work energy power module for iit main

7/29/2019 Work Energy Power Module for IIT Main

1/46

1

WORK, ENERGY & POWER

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

WORK

When a force is applied 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 displacement

S 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 + Fy j + Fz k and

S = x i + y j + z k then W =

F .

S = Fxx+Fyy+Fzz

Positive and Negative work : The work is said to be positive if the angle is acute ( < 900) and

negative if the angle

is obtuse (

> 900

). If the angle between

F

and

S

is 900

then work done by theforce 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 depends on frame of reference. With change of frame of reference inertial force does not

change while displacement may change, so the work done by a force will be different in different frames.

Illustration 1 :

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

is 6 j m. What is the work done by the forceF ?

Solution :

The work done

F .

x


2/46

2

= ji 25 . j6 = - 12 Joule

Illustration 2 :

A load of mass m = 3000 kg is lifted by a rope with an acceleration a = 2 m/s 2. Find the work done

during the first one and a half seconds from the beginning of motion.

Solution :

The height to which the body is lifted during the first 't' second is h =2

1at2 tension in the rope T =

mg + ma

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

221 at = 3000 (10 + 2)

251221 .xx

= 81 KJ

WORK DONE BY A SPRING FORCE :

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

equilibrium position. If x be the displacement of the free end of the spring from its equilibrium position

then, the magnitude of the spring force is FS = - kx

The negative sign indicates that the force is restoring.

The work done by the spring force for a displacement from xi to xf is given by

Ws = fx

ix

kxdx Ws = 222

1ifxxk

WORK DONE BY FRICTION :

Work done by friction may be zero, positive or negative depending upon the 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 pulled 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

negative work on top block and positive work on the lower block


3/46

3

WORK DONE BY GRAVITY :

Here the force of gravity is Fg = - mg j and the displacement is given by

S = x i + y j + z k

Work done by gravity is Wg = gF .

S = - mg y

y = yf - 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 also changes. For

example, if you calculate work from a non inertial frame work due to pseudo force has to be included.

Again displacement from the inertial frame of reference will be different from ground frame.

CONSERVATIVE AND NON CONSERVATIVE FORCES :

In Conservative force field the work done 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 depends on path followed. Such forces are called non- Conservative

forces. Example is frictional force.


4/46

4

Illustration 3 :

A train is moving with a constant speed "v". A box is pushed by a worker applying a force "F" on the

box in the train slowly by distance "d" on the train for time "t". Find the work done by "F" from the train

frame as well as from the ground frame.

Solution :

As the box is seen from the train frame the displacement 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 displacement on the train is opposite to 'F')

As the box is seen from ground frame,

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

direction of motion of the train )

= d - vt (if the displacement is opposite to

direction of motion of the train)

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

Illustration 4 :

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

friction "" which in turn is placed on a smooth surface. 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 plank frame.

b) The work done by friction on the smaller block in the plank frame.

c) Find the final velocity of the plank

Solution :

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

plank.

gm

mgap

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

m

m 0v


5/46

5

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

V20

Velocity of the plank during 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 done by friction on the block = 18

32

0

g

V.mgcos.S.F = 2

08

3mv

(b) From the Plank frame

Work done by friction on smaller block = -mgl

g2

V0 20

2

mV

g

20

work done by friction from the Plank frame =2

mV20

(c) Final velocity of the block

= Velocity of the plank =2

0V

WORK ENERGY THEOREM :

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

constant force F acts through a displacement x, it does work W = Fx

22if

vv + 2 ax

W =2

22if

m vv =2

1m 2

fv -

2

1m 2

iv

mg

pma

m


6/46

6

The quantity k =2

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

energy posses by the particle by virtue of its motion.

Thus the equation takes the form KKKW if

The work done by a force changes the kinetic energy of the particle. This is called the work -

Energy Theorem.

Illustration 5 :

The velocity of an 800 gm object changes from

0v = 3 i - 4 j tofv = -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 between a 900 kg car and pavement is 0.8. If the car is moving at

25 m/s along level pavement, 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 =?

