notes on work energy power for iit(mains+ advance)

7/29/2019 Notes on Work Energy Power for IIT(Mains+ Advance)

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WORK, ENERGY & POWER

Syllabus : Kinetic and potential energy; Work and Power, Conservation of mechanical energy, work energy

principle.

WORK

When a force is app lied at a point and the point gains some energy. Then the work is

said to be done by the force.

The work W done by a constant forceF when its point of application undergoes a

displacementS is measured as

W =F .

S = |

F | |

S | cos

Where is the angle betweenF and

S . Work is a scalar quantity and its SI unit is N-m or

joule (J).

Only the component (Fcos) of the force F which is along the displacement contributes

to the work done. IfF = Fx i + Fyj + Fz k and S = x i + y j + z k then W =

F . S = Fx

x+Fyy+Fzz

Positive and N egative work : The work is said to be pos itive if the angle is acute ( 900). If the angle betweenF an d

S is 900 then

work done by the force is zero.

If the force is variable then the work done by the variable force is given by dW =F .

dS

or

W = 2

1

S

S

dS.F

Work d epend s on fram e of reference. With change of frame of reference inertial force

does not change w hile displacement m ay change, so the w ork d one by a force will be different

in different frames.


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Illustra tion 1 :

A particle of mass 2 kg moves under the action of a constant forceF = ( )ji 25 N. If its

displacement is 6 jm. What is the work d one by the forceF ?

Solution :

The work doneF .

x

= ( )ji 25 . j6 = - 12 Joule

Illustration 2 :

A load of mass m = 3000 kg is lifted by a rop e with an acceleration a = 2 m/ s2. Find the work

done du ring the first one and a half seconds from the beginning of motion.

Solution :

The height to which the body is lifted d ur ing the first 't' second is h =21 at 2 tension in the

rope T = mg + ma

Work done = T.h = m(g +a)

2

2

1at = 3000 (10 + 2) ( )

2

5122

1.xx

= 81 KJ

WORK DONE BY A SPRING FORCE :

Whenever a spring is stretched or compressed, the sp ring force always tend to restore it

to the equilibrium pos ition. If x be the displacement of the free end of the spring from its

equilibrium position then, the m agnitud e of the spring force is FS = - kx

The negative sign ind icates tha t the force is restoring.

The work d one by the sp ring force for a displacement from xi to xf is given by


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Ws = f

x

ix

kxdx Ws = ( )222

1if

xxk

WORK DONE BY FRICTION :

Work d one by friction may be zero, positive or negative depen ding upon t he situations:

When a block is pulled by a force F and the block does not move, the work done

by friction is zero.

When a block is pu lled on a stationary surface, the work done by the kinetic friction

is negative.

When one block is placed on another block and is pulled by a force then friction

force does n egative w ork on top block and p ositive work on the lower block

WORK DONE BY GRAVITY :

Here the force of grav ity is Fg = - mg j and the displacemen t is given by

S = x i + y j + z k

Work done by gravity is Wg = gF . S = - mg y

y = y f- yI = - h

Wg = + mgh

If the block moves in the upward direction, then the work done by gravity is negative

and is given by Wg = - mgh .

DEPENDENCE OF WORK ON FRAME OF REFERENCE :

Work depends upon the frame of reference from where it is calculated. As the

displacement as well as force, depends upon the deferent frames of reference. Therefore, the

work a lso changes. For examp le, if you calculate work from a non inertial fram e work d ue to

pseu do force has to be included . Again displacement from the inertial frame of reference will be

different from ground frame.


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CON SERVATIVE AND NO N CONSERVATIVE FORCES :

In Conservative force field the w ork d one by the force is independent on path followed

and depends only on initial and final co-ordinates. Such forces are known as Conservative

forces.

Examples are gravitational, electrostatic forces.

If the work done dep ends on path followed. Such forces are called non-

Conser vative forces. Example is frictional force.

Illustration 3 :

A train is moving w ith a constant speed "v". A box is pushed by a worker ap plying a force "F"

on the box in the train slowly by distance "d" on the tra in for time "t". Find the w ork done by

"F" from the train frame as well as from the ground frame.

Solution :

As the box is seen from th e train frame th e d isplacement is only 'd' if the force direction

is same as the direction of motion of the box.

Then the work done = F.d = Fdcos00 = Fd

= Fdcos1800 = -Fd

(if the disp lacement on the train is opposite to 'F')

As the box is seen from grou nd frame,

the d isplacement of the box = vt + d (if the d isplacement is along the

direction of motion of the train )

= d - vt (if the disp lacement is opp osite to

direction of motion of the train)

then work done = F. (vt + d) = Fvt + Fd OR = F.(d-vt) = Fd - Fvt

Illustra tion 4 :

A block is (mass m) placed on the rough surface of a plank (mass m) of coefficient of

friction "" which in tu rn is placed on a smooth su rface. The block is given a velocity

v0 with respect to the plank which comes to rest with respect to the plank. Find the

a) The total work done by friction in the p lank frame.

b) The w ork d one by friction on the smaller block in the plank frame.

c) Find the final velocity of the p lank

m

m 0v


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Solution :

The acceleration of the plank = Friction force applied by the block on th e plank / mass of

the plank.

gm

mga p =

=

(a) Pseudo force acting on the block = g (back wards)

Force of friction is mg ( acting backwards)

From the plank frame time needed to stop the block is given by

O = atV +0

( )ga = 2

t =g

V

20

Velocity of the plank du ring this time is tauV ppp +=

=22

00V

g

Vg =

Displacement of the block = S = g

V

a

VV

=

8

3

2

22

0

2

02

0

Work d one by friction on the block = ( )18

32

0

=g

V.mgcos.S.F = 2

08

3mv

(b) From the Plank frame

Work d one by friction on sm aller block = -mgl

g2V0 20

=l

2mVg

20= l

work d one by friction from the Plank frame =2

mV20

(c) Final velocity of the block

= Velocity of the p lank =2

0V

mg

pma

m


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WORK ENERGY THEOREM :

Now we have to study which physical quantity changes when work is done on a

pa rticle. If a constan t force F acts through a disp lacemen t x, it does work W = Fx

Q 22if

vv = + 2 ax

W =2

22if

m vv =

2

1m

2f

v -2

1m

2i

v

The quan tity k =2

1m v2 is a scalar and is called the kinetic energy of the par ticle. It is

the energy posses by the p article by virtue of its motion.

Thus the equ ation takes the form KKKW if == The work d one by a force chan ges the kinetic energy of the par ticle. This is called the

work -Energy Theorem.

Illustration 5:

The velocity of an 800 gm object changes from

0v = 3 i - 4 j to

fv = -6 j + 2 k m/ s. What is

the change in K.E of the body?

Solution:

Here m = 800gm = 0.8 kg

ov = ( )

22 43 + = 5 ( ) ( )22 26 +=

fv = 40

change in K.E =2

1x 0.8

2

0

2

vvf = ( ) Jx.x 62540802

1=

Illustration 6 :

The coefficient of sliding friction betw een a 900 kg car and pavem ent is 0.8. If the car is moving

at 25 m/ s along level pavem ent, when it begins to skid to a stop, how far will it go before

stopping?

Solution :

Here m = 900kg = 0.8, v = 25 m/ s S =?


