16469890 formula booklet physics xi

5/27/2018 16469890 Formula Booklet Physics XI

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Call 1600-111-533 (toll-free) for info. Formula Booklet Physics XI

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1

Dear students

Most students tend to take it easy after the board examinations of Class X. The summer

vacations immediately after Class X are a great opportunity for the students to race ahead of

other students in the competitive world of IITJEE, where less than 2% students get selected

every year for the prestigious institutes.

Some students get governed completely by the emphasis laid by the teachers of the school in

which they are studying. Since, the objective of the teachers in the schools rarely is to equip the

student with the techniques reqired to crack IITJEE, most of the students tend to take it easy in

Class XI. Class XI does not even have the pressure of board examinations.

So, while the teachers and the school environment is often not oriented towards the serious

preparation of IITJEE, the curriculum of Class XI is extremely important to achieve success in

IITJEE or any other competitive examination like AIEEE.

The successful students identify these points early in their Class XI and race ahead of rest of

the competition. We suggest that you start as soon as possible.

In this booklet we have made a sincere attempt to bring your focus to Class XI and keep your

velocity of preparations to the maximum. The formulae will help you revise your chapters in a

very quick time and the motivational quotes will help you move in the right direction.

Hope youll benefit from this book and all the best for your examinations.

Praveen Tyagi

Gaurav Mittal

Prasoon Kumar


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2

CONTENTS

Description Page Number

1. Units, Dimensions & Measurements 03

2. Motion in one Dimension & Newtons Laws of Motion 04

3. Vectors 06

4. Circular Motion, Relative Motion, and Projectile Motion 07

5. Friction & Dynamics of Rigid Body 09

6. Conservation Laws & Collisions 12

7. Simple Harmonic Motion & Lissajous Figures 14

8. Gravitation 18

9. Properties of Matter 20

10. Heat & Thermodynamics 25

11. Waves 30

12. Study Tips 35


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3

UNITS, DIMENSIONS AND MEASUREMENTS

(i) SI Units:(a) Time-second(s);(b) Length-metre (m);(c) Mass-kilogram (kg);

(d) Amount of substancemole (mol); (e) Temperature-Kelvin (K);(f) Electric Current ampere (A);(g) Luminous Intensity Candela (Cd)

(ii) Uses of dimensional analysis(a) To check the accuracy of a given relation(b) To derive a relative between different physical quantities(c) To convert a physical quantity from one system to another system

c

2

1

b

2

1

a

2

1122221

T

Tx

L

Lx

M

Mnnorunun !

"

#$%

&!"

#$%

&!"

#$%

&==

(iii) Mean or average value: N

X...XXX N21

++=

(iv) Absolute error in each measurement: |Xi| = | XXi|

(v) Mean absolute error: Xm=N

|X| i

(vi) Fractional error=X

X

(vii) Percentage error= 100xX

X

(viii) Combination of error: If =c

ba

Z

YX, then maximum fractional error in is:

++=|c|

YY|b|

XX|a|


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4

MOTION IN ONE DIMENSION & NEWTONS LAWS OF MOTION

(i) Displacement: | displacement | distance covered

(ii) Average speed:

2

2

1

1

21

21

21

v

s

v

s

ss

tt

ssv

+

+=

+

+=

(a) If s1= s2= d, then21

21

vv

vv2v

+= = Harmonic mean

(b) If t1= t2, then2

vvv 21

+= =arithmetic mean

(iii) Average velocity: (a)12

12av

tt

rrv

=

; (b) v|v| av

(iv) Instantaneous velocity: |v|anddt

rdv

= = v = instantaneous speed

(v) Average acceleration:12

12av

tt

vva

=

(vi) Instantaneous acceleration: dt/vda

=

In one dimension, a = (dv/dt) = vdx

dv'(

)*+

,

(vii) Equations of motion in one dimension:(a) v = u + at;

(b) x = ut +2

1at

2;

(c) v2 u2+ 2ax;

(d) x = vt 2

1at

2;

(e) '(

)*+

, += 2

uvx t;

(f) ;at2

1utxxs 20 +==

(g) v2= u

2+ 2a (xx0)

(viii) Distance travelled in nth second: dn= u +2

a(2n1)

(ix) Motion of a ball: (a) when thrown up: h = (u2/2g) and t = (u/g)

(b) when dropped: v = (2gh) and t = (2h/g)