K.E = work done against friction 2

2

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

s =g

v

2

2

=

10802

25 2

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 much work is done by the push of air on the object ? (g = 10 m/s2)


7/46

7

Solution :

Let upward push of air be F

The resultant downward 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 done by push of air = 75 x 20 = 15 Joule

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 magnitude of work done against the associated conservative force

du = - rd.F

For Example :

(i) If an object is placed at any point in gravitational field work is to be done against gravitationalfield force. The magnitude 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.

ii) The magnitude of work done against the spring force to compress it gives the measure of elastic

potential energy, which is U =2

1k x2

iii) A charged body in any electrostatic field will have electrostatic potential energy. The change in

potential energy of a system associated with conservative internal force as U2-U1= - W= 2

1

F . dr

CONSERVATION OF MECHANICAL ENERGY :

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

work energy theorem

Wnet = k


8/46

8

Where Wnet is the sum of work done by all the forces acting on the mass. If the system is

subjected to only conservative forces then Wnet = 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 + Uf - (ki + Ui) = 0

kf + Uf = ki + Ui

E = 0, integrating both sides E = constant.

Illustration 8 :

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

angle. Find its speed when 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 + Ui = Kf + Uf

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 which work is done. If an amount of work W is done in a time

interval t, then average power is defined to be

'v

H

A

v

B


9/46

9

Pav =t

W

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

The instantaneous power is the limiting value of Pav as t 0 that is P = dtdW

Instantaneous power may also be written as P = v.F

dt

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 which 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 unidirectionally 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 delivered by the engine

P =

t

vvm

t

K

21

22

21

=

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 radial force on the particle is T-mg cos.

T - mg cos =r

mv2

O

TP

cosmg

sinmg


10/46

10

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 should be greater than or equal to zero (T > 0) at = for critical situation

T = 0 and =

mg =R

mv min2

2minv = gR

minv = gR

Now conserving energy between the lowest and the highest point

2

1 Rmgmvmu

minmin2

2

1 22

gRgRgRumin

542

gRumin 5

If gRumin 5 the particle will complete the circle. At u = gR5 , velocity at highest point is

v = gR and tension in the string is zero.

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 particle will follow a parabolic path.

Above conditions are applicable even if a particle moves inside a smooth spherical shell of radius

R. The only difference is that the tension is replaced by the normal reaction N.

Illustration 10 :

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

horizontally with speed g . Find the speed of the particle and the inclination of the string to the

vertical at the instant of the motion when the tension in the string is equal to the weight of the

particle.

Solution :

Let T = mg at an angle as shown in figure

h = l (1 - cos)

Conserving mechanical energy between

A gu

cosmg

sinmg

B

h


11/46

11

A and B2

1mu2 =

2

1mv2 + mgh

u2 = v2+ 2gh v2 = u2 - 2gh . (i)

T - mg cos =

2mv T= mg cos +

2mv

mg = mg cos +

2mv

v2 = gl (1- cos) . (ii)

From (i) and (ii) u2 - 2gl (1 - cos) = gl (1 - cos)

cos =

3

2 = cos-1

3

2

putting the value of cos in equation (ii)

v2 = gl

3

21 =

3

g v =

3

g

Equilibrium : As we have studied earlier a body is said to be in translational equilibrium if net

force acting on the 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 maximum or

minimum or constant at that position.

At the stable equilibrium position P.E is minimum

At the unstable equilibrium 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 discuss type of equilibrium.


12/46

12

Solution: For Equilibrium F = 0

F = - 0)2x(2

dx

dU

x = 2

and 0dx

Ud2

2

it is Stable equilibrium position at x= 2 and P.E at that position is 20 units.

* * * * *


13/46

13

WORKED OUT OBJECTIVE PROBLEMS

EXAMPLE : 01

A particle moves with a velocity 5 i - 3 j + 6 k m/s under the influence of a constant forceF = kji 201010 N. The instantaneous power applied to the particle isA) 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 +20k)

= 50 - 30 + 120 = 140 J/S

EXAMPLE : 02

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

compressed by 5 cm. The greatest possible horizontal range of the ball for this compression is

(g = 10 m/s2)

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

Solution :

R max =g

u2

=

mg

mu2

2

1 2 =mg

kx

mgkx

22 2

2

1

22 kx2

1mu

2

1

=

mx.

.10

100150

050600 2 .