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K.E = work done against friction2

2

1mv = F.s = N.s = mgs

s =

g

v

2

2

=( )

10802

252

x.x

~ 39 m

Illustration 7 :

An object of mass 10kg falls from rest through a vertical distance of 20m and acquires a

velocity of 10 m/ s. How m uch work is done by the push of air on the object ? (g = 10 m/ s2)

Solution :

Let up ward pu sh of air be F

The resultant down ward force = mg - F

As work done = gain in K.E

(mg - F) x S =2

2

1mv

(10 x 10 - F) x 20 =2

1x 10 x (10)2 F = 75 N

Work d one by p ush of air = 75 x 20 = 15 Jou le

This work done is negative.

POTENTIAL ENERGY :

Potential energy of any body is the energy possessed by the body by virtue of its

position or the state of deformation. With every potential energy there is an associated

conservative force. The potential energy is measured as the m agnitud e of work d one against

the associated conservative force

du = - rd.F

For Example :

(i) If an object is placed at any point in grav itational field work is to be done against

gravitational field force. The magnitud e of this work done against the gravitational

force gives the measure of gravitational potential energy of the body at that position

which is U = mgh. Here h is the height of the object from the reference level.


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ii) The magn itude of work done against the spring force to compress it gives the measu re

of elastic poten tial energy, wh ich is U =2

1k x2

iii) A charged body in any electrostatic field will have electrostat ic poten tial energy. The

change in potential energy of a system associated with conservative internal force as U2-

U1= - W= 2

1

F . d r

CONSERVATION OF MECHANICAL ENERGY :

Change in potential energy U = - WC where WC is the work done by conservative

forces. From work energy theorem

Wnet = kWhere Wnet is the sum of work d one by all the forces acting on the m ass. If the system is

subjected to only conservative forces then Wne t = WC = k

U = - k U + k = 0

The above equation tells us that the total change in potential energy plus the total

change in kinetic energy is zero, if only conservative forces are acting on the system.

(k+U) = 0 or E = 0 where E = k + U

When only conservative forces act, the change in total mechanical energy of a system

is zero. i.e if only conservative forces perform work on and within a system, the total

mechanical energy of the system is conserved.

kf+ U f- (ki + U i) = 0

kf+ U f = ki + U i

Q E = 0, integrating both sides E = constan t.

Illustra tion 8 :

A projectile is fired from the top of a 40m. high cliff with an initial speed of 50 m/ s at an

un know n angle. Find its speed wh en it hits the ground .

Solution :

Taking ground as the reference level we can conserve the

mechanical energy between the points A and B

(K + U) = 0 Ki + U i = Kf+ U f

'v

H

A

v

B


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2

1mv2 + mgH =

2

1mv' 2 + 0

2

1(50)2 + 40 x 10 =

2

1v' 2

(1250 + 400) x 2 = v' 2

v' 2 = 3300

v' ~ 58 m/ s

POWER

Power is defined as the rate at wh ich work is done. If an amou nt of workW is done in

a time interval t, then average pow er is defined to be

Pav =t

W

The S.I. unit of pow er is J/ S or watt (W). Thus 1 W = 1 J/ S

The instantaneous p ower is the limiting value of Pav as t 0 that is P =dt

dW

Instantaneous p ower m ay also be written as P =

= v.Fdt

dWSince work and

energy are closely related, a more general definition of power is the rate of energy

transfer from one body to another, or the rate at wh ich energy is transformed from one

form to another , i.e. P =dt

dE.

Illustration 9 :

A car of mass 500 kg moving with a speed 36km/ hr in a straight road unid irectionally doubles

its speed in 1 minute. Find the average power delivered by the engine.

Solution :

Its initial speed V1 = 10 m/ s then V2 = 20 m/ s

k =2

1m

21

22

2

1mvv

Power d elivered by the engine


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P =

( )t

vvm

t

K

=

21

22

2

1

= ( )60

10205002

1 22

x

= 1250 W.

MOTION IN A VERTICAL CIRCLE :

A particle of mass 'm' is attached to a light and inextensible

string. The other end of the sting is fixed at O and the particle moves

in vertical circle of radius 'r' equal to the length of the string as shown

in the fig. At the point P, net rad ial force on the particle is T-mg cos.

T - mg cos =r

mv2

T = mg cos +r

mv2

The particle will complete the circle if the string does not slack even at the highest point

( = ). Thus, tension in the string shou ld be greater than or equal to zero (T > 0) at = for

critical situat ion T = 0 and =

mg =R

mv min2

2min

v = gR

minv = gR

Now conserving energy between the lowest and th e highest point

2

1 ( )Rmgmvmu

minmin2

2

1 22 +=

gRgRgRumin 542 =+=

gRumin 5=

If gRumin 5 the particle will comp lete the circle. At u = gR5 , velocity at h ighest

point is v = gR and tension in the string is zero.

O

T Pcosmg

sinmg


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If u < gR5 , the tension in the string become zero before reaching the highest point and at

that point the particle will leave the circular path. After leaving the circle the part icle will follow a

parabolic path.

Above conditions are ap plicable even if a particle moves inside a smooth spherical shell

of rad ius R. The only d ifference is that the tension is replaced by the norm al reaction N.

Illustra tion 10 :

A heavy particle hanging from a fixed point by a light inextensible string of length l is

projected horizontally w ith speed lg . Find the speed of the particle and the inclination of

the string to th e vertical at the instant of the m otion when the tension in the string is equal to

the weight of the particle.

Solution :

Let T = mg a t an angle as shown in figure

h = l (1 - cos)

Conserving m echanical energy betw een

A and B2

1mu2 =

2

1mv2 + mgh

u 2 = v 2+ 2gh v2 = u 2 - 2gh . (i)

T - mg cos =l

2mv T= mg cos +

l

2mv

mg = m g cos +l

2mv

v2 = g l (1- cos) . (ii)

From (i) and (ii) u 2 - 2gl (1 - cos) = g l (1 - cos)

cos =3

2 = cos-1

3

2

pu tting the value of cos in equ ation (ii)

v2 = g l

3

21 =

3

lg v =

3

lg

A lgu =

cosmg

sinmg

B

h


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Equ ilibrium : As we have stud ied earlier a body is said to be in translational

equ ilibrium if net force acting on th e body is Zero.

Fnet = 0

If the forces are Conservative F = -dr

dU

0dr

dU=

At Equilibrium slope of U and r graph is Zero (or) Potential energy either m aximu m

or minimu m or constant at that position.

At th e stable equilibrium position P.E is minimum

At the unstable equ ilibrium position P.E is maximum

Illustration 11:the P.E of a Conservative system is given as U = 10 + (x-2)2. Find the

equilibrium position and d iscuss type of equilibrium .

Solution: For Equilibrium F = 0

F = - 0)2x(2dx

dU==

x = 2

an d 0dx

Ud2

2

<

it is Stable equilibrium p osition at x= 2 and P.E at that position is 20 un its.

* * * * *


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WORKED OUT OBJ ECTIVE PROBLEMS

EXAMPLE : 01

A par ticle moves w ith a velocity 5i - 3 j + 6 k m/s un der the in fluence of a constant force

F = ( )kji 201010 ++ N. The instantaneous pow er app lied to the particle is

A) 200 J/S B) 40 J/S C) 140 J/S D) 170 J/S

Solution :

P =F .

V = (5 i - j3 +6 k) . (10 i + 10 j+20 k)

= 50 - 30 + 120 = 140 J/ S

EXAMPLE : 02

A 15 gm ball is shot from a spring gun w hose spring has a force constant of 600 N/m. The

spring is comp ressed by 5 cm. The greatest possible horizontal range of the ball for this

comp ression is

(g = 10 m/s2)

A) 6.0 m B) 12.0 m C) 10.0 m D) 8.0 m

Solution :

R ma x =g

u2=

mgmu

2

2

1 2=

mg

kx

mgkx

22 2

2

1=

= 22 kx

2

1mu

2

1Q

=( )

mx.