(x) Resultant force: F = (F12+ F2

2+ 2F1F2cos )

(xi) Condition for equilibrium: (a) ;)FF(F 213 += (b) F1+ F2F3|F1 F2|

(xii) Lamis Theorem:( ) ( ) ( )

=

= sin

R

sin

Q

sin

P

(xiii) Newtons second law: ''(

)**+

,==

dt/pdF;amF

(xiv) Impulse: ==

12 ppandtFp Fdt

(xv) Newtons third law:

-21


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5

(a)

= 1212 FF

(b) Contact force: 2112 FFmM

mF =

+=

(c) Acceleration: a =mM

F

+

(xvi) Inertial mass: mI= F/a

(xvii) Gravitational mass: mG= GI2

mm;GM

FR

g

F==

(xviii) Non inertial frame: If

0a be the acceleration of frame, then pseudo force

= 0amF

Example: Centrifugal force = rmr

mv2

2=

(xix) Lift problems: Apparent weight = M(g a0)(+ sign is used when lift is moving up while sign when lift is moving down)

(xx) Pulley Problems:(a) For figure (2):

Tension in the string, T = gmm

mm

21

21

+

Acceleration of the system, a = gmm

m

21

2

+

The force on the pulley, F = gmm

mm2

21

21

+

(b) For figure (3):

Tension in the string, gmm

mm2T

21

21

+=

Acceleration of the system, gmmmma

12

12

+=

The force on the pulley, gmm

mm4F

21

21

+=

a

T

T T

T

a

m2

m1

Fig. 3

m2

m1T

Frictionlesssurface

m2gFig. 2

T

F

(1)

(2)

M

F21

mF12

Fig. 1


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6

VECTORS

(i) Vector addition: )B(ABAandABBAR

+=+=+=

(ii) Unit vector: )A/A(A^

=

(iii) Magnitude: )AA(AA

2

z

2

y

2

x ++= (iv) Direction cosines: cos = (Ax/A), cos = (Ay/A), cos = (Az/A)

(v) Projection:

(a) Component ofA along

B =

^

B.A

(b) Component ofB along

A =

B.A

^

(c) IfA = ,jAiA

^

y

^

x + then its angle with the xaxis is = tan1

(Ay/Ax)

(vi) Dot product:

(a)B.A = AB cos , (b) zzyyxx BABABAB.A ++=

(vii) Cross product:

(a) =BxA AB sin

^

n ;

(b) ;0AxA =

(c)BxA =

zyx

zyx

^^^

BBB

AAA

kji

(viii) Examples:

(a) ;r.FW

= (b) ;v.FP

= (c) ;A.E

= (d) ;A.B

=

(e) ;rxwv

= (f) ;Fx

= (g) ''(

)**+

,=

BxvqF m

(ix) Area of a parallelogram: Area = |BxA|

(x) Area of a triangle: Area = |BxA|2

1

(xi) Gradient operator:z

ky

jx

iV^^^

+

+

=

(xii) Volume of a parallelopiped: ''(

)**+

,=

CxB.AV


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7

CIRCULAR MOTION, RELATIVE MOTION & PROJECTILE MOTION

(i) Uniform Circular Motion:

(a) v = r;(b) a = (v

2/r) = 2r ;

(c) F = (mv2/r);

(d) ;0v.r =

(e) 0a.v =

(ii) Cyclist taking a turn: tan = (v2/rg)

(iii) Car taking a turn on level road: v = (srg)

(iv) Banking of Roads: tan = v2/rg

(v) Air plane taking a turn: tan = v2/r g

(vi) Overloaded truck:

(a) Rinnerwheel< Router wheel

(b) maximum safe velocity on turn, v = (gdr/2h)

(vii) Nonuniform Circular Motion:(a) Centripetal acceleration ar= (v

2/r);

(b) Tangential acceleration at= (dv/dt);

(c) Resultant acceleration a= )aa( 2t2r+

(viii) Motion in a vertical Circle:

(a) For lowest point A and highest point B, TA TB= 6 mg; v2A= v

2B+ 4g!; vA(5g!); and vB

(g!)(b) Condition for Oscillation: vA< (2g!)

(c) Condition for leaving Circular path: (2g!) < vA< (5g!)