[ 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 when the spring regains its length]

EXAMPLE : 03

Force acting on a particle is (2 i + 3 j ) N. work done by this force is zero, when 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 parallel to the line y = 3/2 x + c

and the given line can be written as y =

3

5

3

xk


14/46

14

as the work done is zero force is perpendicular to the displacement

32

3 k= - 1

k = 2

EXAMPLE : 04

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

32t

watt. Here 't' is in second. If

velocity of particle at t = 0 is v = 0. The velocity of particle 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

1

mv2 =

2

02

3

t2 dt v2 =

2

0

3

2

t

m = 2 kg v = 2 m/s

EXAMPLE : 05

A particle of mass 'm' is projected with velocity 'u' at an angle with horizontal. During the periodwhen the particle descends from highest point to the position where its velocity vector makes an

angle /2 with horizontal, work done by the gravity force isA) 1/2 mu2 tan2 /2 B) 1/2 mu2 tan2C) 1/2 mu2 cos2 tan2/2 D) 1/2 mu2 cos2/2 sin2

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

EXAMPLE : 06

A body of mass 1 kg thrown upwards with a velocity of 10 m/s comes to rest (momentarily) after

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

)

2/

cosu

V

u


15/46

15

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

Solution :

From work energy theorem Wgr + Wair drag = k

- mgh + Wair drag = 0 -2

1mu2

Wair drag = mgh -2

1mu2 = (40 - 50) J = - 10 J

EXAMPLE : 07

The potential 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 at x = k

E2is

A) zero B)M

E2C)

m

ED)

m

E

2

Solution :

Potential energy of particle 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 pulled down by applying

a constant Force 'F' and if maximum displacement of block from its initial position of rest is then

A)K

F< K2). When they are stretched by the

same force :


19/46

19

A) no work is done 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 spring D) more work is done in case of first spring

18. The kinetic energy K of a particle moving in a straight line depends upon the distance s as K = as2

where a is a constant. The force acting on the particle is

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

19. A particle moves in a straight line with a retardation proportional to its displacement. Its loss of

kinetic energy for any displacement x is proportional to

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

20. A particle falls from rest under gravity. Its potential energy (PE) with respect to the ground and

its kinetic energy (KE) are plotted against time (t). Choose the correct graph.

A) B) C) D)

21. Choose the wrong option

A) If conservative forces are doing negative work then potential energy will increase and kinetic

energy will decrease.

B) If kinetic energy is constant it means work 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. Instantaneous power of a constant force acting on a particle moving in a straight line under the

action of this force :

A) is constant B) increases linearly with time

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

23. Suppose y represents the work done and x the power, then dimensions of2

2

dx

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 dependent on the initial and final positions D) None of these

25. Identify the correct statement for a non-conservative force

A) A force which is not conservative is called a non-conservative force


20/46

20

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

C) The word done by this force along a closed path is zero

D) The work done by this force is always negative

26. The figure shows a plot of potential 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 shown in figure. A single force acts on the body. Find correct

statement

A) In moving from C to D, work done by the force on the body is positive

B) In moving from B to C, work done by the force on the body is positive

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

D) In moving from O to A, work done by the body and 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 particle moving in a straight line is as shown in

figure. Select the correct alternative(s).

A) Work done by all the forces in region OA and BC is positive

B) Work done by all the forces in region AB is zero

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

D) Work done by all the forces in region OA is negative

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


21/46

21

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

LEVEL - II

1. A particle of mass m is moving in a circular path of radius r under the influence ofcentripetal force F C/r2. The total energy of the particle 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 delivered to the 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 = 10 5w

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

distance covered by the body in time t is proportional to

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

Sol: P = Fv = constant or ma . at = constant or a2t = constant

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

S t2/ t or S t3/2

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

then after collision the 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 influence of a force F = CX from X = 0 to X = X1. The work done

in this process will be

a)2

CX 21 b) 21CX c)31CX d) 0

Sol: W = 22x

0

2x

1x

Cx2

1dxcxFdx


22/46

22

6. A uniform chain of mass M and length L lies on a horizontal table such that one third of its

length hangs from the edge of the table. The work done is pulling the hanging part on the