.10

100150

0506002

= .

[ Note : The actual value of 'u ' will be less than the calculated value as some part of 1/ 2kx2 is

used up in doing work against gravity wh en the spring regains its length]

EXAMPLE : 03

Force acting on a particle is (2 i + 3 j ) N. work d one by this force is zero, wh en a particle is

moved on the line 3y + kx = 5 Here value of k is

A) 3 B) 2 C) 1 D) 4

Solution :

Force is pa rallel to the line y = 3/ 2 x + c


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and the given line can be written as y =3

5

3+ x

k

as the work d one is zero force is perpendicular to the d isplacement

323 k = - 1

k = 2

EXAMPLE : 04

Power supplied to a particle of mass 2 kg varies with time as p =2

3 2twatt. Here 't' is in

second. If velocity of part icle at t = 0 is v = 0. The velocity of part icle at time t = 2 second

will be

A) 1 m/s B) 4 m/s C) 2 m/s D) 2 2 m /s

Solution :

kf- ki = 2

0

dtP 2

1mv2 =

2

02

3t2 dt v2 =

2

0

3

2

t

Q m = 2 kg v = 2 m/ s

EXAMPLE : 05

A particle of mass 'm ' is projected w ith velocity 'u' at an angle with horizontal. Durin g the

period when the particle descends from highest point to the position where its velocity

vector makes an angle /2 with hor izon ta l, work don e b y the grav ity force i s

A) 1/2 m u 2 tan 2 /2 B) 1/2 mu 2 tan 2

C) 1/2 mu 2 cos 2 tan 2/2 D) 1/2 mu 2 cos2/2 sin 2

Solution :

As horizontal component of velocity does not change v cos / 2 = ucos

v =

2

cos

cosu

Wgravity = K =2

1mv2 -

2

1m (u cos)2

=2

1mu2 cos2 tan2

2

2/

cosu

V

u


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EXAMPLE : 06

A body of mass 1 kg throw n up ward s with a velocity of 10 m/s comes to rest (momen tarily)

after moving up 4 m. The work d one by air drag in this process is (g = 10 m/s2)

A) 10 J B) - 10 J C) 40 J D) 50 J

Solution :

From work energy theorem Wgr + W air drag = k

- mgh + Wair drag = 0 -2

1mu 2

Wair drag = mgh - 21

mu2

= (40 - 50) J = - 10 J

EXAMPLE : 07

The p otential energy of particle of mass 'm' is given by U =2

1kx2 for x < 0 and U = 0 for x >

0. If total mechanical energy of the particle is E. Then its speed a t x =k

E2is

A) zero B)M

E2C)

m

ED)

m

E

2

Solution :

Poten tial energy of par ticle at x =k

E2is zero K.E = E

2

1mv2 = E or v =

m

E2

EXAMPLE : 08

A block is suspended by an ideal spring of force constant k. If the block is pu lled d own b y

app lying a constant Force 'F' and if maximum displacemen t of block from its initial position

of rest is then

A)K

F< K2). When they ar e stretched by

the sam e force :

A) no work is don e in case of both the springs B) equal work is done in case of both the

springs

C) more work is done in case of second spr ing D) m ore work is d one in case of first

spring

18. The kinetic energy K of a particle moving in a straight line depends up on the d istance s

as K = as2 wh ere a is a constan t. The force acting on the particle is

A) 2as B) 2mas C) 2a D) 2as

19. A par ticle moves in a straight line with a retardation proportional to its displacement. Its

loss of kinetic energy for any d isplacement x is proportional to

A) x B) x2 C) ln x D) ex

20. A pa rticle falls from rest un der gravity. Its po tential energy (PE) with respect to the

groun d and its kinetic energy (KE) are plotted ag ainst time (t). Choose the correct grap h.


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A) B) C)

D)

21. Choose the wron g option

A) If conservative forces are d oing negative w ork then potential energy w ill increase and

kinetic energy w ill decrease.

B) If kinetic energy is constant it means w ork done by conservative forces is zero.

C) for change in potential energy only conservative forces are responsible, but for

change in kinetic energy other than conservative forces are responsible

D) all of the above are wrong

22. Instan taneou s power of a constant force acting on a par ticle mov ing in a straigh t line

un der the action of this force :

A) is constant B) increases linear ly with time

C) decreases linearly with time D) either increases or decreases linearly with time.

23. Sup pose y represents the work done and x the pow er, then dimensions of2

2

x

ydwill be :

A) 421 TLM B) 232 TLM C) 442 TLM D) 63TML

24. Choose the correct statement Work done by a variable force

A) Is defined as S.F B) Is independent of

path

C) Is always dep endent on the initial and final positions D) Non e of these

25. Iden tify the correct statemen t for a non -conservative force

A) A force wh ich is not conservative is called a non-conservative force

B) The w ork done by this force depends on the path followed

C) The word don e by this force along a closed p ath is zero

D) The work done by this force is always negative


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26. The figu re show s a plot of po tential energy function, u(x) = kx2 where x is the

displacement and k is a constant. Identify the correct conservative force

function F(x)

27. A plot of velocity versus time is show n in figu re. A single force acts on the body. Find

correct statemen t

A) In m oving from C to D, work done by the force on th e body is positive

B) In m oving from B to C, work don e by the force on the bod y is positive

C) In moving from A to B, the body does work on the system and is negative

D) In m oving from O to A, work don e by the body an d is negative

28. The force acting on a body moving along x-axis varies with the position of the particle

as shown in the figure. The body is in stable equilibrium at

A) x = x1 B) x = x2

C) both x1 and x2 D) neither x1 and x2

29. Displacement time graph of a par ticle moving in a straight line is as shown in

figu re. Select the correct alterna tive(s).

A) Work d one by a ll the forces in region OA and BC is positive

B) Work d one by a ll the forces in region AB is zero

C) Work done by all the forces in region BC is negative

D) Work don e by all the forces in region OA is negat ive

KEY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

B D C A B C C D C D AD BD AB B A

16 17 18 19 20 21 22 23 24 25 26 27 28 29

B C A B B D B A C B B A B B


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LEVEL - II

1. A particle of mass m is moving in a circular path of rad ius r un der the influence of

centripetal force F C/ r2. The total energy of the part icle is

a)r2

C b)

r2

Cc) C x 2r d) Zero

Sol: Fcentipetal F =2

2

r

C

r

mv= ; v =

r

CdrrCFdr 2 ==

; E1 = EK + v = C/ 2r C/ r

= -C/ 2r

2. Water from a stream is falling on the blades of a turbine at the rate of 100kg/ sec. If

the height of the stream is 100m then the power d elivered to th e turbine is

a) 100 kw b) 100 w c) 10 kw d) 1 kw

Sol: P = w/ 1 = (m/ g) gh = 100 x 10 x 100 = 105w

3. A body is being moved along a straight line by a machine delivering a constant

pow er. The distance covered by the body in time t is proportional to

a) 1 b) t 3/ 2 c) t3/ 4 d) t2

Sol: P = Fv = constant or m a . at = constant or a2t = constant

QS = 1/ 2at2 or S at2 But a 1/ t

S t2/ t or S t3/ 2

4. A ball is drop ped from a height of 10m. If 40% of its energy is lost un collision with

the earth then after collision th e ball will rebound to a height of

a) 10m b) 8m c) 4m d) 6m

Sol: 2

2

1

2

1

h

10

60

100or

h

h

u

u==

5. A particle moves under the influen ce of a force F = CX from X = 0 to X = X1. The

work done in this process w ill be

a)2

CX21 b) 21CX c)