(ix) Relative velocity: ABBA vvv

=

(x) Condition for Collision of ships: 0)vv(x)vr( BABA =

(xi) Crossing a River:(a) Beat Keeps its direction perpendicular to water current

(1) vR=( ;)vv( 2b2w+ (2) = tan

1 );v/v( bw

(3) t=(x/vb) (it is minimum) (4) Drift on opposite bank = (vw/vb)x(b) Boat to reach directly opposite to starting point:

(1) sin = (vw/vb); (2) vresultant= vbcos ; (3) t= cosvx

b

(xii) Projectile thrown from the ground:

(a) equation of trajectory: y = x tan 22

2

cosu2

xg

(b) time of flight:g

sinu2T

=

(c) Horizontal range, R = (u2sin 2/g)

(d) Maximum height attained, H = (u2sin

2/2g)


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8

(e) Range is maximum when = 450(f) Ranges are same for projection angles and (900)(g) Velocity at the top most point is = u cos (h) tan = gT2/2R(i) (H/T

2) = (g/8)

(xiii) Projectile thrown from a height h in horizontal direction:(a) T = (2h/g);(b) R = v(2h/g);(c) y = h (gx

2/2u

2)

(d) Magnitude of velocity at the ground = (u2+ 2gh)

(e) Angle at which projectiles strikes the ground, = tan1u

gh2

(xiv) Projectile on an inclined plane:

(a) Time of flight, T =( )

0

0

cosg

sinu2

(b) Horizontal range,( )

0

0

=

cosg

cossinu2R

2

2


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9

FRICTION

(i) Force of friction:

(a) ssN (self adjusting); (s)max= sN(b) k= kN (k= coefficient of kinetic friction)(c) k< s

(ii) Acceleration on a horizontal plane: a = (F kN)/M

(iii) Acceleration of a body sliding on an inclined plane: a = g sin (1 kcot t2)

(iv) Force required to balance an object against wall: F = (Mg/s)

(v) Angle of friction: tan = s(s= coefficient of static friction)

DYNAMICS OF RIGID BODIES

(i) Average angular velocity:ttt

1

=

=

2

12

(ii) Instantaneous angular velocity: = (d/dt)(iii) Relation between v, and r : v=r; In vector form

= rxv ; In general form, v = r sin

(iv) Average angular acceleration:ttt 12

=

= 12

(v) Instantaneous angular acceleration: = (d/dt) = (d2/dt2)

(vi) Relation between linear and angular acceleration:

(a) aT= r and aR= (v2/r) = 2R

(b) Resultant acceleration, a = )aa( 2R2T+

(c) In vector form,

''

(

)**

+

,===+=

rxxuxaandrxawhere,aaa RTRT

(vii) Equations for rotational motion:

(a) = 0+ t;

(b) = 0t +2

1t2;

(c) 2 02= 2

(viii) Centre of mass: For two particle system:

(a) ;mm

xmxmx

21

2211CM +

+=

(b)21

2211CM

mm

vmvmv

+

+=

(c)21

2211CM

mm

amama

+

+=

Also2

CM2

CMCM

CMCM

dt

xd

dt

dvaand

dt

dxv ===

(ix) Centre of mass: For many particle system:


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10

(a) ;M

xmX iiCM

=

(b) ;M

rmr

iiCM

=

(c) ;dt

rd

v

CM

CM

=

(d) ;dt

vda

CMCM

=

(e)

== iiCMCM vmvMP ;

(f) .FamaMF iiiCMext

=== If ;constantV,0a,0F CMCMext ===

(g) Also, moment of masses about CM is zero, i.e., 2211ii rmrmor0rm ==

(x) Moment of Inertia: (a) I= miri2

(b) I= r2, where = m1m2/(m1+ m2)

(xi) Radius of gyration: (a) K = (I/M) ; (b) K = [(r12+ r22+ + rn2)/n] = root mean square distance.