table will be

a)3

MgL b) MgL c)9

MgL d)18

MgL

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

7. A body of mass 2kg moves under 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 seconds will be

a) 1600 Joule b) 160 Joule c) 16 Joule d) 1.6 Joule

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 weight is supported by the child. If the child is at one end of the rod

then the distance of man from another end will be

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

Sol:

x

2

L

4

w3

9. An electric motor produces a tension of 4500N in a load lifting cable and rolls it at the rate of

2m/s. The power 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

function of time t will be

a)2

22

e2tmv b) 2

2

ttmv

21 c) 2t

t2mv d)

t2mvt

2

Sol: Acceleration produced in a body a =1t

v; W =

2

1ma2t2 = 2

21

2

tt

mv

2

1

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

exerted by the engine of the car is

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


23/46

23

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 his kinetic energy becomes equal to that of the boy. The

ratio of the velocity of the boy and that the man is

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

Sol: According to question

22 U2

M

2

1x

2

1Mv2

1

13. A bomb of mass 9 kg explodes into 2 pieces of 3kg and 6kg. The velocity of 3 kg piece is 16

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

a) 768 Joule b) 786 Joule c) 192 Joule d) 687 Joule

Sol: m1v1 = m2v2; 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 upto a height equal to the radius of earth Re, will be

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

Sol:2

mgR

R2

GMm

15. A person of mass 60kg carries a 15 kg body on the top of a building 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 Joule b) 70 Joule c) 115 Joule d) 270 Joule

Sol: W =

dx7x2x3Fdx

s

0

22x

1x

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

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

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

Sol: 100xmM

m100x

E

E

21

2


24/46

24

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

compression in the spring will be

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

Sol: mv2 = kx2

20. The relation between time and displacement of a particle moving under the influence of a

force F is t = x +3 where x is in meter and t in second. The displacement of the particle

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 around the earth at a height of 100 km in 108

minutes. The work done by the gravitational force of earth will be

a) 108 x 100 x 10 Joule b)100

10x108Joule c) 0 Joule d)

108

10x100Joule

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

22. A particle moves in a potential region given by u = 8x2 4x + 400 Joule. 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 power of first to that of second is

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


25/46

25

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

8

E

9

EE

E

EE

iK

iK

iK

1K

FK1K

25. The potential energy function of a diatomic molecule is given as u(r) = 612 r

b

r

a

, where 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 pump pulls 1000 kg water per minute from a 15 m deep well and provides 4 m/s velocity

to it. The power of pump 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 weighing 80N is moved up a slope of angle 60 0 with the horizontal through a

displacement 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 path PQR in a conservator force field (figure) amounts work done in carrying a body

from P to Q and from Q to R are 5 Joule and 2 Joule respectively. The work done in carrying

the body from P to R will be

a) 7 Joule

b) 3 Joule

c) 21 Joule

d) Zero

Sol: WPR = WPQ + WQR

29. Two particles each of mass m and traveling with velocities u1 and u2 collide perfectly

inelastically. The loss of energy will be

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


26/46

26

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 makes 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 m2 = 5kg are moving on a frictionless surface with velocities 10m/s and 3 m/s respectively. m2 is ahead of m1. An ideal spring of spring constant k = 1120

N/m is attached on the backside of m2. The maximum compression of the spring will be, if

on collision the two bodies stick together.

a) 0.51 m b) 0.062 m

c) 0.25 m d) 0.72 m

Sol: E = 222121

21 kx2

1uu

mm

mm

2

1

34. A body at rest explodes all of a sudden 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 part then the ratio k2/k3 will be

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

Sol: Conceptual


27/46

27

35. A block falls down from a table 0.5m high. 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 kgSol: mgh = kx2

36. The mass of a bucket full of water is 15 kg. It is being pulled up 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 the

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 done in the first case is w1 and in the second case w2, then

a) w2 = w1 b) w2 = 2w1 c) w2 = 3w1 d) w2 = 4w1

Sol: w1 = kx2, w3 = 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 joule b) 50 joule c) 100 joule d) 200 joule

Sol: v = sinugt2tgu 222

39. A 10 kg block is pulled in the vertical plane along a frictionless surface in

the form of an arc of a circle of radius 10m. The applied 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 surfaces are smooth. When the block collides with the wedge, the velocity of

centre of mass of block wedge system will be

a) u b)1

u


28/46

28

c) u(1 + ) d) 0

Sol: mu = (m + m)v cm

41. In the above problem, the minimum value of u for which the block will reach the top of the

wedge, will be

a)