31CX d) 0

Sol: W = 22x

0

2x

1x

Cx2

1dxcxFdx ==


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6. A un iform chain of mass M and length L lies on a horizontal table such that one

third of its length han gs from the edge of the table. The work d one is pu lling the

hanging par t on the table will be

a)3

MgLb) MgL c)

9

MgLd)

18

MgL

Sol: W = M/ 3 . g . 1/ 6

7. A body of mass 2kg moves un der the influence of a force. Its position x changes

with time according to the relation x = t3/ 3 where x is in meter and t in seconds. The

work done by this force in first two second s will be

a) 1600 Jou le b) 160 Jou le c) 16 Jou le d) 1.6 Jou le

Sol: W = mv22 mv12

8. A man and a child are holding a uniform rod of length L in the horizontal direction

in such a way that one fourth w eight is supp orted by the child. If the child is at one

end of the rod th en the d istance of man from another end will be

a) 3L/ 4 b) L/ 4 c) L/ 3 d) 2L/ 3

Sol:

x2

L

4

w3

9. An electric motor p rod uces a tension of 4500N in a load lifting cable and rolls it at

the rate of 2m/ s. The pow er of the motor is

a) 9 kw b) 15 kw c) 225 kw d) 9 x 103 HP

Sol: P = Fv = 4500 x 2 = 9 kw

10. A body of mass m is accelerated to velocity v in time et1. The work done by the

force as a fun ction of time t will be

a)2

22

e2

tmvb) 2

2

tt

mv

2

1

c) 2tt2

mvd)

t2

mvt 2

Sol: Acceleration produ ced in a bod y a =1t

v; W =

2

1ma2t2 = 2

21

2

tt

mv

2

1

11. A motor of 100 HP is moving w ith a constant velocity of 72 km/ hour . The forward

force exerted by the engine of the car is


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25

a) 3.73 x 103 N b) 3.72 x 102 N c) 3.73 x 101 N d) None of

the above

Sol: F = P/ v

12. The kinetic energy of a man is half the kinetic energy of a boy of half of his mass. If

the man increases his speed by 1m/ s, then h is kinetic energy becomes equal to tha t

of the boy. The ratio of the velocity of the boy and that th e man is

a) 2/ 1 b) 1/ 2 c) 3/ 4 d) 4/ 3

Sol: According to question

= 22 U

2

M

2

1x

2

1Mv

2

1

13. A bomb of mass 9 kg explod es into 2 pieces of 3kg an d 6kg. The velocity of 3 kg

piece is 16 m/ s. The kinetic energy of 6kg piece is

a) 768 Jou le b) 786 Jou le c) 192 Jou le d) 687 Jou le

Sol: m1v1 = m 2v2;2222K

vm2

1E =

14. The increase in the potential energy of a body of mass m, when it is carried from the

surface of earth up to a height equ al to the radius of earth Re, will be

a) mgRe b) mgRe/ 2 c) mgRe/ 4 d) 2mgRe

Sol:2

mgRR2

GMm =

15. A person of mass 60kg carries a 15 kg body on the top of a build ing 10m high in 3

minutes. His efficiency is

a) 40% b) 30% c) 20% d) 10%

Sol: M = 100xmM

m

+

16. A force F = (3x2 + 2x 7)N acts on a 2 kg body as a result of which the body gets

displaced form x = 0 to x = 5m. The work done by the force will be

a) 35 Jou le b) 70 Jou le c) 115 Jou le d) 270 Jou le

Sol: W = ( )dx7x2x3Fdxs

0

22x

1x

+=

17. A 50 gm bullet moving w ith a velocity of 10 m/ s gets embedded into a 950 gm

stationary body. The loss in kinetic energy of the system will be


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26

a) 5% b) 50% c) 100% d) 95%

Sol: 100xmM

m100x

E

E

21

2

+=

18. A crane lifts 300 kg weight from earths surface upto a height of 2m in 3 seconds.

The average power generated by it will be

a) 1960 watt b) 2205 watt c) 4410 watt d) 0 watt

Sol: P = w/ t = mgh/ t

19. A block of mass 16kg is moving on a frictionless horizontal surface with velocity

4m/ s and comes to rest after pressing a spring. If the force constant of the spring is

100 N / m then the comp ression in the spring will be

a) 3.2 m b) 1.6 m c) 0.6 m d) 6.1 m

Sol: m v2 = kx2

20. The relation betw een time and disp lacement of a p article moving un der the

influence of a force F is t = x +3 wh ere x is in meter and t in second . The

displacement of the p article when its velocity is zero will be

a) 1 m b) 0 m c) 3 m d) 2 m

Sol: t = x + 3 or x = (t 3)2; v = dx/ dt

21. A 0 kg satellite completes one revolution arou nd the earth at a height of 100 km in

108 minu tes. The work done by the gravitational force of earth w ill be

a) 108 x 100 x 10 Jou le b)100

10x108Jou le c) 0 Jou le d)

108

10x100

Joule

Sol: W = Fd cos = Fd cos 900 = 0

22. A particle moves in a potential region given by u = 8x2 4x + 400 Jou le. Its state of

equilibrium will be

a) x = 25 m b) x = 0.25 m c) x = 0.025 m d) x = 2.5 m

Sol: F = - du/ dx

23. Two men with weights in the ratio 5 : 3 run up a stair case in time in the ratio 11 : 9.

The ratio of pow er of first to that of second is


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a) 15/ 11 b) 11/ 15 c) 11/ 9 d) 9/ 11

Sol: P =t

w

t

wh

t

mgh=

24. A moving particle of mass m collides head on with another stationary particle of

mass 2m. What fraction of its initial kinetic energy will m lose after the collision?

a) 9/ 8 b) 8/ 9 c) 19/ 18 d) 18/ 19

Sol: mu + 2m x 0 = (m + 2m)v;9

8

E

9

EE

E

EE

iK

iK

iK

1K

FK1K =

=

25. The potential energy fun ction of a d iatomic molecule is given as u (r) =612 r

b

r

a ,

wh ere a and b are positive constants and r is inter atomic distance. The equilibrium

between two atoms is

a)6/1

a

b

b)6/1

b

a

c)6/1

a2

b

d)6/1

b

a2

Sol: 0r

b6

r

a12

dr

du713

=+=

26. A pu mp pu lls 1000 kg water per m inute from a 15 m deep w ell and provides 4 m/ s

velocity to it. The pow er of pu mp is (g = 10 m/ s2)

a) 2.6 kw b) 2.6 w c) 0.6 w d) 0.6 kw

Sol:t

mv2/1mgh

t

wP

2+==

27. A body w eighing 80N is moved up a slope of angle 600 with the horizontal through

a disp lacemen t of 1m. The energy loss due to friction is 20%. The energy gained by

the body will be

a) 332 J b) 64 J c) 340 J d) 80 J

Sol: W = mg sin d

28. For the p ath PQR in a conservator force field (figure) amou nts w ork d one in

carrying a body from P to Q and from Q to R are 5 Joule and 2 Joule respectively.