(xii) Kinetic energy of rotation: K =2

1I2 or I= (2K/2)

(xiii) Angular momentum: ( ) ( ) dvmcsinrpL(b);pxrLa ;==

(xiv) Torque: ( ) ( ) ==

sinFrb;Fxra

(xv) Relation between and L: ''(

)**+

,=

dt/dL ;

(xvi) Relation between L and I: (a) L = I; (b) K = 21 I2= L2/2I

(xvii) Relation between and :(a) = I,(b) If = 0, then (dL/dt)=0 or L=constant or, I=constant i.e., I11= I22

(Laws of conservation of angular momentum)

(xviii) Angular impulse: tL =

(xix) Rotational work done: W = = avd

(xx) Rotational Power:= .P

(xxi) (a) Perpendicular axes theorem: Iz= Ix+ Iy(b) Parallel axes theorem: I= Ic+ Md

2

(xxii) Moment of Inertia of some objects

(a) Ring: I= MR2(axis); I=

2

1MR

2(Diameter);

I= 2 MR2(tangential to rim, perpendicular to plane);

I= (3/2) MR2(tangential to rim and parallel to diameter)

-


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11

(b) Disc: I=2

1MR

2(axis); I=

4

1MR

2(diameter)

(c) Cylinder: I= ( )axisMR2

1 2

(d) Thin rod: I= (ML2/12) (about centre); I= (ML

2/3) (about one end)

(e) Hollow sphere : Idia= (2/3) MR2; Itangential = (5/3) MR

2

(f) Solid sphere: Idia= (2/5) MR2; Itangential = (7/5) MR2

(g) Rectangular:12

bMI

22

C

+=

!(centre)

(h) Cube: I= (1/6) Ma2

(i) Annular disc: I= (1/2) M ( 2221 RR + )

(j) Right circular cone: I= (3/10) MR2

(k) Triangular lamina: I= (1/6) Mh2(about base axis)

(l) Elliptical lamina: I= (1/4) Ma2(about minor axis) and I= (1/4) Mb

2(about major axis)

(xxiii) Rolling without slipping on a horizontal surface:

!!"

#

$$%

&+=+= 2

2

222

R

K1MV

2

1I

2

1MV

2

1K ("V = Rand I= MK2)

For inclined plane

(a) Velocity at the bottom, v =''

(

)

**

+

,+

2

2

R

K1gh2

(b) Acceleration, a = g sin ''

(

)

**

+

,+

2

2

r

K1

(c) Time taken to reach the bottom, t = ''

(

)**

+

, + singR

K1s22

2

(xxiv) Simple pendulum: = T = 2(L/g)

(xxv) Compound Pendulum: T = 2(I/Mg !), where != M (K2+ !2)Minimum time period, T0= 2(2K/g)

(xxvi) Time period for disc: T = 2(3R/2g)Minimum time period for disc, T = 2(1.414R/g)

(xxvii)Time period for a rod of length L pivoted at one end: T = 2(2L/3g

The heights by great men reached and kept

were not attained by sudden flight,

but they, while their companions slept

were toiling upwards in the night.


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12

CONSERVATION LAWS AND COLLISIONS

(i) Work done: (a) ;d.FW

= (b) ;= cosFdW (c) =W

(ii) Conservation forces: 0rd.F(c);rd.F(b)rd.F)a( =

For conservative forces, one must have:FxV = 0

(iii) Potential energy: (a) ( ) ( ) UVFc;dU/dXF(b)W;UV ===

(iv) Gravitational potential energy: (a) U = mgh ; (b)( )hRGMm

U+

=

(v) Spring potential energy: ( ) ( ) ( )21222 xxKUb;KxUa ==

(vi) Kinetic energy: (a) K = W = ( ) 22i2f mvKb;mvmv =

(vii) Total mechanical energy: = E = K + U

(viii) Conservation of energy: K = U or, K+ U= Ki+ Ui

In an isolated system, Etotal= constant

(ix) Power: (a) P = (dw/dt) ; (b) P = (dw/dt) ; (c) P =v.F

(x) Tractive force: F = (P/v)

(xi) Equilibrium Conditions:(a) For equilibrium, (dU/dx) = 0(b) For stable equilibrium: U(x) = minimum, (dU/dx) = 0 and (d

2U/dx

2) is positive

(c) For unstable equilibrium: U(x) = maximum, (dU/dx) = 0 and (d2

U/dx2

) is negative(d) For neutral equilibrium: U(x) = constant, (dU/dx) = 0 and (d

2U/dx

2) is zero

(xii) Velocity of a particle in terms of U(x): v = ( )[ ]xUEm

2

(xiii) Momentum:

(a) ( ) ''(

)**+

,==

dt/pdFb;vmp ,

(b) Conservation of momentum: If ,ppthen,0F ifnet

==

(c) Recoil speed of gun, BG

BG x v

m

mv =

(xiv) Impulse: tFp av=

(xv) Collision in one dimension:(a) Momentum conservation : m1u1+ m2u2= m1v1+ m2v2(b) For elastic collision, e = 1 = coefficient of restitution(c) Energy conservation: m1u1