1

1gh2 b) gh2 c) gh2 d)

1

1gh2

Sol: mu2 = mgh + (m + m)V2cm

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

The density of liquid is d and area of cross section of the tube is a. The work done in

pumping the liquid will be

a) dgha

b) dgh2a

c) 2gdh2a

d) 4dgh2a

Sol: W = 2ahdgh 2(ahdg h/2) = dgh2a

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

average pressure of 10cm of mercury. The pulse frequency of the heart is 72 per minute.

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 30 0 with the horizontal. The block

moves with an acceleration of 1 m/s2. The power delivered by the pulling 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 + ma

45. A particle of mass m is moving in a circular path of constant radius r. The centripetal

acceleration of the particle (ac) is varying with time t according to following relation ac = k2n2

where k is a constant. The power delivered to the particle by the forces acting on it will be

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

Sol: ac = v2/r = k2n2; w = mv 2/2 mv12 = m k2r2t2 = 0; P = dw/dt


29/46

29

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

1m. It slides down the track and reaches B with a speed of 4m/s and finally stops at C at a

distance of 3m from B. The work done against the force of friction is

a) 2 joule b) 5 joulec) 10 joule d) 20 joule

Sol: W =

2C2B

2B mv

2

1mv2

1mv2

1mgh

47. A man pulls 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) gh

2

mM

d) (M + m) gh

Sol: wgh2

mM

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

point j3i2r2 . During that time, the particle experiences the action of two forces

k2j7i2Fandj4i3F 2i . The work done by the forces on the particle will 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 body. 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 two equal parts at the highest point of its path.

One part refracts its path and reaches back upto the cannon. The velocity of the second part

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


30/46

30

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

equal parts. Both parts continue to move in the seme direction. If one of the parts moves at

40 m/s, the energy produce in the process is

a) 200 J b) 500 J c) 700 J d) JSol: mv = m1 v1 + m2 v2; E = m1v12 + m2 v22 mv2

52. Two identical 5 kg blocks are moving with same speed of 2 m/s towards each other along a

frictionless horizontal surface. The two blocks collide, stick together and come to rest. The

work done by the external forces is

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

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

53. In the above problem, the work done 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

displacement s has been shown 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 wooden plank. After passing through the

plank its speed becomes 125m/s. Another bullet of the same mass and size strikes the plank with

a speed of 90m/s. It speed after passing through the plank would 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 potential energy of a particle moving in x-y plane is given by U = x2 + 2y. The force acting on

the particle at (2, 1) is

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

power available from the current in the river is

A) 0.5MW B) 1.0MW C) 1.5MW D) 2.0MW


31/46

31

59. A 5kg brick of dimensions 20cm x 10cm x 8cm is lying on the largest base. It is now made to stand

with length vertical. If g = 10m/s2, then the amount of work done is

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

60. The displacement x of a particle moving in one dimension, under the action of a constant force is

related to time t by the equation t = x +3, where x is in metres and t in seconds. The work done

by the force in first 6 seconds is

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

61. A body of mass m was slowly pulled up the hill by a force F which at

each point was directed along the tangent of the trajectory. All surfaces

are smooth. Find the work performed by this force

A) mg B) -mg

C) mgh D) zero

62. A rope ladder with 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 mg (l -h)

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

63. Two springs s1 and s2 have negligible masses and the spring constant of s1 is one-third

that of s2. When a block is hung from the springs as shown, the springs came to theequilibrium again. The ratio of work done is stretching s1 to s2 is

A) 1/9

B) 1/3

C) 1

D) 3

64. A light 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 spring is

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

65. A block of mass 1kg is permanently attached with a spring of spring constant k = 100N/m. The

spring is compressed 0.20m and placed on a horizontal smooth surface. When the block is

released, it moves to a point 0.4m beyond the point when the spring is at its natural length. The

work done by the spring in changing from compressed state to the stretched state is

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


32/46

32

66. A chain of length l and mass m lies on the surface of a smooth sphere of radius R with one end

tied on the top of the sphere. If = 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 particle is given by F = 2i + xyj + xz2k, how much work is done when the

particle moves parallel to Z-axis from the point (2, 3, 1) to (2, 3, 4) ?