The work d one in carrying the body from P to R will be

a) 7 Joule

b) 3 Joule


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28

c) 21 Joule

d) Zero

Sol: WPR = W PQ + W QR

29. Two p articles each of m ass m and traveling w ith velocities u1 and u2 collide

perfectly inelastically. The loss of energy w ill be

a) m(u 1 u2)2 b) m(u 1 u2)2 c) m(u1 u 2)2 d) 2m(u1

u2)2

Sol: E = ( )22121

21 uumM

mm

2

1

+

30. Two protons are situated at a distance of 100 fermi from each other . The potential

energy of this system will be in ev

a) 44 b) 1.44 x 103 c) 1.44 x 102 d) 1.44 x 104

Sol: U = kq2/ r

31. In order to reduce the kinetic energy of a body to half its initial value, its speed will

have to be changed by the following factor, of its initial speed

a) 1/ 2 times b) 2 times c) 1/ 2 times d) 2 times

Sol: E = mv2; v = F

32. A body of mass M and moving with velocity u m akes a head on elastic collision with

another stationary body of m. If A = m/ M, then the ratio (f) of the loss of energy of

M to its initial energy will be

a) f = A(A + 1)2 b) f =( )21A

A

+c) f =

( )21A

uA

+d) f =

( )21A

A4

+

Sol: f =( ) ( )22 A1

A4

mM

Mm4

+=

+

33. Two masses m1 = 2kg and m 2 = 5kg are moving on a frictionless surface with

velocities 10 m/ s and 3 m/ s respectively. m2 is ahead of m 1. An ideal spring of

spring constant k = 1120 N/ m is attached on th e backside of m2. The maximu m

compression of the spr ing w ill be, if on collision the two bodies stick together.


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29

a) 0.51 m b) 0.062 m

c) 0.25 m d) 0.72 m

Sol: E = ( ) 222121

21 kx

2

1uu

mm

mm

2

1=

+

34. A body a t rest explod es all of a sud den in three equal parts. The moments of two

parts are Pi and 2Pj and their kinetic energies are k1 and k2. If 3P and k3 are the

momentum and kinetic energy respectively of the third p art then the ratio k2/ k3 will

be

a) 2/ 5 b) 3/ 5 c) 4/ 5 d) 1/ 5

Sol: Conceptual

35. A block falls down from a table 0.5m h igh. It falls on an ideal vertical spring of

constant 4 x 102 N/ m. Initially the spring is 25 cm long and its length becomes 10

cm after compression. The mass of the block is (g = 10m/ s2)

a) 0.5 kg b) 2 kg c) 1.2 kg d) 0.9 kg

Sol: mgh = kx2

36. The mass of a bucket full of water is 15 kg. It is being pulled u p from a 15m deep

well. Due to a hole in the bucket 6 kg water flows out of the bucket. The work done

in drawing th e bucket out of the well will be

a) 900 joule b) 1500 joule c) 1800 joule d) 2100 joule

Sol: W = mgh = kg122

915=

+

37. A spring of force constant k is first stretched by a lens x and then again by a further

length x. The work d one in the first case is w1 and in the second case w 2, then

a) w 2 = w 1 b) w 2 = 2w 1 c) w 2 = 3w 1 d) w 2 = 4w 1

Sol: w1 = kx2, w 3 = k(2x2)

38. A 2k body is projected, at an angle of 300 with the horizontal, with a velocity of

10m/ s. The kinetic energy of the body after 1 second will be

a) 10 jou le b) 50 joule c) 100 joule d) 200 jou le

Sol: v = + sinugt2tgu 222


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30

39. A 10 kg block is pu lled in the vertical plane along a frictionless surface in the form of

an arc of a circle of rad ius 10m. The app lied force is of 200N as shown in the figure.

If the block started from rest to A, the velocity at B would be

a) 1.732 m/ s b) 17.32 m/ s

c) 173.2 m/ s d) none of these

Sol: mx2 = 200 cos 300 x 35

40. A block of mass m is pushed towards a movable wedge of mass m and height h

with a velocity u. All sur faces are smooth. When the block collides with the wed ge,

the velocity of centre of mass of block wedge system w ill be

a) u b) +1u

c) u(1 + ) d) 0

Sol: mu = (m + m)v cm

41. In the above problem, the minimu m value of u for wh ich th e block will reach the top

of the wed ge, will be

a)

+1

1gh2 b) gh2 c) gh2 d)

1

1gh2

Sol: m u2 = mgh + (m + m)V2cm

42. A liquid in a U tube is changed from position (a) to position (b) with the help of a

pu mp . The density of liquid is d and area of cross section of the tube is a. The work

done in p um ping the liquid w ill be

a) dgha

b) dgh2a

c) 2gdh2a

d) 4dgh2a

Sol: W = 2ahdgh 2(ahd g h/ 2) = dgh 2a


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31

43. The human heart discharges 75cc of block through the arteries at each beat against

an average pressu re of 10cm of mercury. The pu lse frequen cy of the hear t is 72 per

minu te. The rate of working of heart is

a) 2.35 w b) 3.29 w c) 1.19 w d) 9.11 w

Sol: P = hdgdt

dv

44. A block of mass 1kg is pulled up on an incline of angle 300 with the horizontal. The

block moves with an acceleration of 1 m/ s2. The pow er delivered by the pu lling

force at t = 4s will be

a) 12 w b) 36 w c) 24 w d) 48 w

Sol: F mg sin = ma or F = mg sin + m a

45. A par ticle of mass m is moving in a circular p ath of constant r ad ius r. The

centripetal acceleration of the particle (ac) is varying with time t according to

following relation ac = k2n2 wh ere k is a constant. The pow er delivered to the

particle by the forces acting on it will be

a) mk2 r2 t2 b) m 2k2 r2t2 c) m2k2 rt d) mk2r2t

Sol: ac = v 2/ r = k2n2; w = mv 2/ 2 mv12 = m k2r2t2 = 0; P = dw/ d t

46. A block of mass 2kg is released from A on a track that is a on quadrant of a circle of

rad ius 1m. It slides dow n the track and reaches B with a speed of 4m/ s and finally

stops at C at a d istance of 3m from B. The w ork d one against the force of friction is

a) 2 jou le b) 5 joule

c) 10 joule d) 20 jou le

Sol: W =

+

2C

2B

2B mv

2

1mv

2

1mv

2

1mgh

47. A man pu lls a bucket full of water from a h metre deep well. If the mass of the rope

is m and mass of bucket full of water is M, then the work done by the man is

a) ghm2

M

+ b) gh

2

mM

+

c) gh2

mM

+ d) (M + m)

gh

Sol: wgh2

mM =

+


32/50

32

48. A particle has shifted along some trajectory in the x y plane from point j2ir1 = to

another point j3i2r2 += . During that time, the part icle experiences the action of two

forcesk2j7i2Fandj4i3F

2i +=+= . The work d one by the forces on the particlewill be

a) 5 joule b) -5 joule c) 10 joule d) -10 joule

Sol: 21 FFF +=

49. A 2kg body is dropped from height of 1m on to a spring of spring

constant 800 kg/ m as shown in the figure. A frictional force

equivalent to 0.4 kg wt acts on the bod y. The speed of the body just

before striking the spring will be

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

c) 3 m/ s d) 4 m/ s

Sol: mgh = mv2 + Ffr h

50. A shell is fired from a cannon with a velocity v and at an angle from the horizontal

to hit a target at a horizontal distance R. It splits in tw o equal parts at the highest

point of its path . One part refracts its path and reaches back up to the cannon . The

