 2 m/s

SECTION GVARIABLE MASS

G-1. If the force on a rocket which is ejecting gases with a relative velocity of 300 m/s, is 210 N. Then therate of combustion of the fuel will bea) 10.7 kg/sec b) 0.07 kg/sec c) 1.4 kg/sec d) 0.7 kg/sec

G-2. A rocket of mass m = 20 kg has M = 180 kg fuel. The uniform exhaust velocity of the fuel is vr = 1.6 km/s.Calculate the minimum rate of consumption of fuel so that the rocket may rise from the ground.a) 5 kg/s b) 10 kg/s c) 1.25 kg/s d) 2.5 kg/s

Centre of Mass & Collision

XIth Class

[ 17 ]

SECTION HRANK DECIDING QUESTIONS (RDQs)

H-1. Find the position of centre of mass of the uniform planner sheet shownin figure with respect to the origin(O)a) (5a/6, 5a/6) b) (6a/5, 5a/6)c) (5a/6, 6a/5) d) (5a/3, 6a/4)

H-2. Seven homogeneous bricks, each of length L, are arranged as shown in figure.Each brick is displaced with respect to the one in contact by L/10. Find the xcoordinate of the centre of mass relative to the origin O shown.a) 21L/35 b) 22L/35c) 23L/35 d) 24L/35

H-3. A uniform disc of radius R is put over another uniform disc of radius 2R of the same thickness anddensity. The peripheries of the two discs touch each other. Locate the centre of mass of the systemfrom the centre of large disc.a) R/4 towards smaller disc b) R/2 towards smaller discc) R/5 towards larger disc d) R/5 towards smaller disc

H-4. A square plate of edge d and a circular disc of diameter d are placed touching each other at themidpoint of an edge of the plate as shown in figure. Locate the centre of mass of the combinationfrom the contact point, assuming same mass per unit area for the two plates.

a)d(4 )2(4 )

right of the contact point b)d(4 )2(4 )

left of the contact point

c)d(4 )2(4 )

right of the contact point d)d(4 )2(4 )

left of the contact point

H-5. A thin uniform sheet of metal of uniform thickness is cut into the shapebounded by the line x = a and y = kx2, as shown. Find the coordinates of thecentre of mass.a) 3/2 a b) 3/9 ac) 3/4 a d) 3/5 a

H-6. A uniform chain of mass m and length hangs on a thread and touches the surfaceof a table by its lower end. Find the force exerted by the table on the chain whenhalf of its length has fallen on the table. The fallen part does not form heap.a) 3/2 mg b) 3/4 mgc) 3/5 mg d) 3/9 mg

H-7. A ball moving transitionally collides with another, stationary, ball of the same mass. At the momentof impact the angle between the straight, line passing through the centres of the balls and thedirection of the initial motion of the striking ball is equal to = 45º. Assuming the balls to besmooth, find the fraction of the kinetic energy of the striking ball that turned into potentialenergy at the moment of the maximum deformation.a) 0.25 b) 0.5 c) 0.6 d) 0.025

H-8. Ball 1 collides head on with an another identical ball 2 at rest. Velocity of second ball after collisionbecomes two times that of 1 after collision. The coefficient of restitution between the two balls is:a) e = 1/3 b) e = 1/2 c) e = 1/4 d) e = 2/3

Centre of Mass & Collision

XIth Class

[ 18 ]

H-9. The spacecraft of mass M moves with velocity V in free space at first, then it explodes breakinginto two pieces. If after explosion a piece of mass m comes to rest, the other piece of space craftwill have a velocity:a) MV/(M–m) b) MV/(M+m) c) mV/(M–m) d) mV/(M+m)

H-10. A bullet of mass m strikes a block of mass M connected to a light spring of stiffness k, with a speedV0. If the bullet gets embedded in the block then, the maximum compression in the spring is:

a)

2/120

2

kmMvm

b)

2/120

kmM2Mmv

c)

2/120

kmM2Mv

d)

2/12

kmMMv

H-11* A ball moving with a velocity v hits a massive wall moving towards the ball with a velocity u. Anelastic impact lasts for a time t.

a) The average elastic force acting on the ball is

tvum

b) The average elastic force acting on the ball is

tvum2

c) The kinetic energy of the ball increases by 2mu(u+v)d) The kinetic energy of the ball remains the same after the collision

H-12. On a frictionless surface, a ball of mass m moving at a speed v makes a head on collision with anidentical ball at rest. The kinetic energy of the balls after the collision is 3/4th of the original. Findthe coefficient of restitution.

a)13 b) 3

4 c)12 d) 4

3H-13. A ball is moving with velocity 2 m/s towards a heavy wall moving towards

the ball with speed 1m/s as shown in fig. Assuming collision to be elastic,find the velocity of the ball immediately after the collision.a) 3 m/s b) 4 m/s c) 6 m/s d) 1 m/s

H-14. A ball is approaching to ground with speed u. If the coefficient of restitutionis e then find out the velocity just after collision.a) eu b) e2uc) u/e d) 2eu

H-15. In the previous question find the impulse exerted by the normal due to ground on the ball.a) mu(1 – e) b) mu(1 + e) c) mu(2 + e) d) emu

H-16. If the chain is lowered at a constant speed v = 1.2 m/s, determine the normalreaction exerted on the floor as a function of time. The chain has a mass of 80kg and a total length of 6 m.a) (16t + 9.8) N vertically upwardb) (16t + 9.8) N vertically downwardc) (16t + 19.2) N vertically downwardd) (16t + 19.2) N vertically upward

H-17. Find the mass of the rocket as a function of time, if it moves with a constant acceleration a, inabsence of external forces. The gas escaps with a constant velocity u relative to the rocket and itsinitial mass was m0.a) at/um e b) t/aum e c) at/u

0m e d) a/tu0m e

Centre of Mass & Collision