2+ m2u2

2= m1v1

2+ m2v2

2

(d) Velocities of 1stand 2

ndbody after collision are:

- 21

xx

Path 1 Path 2 closedpath

-ba -

ba

-

12

12

12

12

12

12

12

12

12


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13

2

12

121

21

122

12

21

21

211 u

mm

mmu

mm

m2 v;u

mm

m2u

mm

mmv ''

(

)**+

,

+

+''

(

)**+

,

+=''

(

)**+

,

++''

(

)**+

,

+

=

(e) If m1= m2= m, then v1= u2and v2= u1(f) Coefficient of restitution, e = (v2v1/u1= u2)(g) e = 1 for perfectly elastic collision and e=0 for perfectly inelastic collision. For inelastic

collision 0 < e < 1

(xvi) Inelastic collision of a ball dropped from height h0(a) Height attained after nth impact, hn= e

2nh0

(b) Total distance traveled when the ball finally comes to rest, s = h0(1+e2)/(1e

2)

(c) Total time taken, t = '(

)*+

,

+

e1

e1

g

h2 0

(xvii) Loss of KE in elastic collision: For the first incident particle

( )%100

K

K,mmIf;

mm

m4m

K

Kand

mm

mm

K

K

i

lost212

21

21

i

lost

2

21

21

i

=

=+

=

''(

)**+

,

+

=

(xviii) Loss of KE in inelastic collision: Klost= Ki K=21

21

mm

mm

2

1

+(u1 u2)

2(1e

2)

Velocity after inelastic collision (with target at rest)

( )1

21

121

21

211 u

mm

e1m vandu

mm

emmv

+

+=''

(

)**+

,

+

=

(xix) Oblique Collision (target at rest):

m1u1= m1v1cos 1+ m2v2cos 2 and m1v1sin 1= m2v2sin 2

Solving, we get: m1u12= m1v1

2+ m2v2

2

(xx) Rocket equation: (a)dt

dMv

dt

dVM el=

(b) V = vrel

loge

''(

)

**+

,

0

b0

M

mM[M

0= original mass of rocket plus fuel and m

b= mass of fuel burnt]

(c) If we write M = M0 mb= mass of the rocket and full at any time, than velocity of rocks atthat time is:V = vrelloge(M0/M)

(xxi) Conservation of angular momentum:

(a) If ext= 0, then L= Li

(b) For planets,min

max

min

max

r

r

v

v=

(c) Spinning skater, I11= I2W2 or = i ''

(

)

**

+

,

I

Ii


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14

SIMPLE HARMONIC MOTION AND LISSAJOUS FIGURES(i) Simple Harmonic Motion:

(a) F = Kx ;

(b) a = m

Kx or a = 2x, where = (K/m);

(c) Fmax= KA and amax= 2A

(ii) Equation of motion: 0xdt

xd2

2

=+ 2

(iii) Displacement: x = A sin (t + )(a) If = 0, x = A sin t ;(b) If = /2, x = A cos t(c) If x = C sin t + D cos t, then x = A sin (t + ) with A= (C2+D2) and = tan1(D/C)

(iv) Velocity:

(a) v = A cos (+ );(b) If =0, v = A cos t;

(c) vmax=A(d) v = (A2 x2);

(e) 1A

v

A

x22

2

2

2

=

+

(v) Acceleration:

(a) a = 2x = 2A sin (t+) ;(b) If =0, a= 2A sin t(c) |amax| =

2A;

(d) Fmax= m 2A

(vi) Frequency and Time period:

(a) = (K/m) ;(b) = ( );m/K

2

1

(c) T = 2K

m

(vii) Energy in SHM: Potential Energy:(a) U = Kx

2;

(b) F = dx

dU;

(c) Umax= m2A

2;

(d) U = m2A2 sin2t

(viii) Energy in SHM: Kinetic energy:

(a) K = mv2;

(b) K= m2(A2x2);

(c) K = m2A2cos2t ;

(d) Kmax= m2A

2

2

1

2

1

2

1

2

1

2

1

2

1

2

1


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15

(ix) Total energy:(a) E = K + U = conserved;