velocity of the second p art just after the explosion will be

a) 3/ 2 v cos b) 2 v cos c) 3 v cos d) 3/ 2 v

cos

Sol: mv cos = m/ 2 v cos + m/ 2 v

51. A block of mass 10 kg m oving on a smooth surface with a speed of 30 m/ s bursts

into two equal parts. Both parts continu e to move in the seme direction. If one of

the parts moves at 40 m/ s, the energy prod uce in the pr ocess is

a) 200 J b) 500 J c) 700 J d) J

Sol: mv = m1 v1 + m 2 v2; E = m1v12 + m2 v22 mv2

52. Two identical 5 kg blocks are moving w ith same speed of 2 m/ s toward s each other

along a frictionless hor izontal surface. The two blocks collide, stick together and

come to rest. The work d one by the external forces is


33/50

33

a) 0 b) 10 J c) 20 J d) none of

these

Sol: As Fext = 0; Wext = 0S.F ext =

53. In the above pr oblem, the work d one by the inertial forces is

a) 0 b) 10 J c) 20 J d) none of

these

Sol: Wint = mv2 + mv2

54. The force-displacement curve for a body moving on a smooth surface

under the influence of foce F acting along the direction of

d isplacemen t s has been show n in fig. If the initial kinetic energy of the

body is 2.5J. its kinetic energy at s = 6m is

A) 7J B) 4.5J C) 2.25J D) 9J

55. A bullet, moving with a speed of 150m/ s, strikes a wood en plank. After passing through

the p lank its speed becomes 125m/ s. Another bullet of the sam e mass and size strikes

the plank with a speed of 90m/ s. It speed after passing throu gh the plank wou ld be

A) 25m/ s B) 35m/ s C) 50m/ s D) 70m/ s

56. A man of mass 60kg climbs a staircase inclined at 450 and having 10steps. Each step is

20cm high. He takes 2 seconds for the first five steps and 3 seconds for the remaining

five steps. The average power of the man is

A) 245W B) 245 2 W C) 235 2 W D) 235W

57. The po tential energy of a particle moving in x-y plane is given by U = x2 + 2y. The force

acting on the p article at (2, 1) is

A) 6N B) 20 N C) 12 N D) 0

58. Water is flowing in a river at 20m/ s. The river is 50m wide and has an average depth of

5m. The p ower available from the current in the river isA) 0.5MW B) 1.0MW C) 1.5MW D) 2.0MW

59. A 5kg brick of d imensions 20cm x 10cm x 8cm is lying on the largest base. It is now

mad e to stand with length vertical. If g = 10m/ s2, then the amoun t of work d one is

A) 3J B) 5J C) 7J D) 9J


34/50

34

60. The displacement x of a particle mov ing in one dimension, under the action of a constant

force is related to time t by th e equa tion t = x +3, where x is in metres and t in seconds.

The work d one by the force in first 6 seconds is

A) 9J B) 6J C) 0J D) 3J

61. A body of mass m was slowly pu lled up the hill by a force F wh ich at

each point w as d irected along the tan gent of the trajectory. All surfaces

are smooth. Find the work performed by this force

A) mg l B) -mg l

C) mgh D) zero

62. A rope ladder w ith a length l carrying a man of mass m at its end, is attached to the

basket of a balloon of mass M. The entire system is in equilibrium in air. As the man

climbs up the ladder into the balloon, the balloon descends by height h. Then the

potential energy of man

A) increases by mg l B) increases by m g (l -h)

C) increases by mgh D) increases by mg (2 l -h)

63. Two springs s1 and s2 have negligible masses and th e spr ing constant of s1 is one-third

that of s2. When a block is hung from the springs as shown, the springs came to the

equilibrium again. The ratio of work done is stretching s 1 to s2 is

A) 1/ 9

B) 1/ 3

C) 1

D) 3

64. A ligh t spring of length l and spring constant 'k' it is placed vertically. A small ball of

mass m falls from a height h as measured from the bottom of the spring. The ball

attaining to maximum velocity when the height of the ball from the bottom of the springis

A) mg/ k B) l-mg/ k C) l + mg/ k D) l - k / mg

65. A block of mass 1kg is perm anently attached with a spring of spring constant k =

100N/ m. The spring is comp ressed 0.20m and placed on a horizontal smooth su rface.

When the block is released, it moves to a point 0.4m beyond th e point when the spring is

at its natura l length. The work d one by the sp ring in changing from compressed state to

the stretched state is


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35

A) 10J B) -6J C) -8J D) 18J

66. A chain of length l and m ass m lies on the surface of a smooth sphere of rad ius R with

one end tied on the top of the sphere. Ifl = R/ 2, then the potential energy of the chain

with reference level at the centre of sphere is give by

A) m R g B) 2m R g C) 2/ m R g D) 1/ m R g

67. If the force acting on a pa rticle is given by F = 2i + xyj + xz2k, how mu ch work is done

when the p article m oves p ara llel to Z-axis from the p oint (2, 3, 1) to (2, 3, 4) ?

A) 42J B) 48J C) 84J D) 36J

68. A un iform chain of length ' l ' and m ass m is placed on a sm ooth table with one-fourth of

its length hanging over the edge. The work that has to be done to pull the whole chain

back onto the tab le is

A)4

1mgl B)

8

1mgl C)

16

1mgl D)

32

1m gl

69. A spr ing, which is initially in its un stretched condition, is first stretched by a length x

and then again by a further length x. The work done in the first case is W 1 and in the

second case is W2

A) W 2 = W1 B) W2 = 2W1 C) W 2 = 3W1 D) W 2 = 4W1

70. A pa rticle of mass m is fixed to one end of a ligh t rigid rod of length ' l ' and rotated in a

vertical circular path about its other end. The minimum speed of the particle at its

highest point must be

1) zero B) lg C) lg5.1 D) lg2

71. A force F acting on a body depend s on its displacement x as F xn. The power d elivered

by F w ill be indep endent of x if n is

A) 1/ 3 B) -1/ 3 C) 1/ 2 D) -1/ 2

72. A particle is mov ing in a conservative force field from point A to B. UA and UB are the

potential energies of the particle at points A and B and W c is the work done in the

process of taking the pa rticle from A to B.

A) W c = UB - UA B) Wc = UA - UB C) U A > U B D) U B > UA

73. A force is given by Mv2/ r when the m ass m oves with sp eed v in a circle of r ad ius r. The

work done by this force in moving the body over upper half circle along the

circum ference is

A) zero B) C) Mv2 D) Mv2/ 2


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36

74. A mov ing railway comp artm ent has a spr ing of constan t 'k' fixed to its front wall. A boy

in the compartment stretches this spring by distance x and in the mean time the

compar tment m oves by a distance s. The work done by boy w.r.t earth is

A) 2kx21 B)

21 (kx) (s+x) C) kxs

21 D) ( )sxskx

21 ++

75. Force acting on a block moving along x-axis is given by :

F = N2x

4

2

+

The block is disp laced from x=-2m to x=+4m, the work done w ill be

A) positive B) negative

C) zero D) may be positive or negative

75. The system is released from rest with both the spr ings in unstretched positions. Mass of

each block is 1 kg and force constant of each spr ings is 10 N/ m. Extension of horizontal

spring in equilibrium is:

A) 0.2m B) 0.4m C) 0.6m D) 0.8m

77. In a projectile motion, if we plot a grap h between power of the force acting on the

projectile and time then it would be like :

A) B) C) D)