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

68. A uniform chain of length ' ' and mass m is placed on a smooth 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

table is

A)4

1mgl B)

8

1mgl C)

16

1mgl D)

32

1mgl

69. A spring, which is initially in its unstretched condition, is first stretched by a length x and then

again by a further length x. The work done in the first case is W1 and in the second case is W2

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

70. A particle of mass m is fixed to one end of a light rigid rod of length ' ' 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 depends on its displacement x as F xn. The power delivered by F willbe independent of x if n is

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

72. A particle is moving 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 Wc is the work done in the process of taking the

particle from A to B.

A) Wc = UB - UA B) Wc = UA - UB C) UA > UB D) UB > UA

73. A force is given by Mv2/r when the mass moves with speed v in a circle of radius r. The work

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

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

74. A moving railway compartment has a spring of constant 'k' fixed to its front wall. A boy in the

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

a distance s. The work done by boy w.r.t earth is

A) 2kx2

1B)

2

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

2

1D) sxskx

2

1

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


33/46

33

F = N2x

4

2

The block is displaced from x=-2m to x=+4m, the work done will be

A) positive B) negative

C) zero D) may be positive or negative

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

block is 1 kg and force constant of each springs 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 graph 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 point 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)

A) gR5

2B) gR2

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

wind speed v, the electrical power output will be proportional 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


34/46

34

LEVEL III

1. A block m is pulled by applying a force F as shown in fig. If the block has

moved up through a distance '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 momentum 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) = p2 / (2mg s) C) = p2 / (2g m2s) D) = p2 (2g m2s2)

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

P. the velocity of the body after time t will be

A)m

Pt2B)

m

Pt2C)

m2

PtD)

m2

Pt

4. The spring shown in fig has a force constant k and the mass of block is m. Initially,

the spring is unstretched when the block is released. The maximum elongation of the

spring on the releasing the mass will be

A) k

mg

B) 2

1

k

mg

C) 2 k

mg

D) 4 k

mg


35/46

35

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

without turning or braking. The friction coefficient is . When

he stops at point B, his horizontal displacement in S. The height

difference h between points 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 kept on a rough inclined surface of inclination fixed in an elevator.

The elevator goes up with a uniform velocity v and the block does not slide on the wedge. The

work 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 plane. The block slides down the plane travels a total

distance d across a rough horizontal surface with coefficient of

kinetic friction k and compresses a spring with force constant

k, a distance x before momentarily coming to rest. The spring then extends and the block travels

back across the rough surface, sliding up the plane. The maximum height h' that the block

reaches on its return is

A) h' = h - 2d B) h' = h - 2d -21 kx2

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

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

that its length is one side and 2 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 point?

A) lg2 B) lg3

2C) lg

3

1D) lg

2

1

9. If a constant power P is applied in a vehicle, then its acceleration increases with time according to

the relation

A) a = tm2

P

B) a = 2/3t

m2

P

C) a = t/1

m2

P

D) a =

mt2

P


36/46

36

10. A body of mass m slides downward along a plane inclined at an angle . The coefficient of

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

expressed as a function of time is

A) mtg2 cos B) mtg2 cos (sin - cos )

C) mtg2 sin D) mtg2 sin (sin - cos )

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

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

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

12. A uniform rope of length ' ' 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 during the period the chain slips completely 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 done in the process is

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


37/46

37

14. A compressed spring of spring 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 ground is

A) x mg

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 the block

and the ground is = 1/5. Find maximum 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

16. Potential energy of a particle moving along x-axis under the action of only conservative forces is

given as : U = 10 + 4 sin(4x). Here U is in Joule and x in meters. Total mechanical energy of the

particle is 16J. Choose the correct option.

A) At x = 1.25m, particle is at equilibrium position. C) both A and B are correct

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

17. A system shown in figure is released from rest. Pulley and spring is masslessand 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 and

g = 10 m/s2)

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

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

18. Two blocks of masses m1 = 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 m1 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 spring of force constant k.

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

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


38/46

38

the plane is (Neglect mass of string and pulley 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 radius 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 power

delivered to the particle by the forces acting on it?

A) 2 pmk2r2t B) mk2r2t C)3

)trmk(s24

D) 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

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

MULTIPLE ANSWER TYPE QUESTIONS

1. The potential energy U for a force field F

is such that U = - kxy, where k is a constant

A) jkxikyF

B) jkyikxF

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

2. A sledge moving over a smooth horizontal 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

runners and road is

A) No work is done by the friction to switch the sledge

from ice to the road

A) B) c) D)


39/46

39

B) A work of2

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

C) The distance covered by the sledge on the road is

2g2

v20 l

D) Total distance moved by the sledge before stopping is

2g2

v20 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 with a constant

speed

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

19. At the position of stable equilibrium

A) dx

dU0 only B)

dx

du= 0 and

2

2

dx

Ud> 0 C) 0

dx

dU and 0

dx

Ud

2

2

D) None of these

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

A) Potential energy decrease in the direction of conservative force

B) Potential energy increase in the direction of conservative 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

COMPREHENSION TYPE QUESTIONS

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


43/46

43

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 appropriate graph U(r) versus r:

3. Which is the most appropriate 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 predict about equilibrium state:

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

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

6. What minimum energy must be added to the molecule to dissociate it, if the distance between the

two atoms is equal to the equilibrium distance found in Q. 4 ?

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

7. For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is 1.13 x 10-

10 m and the dissociation energy is 1.54 x 10-18 J per molecule. Find the values of a and b:

A) a = 6.67 x 10-138 J-m12

b = 2.08 x 10-60 J m6

B) a = 6.41 x 10-78 J-m6

b = 6.67 x 10-138 J-m12

C) a = 6.67 x 10-138 J-m12 D) a = 0

A) B) C) D)

A) B) C) D)


44/46

44

b = 6.41 x 10-78 J m6 b = 6.41 x 10-78 J m6

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

A cutting tool under microprocessor control 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 constant is = 2.50 N. Consider the displacement of the tool from the origin to the

point x = 3.00 m, y = 3.00 m.

8. Calculate the work done 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 byF

if the tool is first moved out along the x-axis to thepoint 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. Compare the work done by F

along these two paths ?

A) Work done on x-axis is zero

B) Work done on y-axis is less than on y-axis

C) Work done on x-axis is more than on y-axis but not zero

D) Data insufficient

11. What can you predict about F

?

A) Force is non-conservative

B) Force is conservative

C) Force is neither conservative nor non-conservative

D) Data insufficient to conclude

Passage III (Q.No: 12 to 16):

A 1200 kg car is travelling at 7.5 m/s in a northerly direction on an icy road. It crashes into a 8000

kg truck moving in the same direction as the car with 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 momentum and energy are not conserved

B) The collision is inelastic, kinetic energy is conserved but momentum is not

C) The collision is not perfectly elastic, momentum is conserved but total energy is not

D) The collision is not perfectly elastic, momentum is conserved but kinetic energy is not


45/46

45

13. What 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 driver turns off onto a smooth road with a

coefficient of friction = = 1/4. He then stops for a snack and then tries to drive off. what is the

value of frictional force when the 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 hill. How does the frictional force on the car now compare to the value when the car was

driving 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 much potential 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

* * *


46/46

MULTIPLE MATCHING TYPE QUESTIONS

1. Match the following:

List - I List - II

a) Area under F - S e) Change in KEb) Work energy theorem f) negative of work done to gravitational force

c) change in PE g) work done by F

d) conservative force h) dx.F

, where F is conservative force

i) gravitational force


List - I List - II

a) KE e) depends on frame of referenceb) work done f) defined for conservative force only

c) PE g) independent on frame of reference

d) spring PE h) same for either compression or elongation for same distance


List - I List - II

a) stable equilibrium e) PE in Max

b) unstable equilibrium f) Fnet = 0

c) 0dx

dF

g) PE is Min

d) 0dx

dF

h) slope of F-x graph is +ve


List - I List - II

a) work done by frictional force e) indepent of path

b) work done by electrostatic force f) non-conservative

c) work done by gravitational force for closed loop g) depends on path

d) for slowly moving body, wc + wn.c equal to h) define PE

i) zero

KEY

1 2 3 4

work energy power module for iit main

Documents