78. A golfer rolls a small ball with speed u along the floor from poin t A. If x = 3R, determine

the required

speed u so that the ball returns to A after rolling on the circular

surface in the vertical plane from B to C and becoming a

projectile at C. (Neglect friction)


37/50

37

A) gR5

2B) gR

2

5

C) gR7

5D) none of these

79. A wind -powered generator converts wind energy into electrical energy. Assume that the

generator converts a fixed fraction of the wind energy intercepted by its blades into

electrical. For w ind speed v, the electrical pow er ou tpu t will be prop ortional to

A) v B)2 C)3 D)4

KEY

54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

A B D B B A C C B D B B C A D

69 70 71 72 73 74 75 76 77 78 79

C A B B A A B B B B C

LEVEL III

1. A block m is pu lled by applying a force F as show n in fig. If the block has

moved up through a d istance 'h', the work done by the force F is

A) 0 2) Fh

C) 2Fh D)2

1Fh

2. A body of mass m, having mom entum p is moving on a rough horizontal surface. If it is

stopped in a distance x, the coefficient of friction between the body and the surface is

given by

A) = p/ (2mg x) B) = p 2 / (2mg s) C) = p 2 / (2g m2s) D) = p 2 (2g m 2s2)

3. A body of mass m moves from rest, along a straight line, by an engine delivering

constan t pow er P. the velocity of the bod y after time t will be

A)m

Pt2B)

m

Pt2C)

m2

PtD)

m2

Pt

4. The spring show n in fig has a force constan t k and the mass of block is m. Initially,

the sp ring is unstretched when the block is released. The maximum elongation of the

spring on the releasing the m ass will be


38/50

38

A)k

mgB)

2

1

k

mgC) 2

k

mgD) 4

k

mg

5. A skier starts from rest at point A and slides dow n the hill,

without turning or braking. The friction coefficient is . Whenhe stops at point B, his horizontal displacement in S. The height

difference h between p oints A and B is

A) h = S/ B) h = S

C) h = S2 D) h = S/2

6. A small block of mass m is kep t on a rou gh inclined surface of inclinat ion fixed in an

elevator. The elevator goes up with a un iform velocity v an d the block d oes not slide on

the w edge. The w ork done by the force of friction on the block in time t will be

A) zero B) mg vt cos2 C) mg vt sin2 D) mg vt sin2

7. A block of mass m starts at rest at height h on a frictionless

inclined p lane. The block slides down the p lane travels a total

distance d across a rough horizontal surface with coefficient of

kinetic friction k and compresses a spring w ith force constant

k, a d istance x before m omentarily coming to r est. The spring then extends and the block

travels back across the rough surface, sliding u p the p lane. The maximum height h' that

the block reaches on its return is

A) h' = h - 2d B) h' = h - 2d -2

1kx2

C) h' = h - 2d + kx2 D) h' = h - 2d - kx2

8. A chain of length 3 l and mass m lies at the top of sm ooth prism such

that its length l is one side and 2 l is on the other side of the vertex.

The angle of prism is 1200 and the prism is not free to move. If the

chain is released. What will be its velocity when the right end of the

chain is just crossing the top -most po int?

A) lg2 B) lg3

2C) lg

3

1D) lg

2

1

9. If a constant pow er P is app lied in a vehicle, then its acceleration increases with time

according to the relation


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39

A) a = tm2

P

B) a = 2/3t

m2

P

C) a = t/1

m2

P

D) a =

mt2

P

10. A body of mass m slides dow nw ard along a p lane inclined at an angle . The coefficient

of friction is . The rate at which kinetic energy plus gravitational potential energy

dissipa tes expressed as a function of time is

A) mtg 2 cos B) mtg 2 cos (sin - cos )

C) mtg 2 sin D) mtg 2 sin (sin - cos )

11. The poten tial energy for a force field F is given by U(x, y) = sin (x + y). The force acting

on th e particle of mass m at (0, / 4) is

A) 1 B) 2 C) 1/ 2 D) 0

12. A un iform rope of length ' l ' and mass m hangs over a horizontal table with two third

part on the table. The coefficient of friction between the table and the chain is . The

work done by the friction d uring the period the chain slips comp letely off the table is

A) 2/ 9 mgl B) 2/ 3 mgl C) 1/ 3 mgl D) 1/ 9 mgl

13. A particle is moving in a force field given by potential U = - (x + y + z) from the point

(1, 1, 1) to (2, 3, 4). The work d one in the process is

A) 3 B) 1.5 C) 6 D) 12

14. A compressed spring of spr ing constant k releases a ball of mass m.

If the height of spring is h and the spring is compressed through a

distance x, the horizontal distance covered by ball to reach grou nd is

A) xmg

khB)

mg

xkh

C) xmg

kh2D)

khx

mg

15. A block of mass m = 2kg is moving with velocity vo

towards a massless unstretched spring of force constant

K = 10 N/ m. Coefficient of friction between th e block

and the ground is = 1/ 5. Find maximu m value of vo so

that after pressing the spring the block does not return back but stops there

permanently.

A) 6 m/ s B) 12m/ s C) 8m/ s D) 10m/ s


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16. Poten tial energy of a particle moving along x-axis un der the action of only conserva tive

forces is given as : U = 10 + 4 sin(4x). Her e U is in Jou le and x in meters. Total

mechanical energy of the pa rticle is 16J. Choose the correct op tion.

A) At x = 1.25m, particle is at equ ilibrium p osition. C) both A and B are correct

B) Maximu m kinetic energy of the particle is 20J D) both A and B are wrong.

17. A system show n in figu re is released from rest. Pu lley and spring is massless

and friction is absent everywhere. The speed of 5 kg block when 2 kg block

leaves the contact with ground is (Take force constant of spring k = 40 N/ m an d

g = 10 m/ s2)

A) 2 m/ s` B) 2 2 m/ s

C) 2m/ s D) 4 2 m/ s

18. Two blocks of masses m 1 = 1 kg and m2 = 2 kg are connected by a

non-deformed light spring. They are lying on a rough horizontal surface. The coefficient

of friction between the blocks and the surface is 0.4 what minimum constant force F has

to be applied in horizontal direction to the block of mass m 1 in order to shift the other

block? (g = 10 m/ s2)

A) 8 N B) 15 N C) 10 N D) 25 N

19. A block of mass m is attached with a massless spr ing of force constan t k.

The block is placed over a rough inclined su rface for which the coefficient of

friction is = . The minimum value of M required to move the block up

the plane is (Neglect mass of string and p ulley and friction in pulley).

A) 3/ 5m B) 4/ 5m C) 6/ 5m D) 3/ 2m

20. A particle of mass m is moving in a circular path of constant rad ius r such that its

centripetal acceleration ac is varying with time t as, ac = k2 r t2 where k is a constant.

What is the pow er delivered to the par ticle by the forces acting on it?

A) 2 pm k2r2t B) mk2r2t C)3

)trmk( s24D) zero

21. A particle, which is constrained to move along the x- axis, is subjected to a force in the

same direction which varies with the distance x of the particle from the origin as

F(x) = -kx + ax2. Here k and a are positive constant. For x 0, the function form of the

potential energy (x) of the particle is


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KEY

1 2 3 4 5 6 7 8 9 10 11

C C A C B C A B D B A

12 13 14 15 16 17 18 19 20 21

A C C D A B A A B

A) B) c) D)


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MULTIPLE ANSWER TYPE QUESTIONS

1. The potential energy U for a force field F is such that U = - kxy, wh ere k is a constan t

A) jkxikyF += B) jkyikxF +=

C) The force F is a conservative force D) The force F is a non-conservative

force

2. A sledge mov ing over a smooth hor izontal surface of ice at a velocity v0 drives out on a

horizontal road and comes to a halt as shown. The

sledge has a length l, mass m and friction between

runn ers and road is

A) No w ork is done by the friction to switch the sledge

from ice to the road

B) A work of2

1 mgl is done against friction while sledge switches completely on to

road

C) The d istance covered by the sledge on th e road is

2g2v20 l

D) Total distance moved by the sledge before stopping is

+ 2g2v20 l

3. A strip of wood of mass M and length l is placed on a smooth horizontal surface. An

insect of mass m starts at one end of the strip and walks to the other end in time t,

moving w ith a constant speed

A) The speed of the insect as seen from the gr ound is UB D) UB > UA

19. At the position of stable equilibrium

A) =dx

dU0 on ly B)

dx

du= 0 and

2

2

dx

Ud> 0 C) 0

dx

dU= an d 0

dx

Ud

2

2

< D) None of

these

20. Choose the correct statement(s) related to the conservative force and poten tial energy.

A) Potential energy d ecrease in the d irection of conservative force

B) Potential energy increase in the d irection of conserva tive force

C) Conservative force does work by lowering its potential energy

D) Conservative force does work by raising its potential energy

KEY

1 2 3 4 5 6 7 8 9 10

AC BCD AC BD AC BC BD AD ABD ABD

11 12 13 14 15 16 17 18 19 20

A BC B ACD ABD ABC ABC BC C BC


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COM PREH ENSIO N TYPE Q UESTION S

Passage I (Q.No: 1 to 7):

The potential energy of two atoms in a diatomic molecule is approximated by U(r)

=612 r

b

r

a , where r is the spacing between atoms and a and b are positive constants.

1. Find the force F(r) on one atom as a function of r:

A) 0 B)713 r

b

r

a12+ C)

612 r

b6

r

a12 D)

713 r

b6

r

a12

2. Which is the most app ropr iate graph U(r) versus r:

3. Which is the most app ropr iate graph F(r) versus r:

4. Find the equilibrium distance between the two atoms:

A) 2a B) 2a/ b C) 2a/ 5b D)

6/1

b

a2

5. From the above conclusion can we pred ict about equilibrium state:

A) the equ ilibrium is stable B) the equilibrium is unstable

C) the equilibrium may be stable D) the equilibrium may be unstable

6. What minimum energy mu st be add ed to the molecule to d issociate it, if the d istance

between th e two atom s is equal to the equilibrium distance found in Q. 4 ?

A) b 2/ a B) 2b2/ a C) b 2/ 4a D) 2a / b

A) B) C) D)

A) B) C) D)


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7. For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is

1.13 x 10-10 m and the d issociation energy is 1.54 x 10-18 J per m olecule. Find the values of

a and b:

A) a = 6.67 x 10-138 J-m 12

b = 2.08 x 10-60 J m6

B) a = 6.41 x 10-78 J-m 6

b = 6.67 x 10-138 J-m12

C) a = 6.67 x 10-138 J-m 12

b = 6.41 x 10-78 J m6

D) a = 0

b = 6.41 x 10-78 J m6

Passage I I (Q.No: 8 to 11) :

A cutting tool und er microprocessor contro l has several forces acting on it. One force is

jxyF 2= , a force in the negative y-direction whose magnitude depend on the

position of the tool. The constan t is = 2.50 N. Consider the d isplacement of the tool

from the origin to the p oint x = 3.00 m, y = 3.00 m.

8. Calculate the work don e on the tool by F if this displacement is along the straight line y

= x that connects these two points ?

A) 2.50 J B) 500 J C) 50.6 J D) 2 J

9. Calculate the work done on the tool by F if the tool is first m oved out along the x-axis to

the point x = 3.00 m, y = 0 and then moved parallel to the y-axis to x = 3.00 m, y = 3.00

m.

A) 67.5 J B) 85 J C) 102 J D) 7.5 J

10. Comp are the work don e by F along these two paths ?

A) Work d one on x-axis is zero

B) Work d one on y-axis is less than on y-axis

C) Work done on x-axis is more than on y-axis but n ot zero

D) Data insufficient

11. What can you pred ict about F ?

A) Force is non -conservative

B) Force is conservative

C) Force is neither conservative nor n on-conservative

D) Data insu fficient to conclude


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Passage I II (Q.N o: 12 to 16):

A 1200 kg car is travelling at 7.5 m/ s in a northerly d irection on an icy road. It crashes

into a 8000 kg tru ck m oving in the sam e d irection as the car w ith a velocity of 3.0 m/ s

before the collision. The speed of the car after the collision is 3.0 m/ s in its original

direction.

12. Which of the following is true regarding the relationship between energy and momentum in the

passage ?

A) The collision is not perfectly elastic, both mom entu m an d energy are not conserved

B) The collision is inelastic, kinetic energy is conserved but m omen tum is not

C) The collision is not perfectly elastic, momen tum is conserved bu t total energy is not

D) The collision is not perfectly elastic, mom entu m is conserved but k inetic energy is not

13. Wha t is the velocity of the truck after the collision ?

A) 7.5 m/ s B) 3.7 m/ s C) 3.0 m/ s D) 1.1 m/ s

14. The car then proceeds to a garage. To get there, the dr iver turn s off onto a smooth road

with a coefficient of friction = = 1/ 4. He then stops for a snack and th en tries to dr ive

off. what is the va lue of frictional force when th e force the car exerts is 300 N ?

A) 0 N B) 100 N C) 300 N D) 4000 N

15. After leaving the garage, the driver of the car follows the same road and eventually has

to go up a h ill. How does the frictional force on the car now comp are to the value when

the car was d riving on level ground ?

A) No change B) It increased C) It decreased

D) The direction of change depends on the angle of elevation

16. If the car is moving up the hill at 5 m/ s and the car is 40m up

the hill as shown in the diagram, how m uch p otential energy

does the car possess at that point ? (g = 9.8 m/ s2).

A) 2.40 x 105 J B) 2.40 x 104 J

C) 4.95 x 105 J D) 4.95 x 104 J

KEY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

D A D D A A C C A A A

* * *


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MULTIPLE MATCHING TYPE Q UESTIO NS

1. Match the following:

List - I List - II

a) Area und er F - S e) Change in KE

b) Work energy theorem f) negative of work d one to gravitat ional force

c) change in PE g) work don e by F

d) conservative force h) dx.F , wh ere F is conservative forcei) gravitationa l force


List - I List - II

a) KE e) depends on frame of reference

b) work d one f) defined for conservative force only

c) PE g) ind epend ent on frame of reference

d) spring PE h) same for either comp ression or elongation for same d istance


List - I List - II

a) stable equ ilibrium e) PE in Max

b) unstable equilibrium f) Fne t = 0

c) 0dx

dF

g) PE is Min

d) 0dx

dF

h) slope of F-x grap h is +ve

4. Match the following:List - I List - II

a) work done by frictional force e) ind epent of path

b) work d one by electrostatic force f) non-conservative

c) work d one by gravitational force for closed loop g) depends on p ath

d) for slowly moving body, w c + w n.c equal to h) define PE

i) zero

KEY

1 2 3 4

a-eg, b-e, c-fh, d-i a-e, b-e, c-ef, d -gh a-fg, b-efh, c-g, d -eh a-fg, b-eh, c-i, d -i

notes on work energy power for iit(mains+ advance)

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