(b) E = (1/2) m2A2;(c) E = Kmax= Umax

(x) Average PE and KE:

(a) < U > = (1/4) m

2

A

2

;(b) < K > = (1/4) m2A2;(c) (E/2) = < U > = < K >

(xi) Some relations:

(a) = ;xx

vv

21

22

22

21

(b) T = 2

22

21

21

22

vv

xx

; (c) A =

( ) ( )22

21

122

21

vv

xvxv

2

(xii) Spring mass system:(a) mg = Kx0;

(b) T = 2g

x2

K

m 0=

(xiii) Massive spring: T = 2 ( )K

3/mm s+

(xiv) Cutting a spring:(a) K = nK ;

(b) T = T0/(n) ;(c) = (n) 0

(d) If spring is cut into two pieces of lengths !1and !2such that !1= n!2, then K1= K,n

1n'(

)*+

, +K2=

(n +1) K and K1!1= K2!2

(xv) Springs in parallel:

(a) K = K1+ K2;(b) T = 2[m/(K1+ K2)](c) If T1= 2(m/K1) and T2= 2(m/K2), then for the parallel combination:

22

21

2

22

21

21

22

21

2and

TT

TTTor

T

1

T

1

T

1+=

+=+=

(xvi) Springs in series:(a) K1x1= K2x2= Kx = F applied

(b)21

21

21 KK

KKKor

K

1

K

1

K

1

+=+=

(c) 222

12

21

TTTor111

+

+

= 22

2

(d) T = 2 ( )

( )2121

21

21

KKm

KK

2

1or

KK

KKm

+=

+

(xvii) Torsional pendulum:

(a) I=C or 0I

C

dt

d

2

2

=+

;

(b) =0sin (t+);(c) = (C/I) ;


16/35




16

(d) =I

C

2

1

;

(e) T = 2(I/C), where C = r4/2!

(xviii) Simple pendulum:

(a) I= = mg!sin or '()

*+

,+

2 !

g

dt

d2

sin = 0 or 0g

dt

d

2

2

=+

! ;

(b) = (g/!) ;

(c) = ( ) ;g/2

1!

(d) T = 2(!/g)

(xix) Second pendulum:(a) T = 2 sec ;

(b) != 99.3 cm

(xx) Infinite length pendulum:

(a) ;

R

11g

12T

e''(

)**+

,+

=

!

(b) T=2g

Re (when !)

(xxi) Anharmonic pendulum: T T0 ''

(

)

**

+

,+

''

(

)

**

+

, +

2

2

0

20

16

A1T

161

!

(xxii) Tension in string of a simple pendulum: T = (3 mg cos 2 mg cos 0)

(xxiii) Conical Pendulum:

(a) v = (gR tan ) ;

(b) T = 2(L cos /g)

(xxiv) Compound pendulum: T = 2 ( )2

/K2

!! +

(a) For a bar: T = 2(2L/3g) ;(b) For a disc : T = 2(3R/2g)

(xxv) Floating cylinder:

(a) K = Ag ;(b) T = 2(m/Ag) = 2(Ld/g)

(xxvi) Liquid in Utube:

(a) K = 2A g and m = AL ;

(b) T = 2(L/2g) = 2(h/g)

(xxvii)Ball in bowl: T = 2[(R r)/g]

(xxviii) Piston in a gas cylinder:

(a) ;V

EAK

2

=

(b) ;EA

mV2T

2=


17/35




17

(c)PA

V2T

2

m= (EP for Isothermal process);

(d)P

V2T m

=

2 (E = P for adiabatic process)

(xxix) Elastic wire:

(a) K = ;AY

!

(b) T =AY

m2 !

(xxx) Tunnel across earth: T = 2(Re/g)

(xxxi) Magnetic dipole in magnetic field: T = 2(I/MB)

(xxxii)Electrical LC circuit: T = 2

LC orLC2

1

=

(xxxiii) Lissajous figures

Case (a): 1= 2= or 1: 2= 1 : 1General equation: =+ sincos

ab

xy2

b

y

a

x 22

2

2

2

For = 0 : y = (b/a) x ; straight line with positive slope

For = /4 : ellipseoblique;2

1

ab

xy2

b

y

a

x2

2

2

2

=+

For = /2 : ellipselsymmetrica;1b

y

a

x2

2

2

2

=+

For = : y = (b/a) x ; straight line with negative slope.

Case (b): For 1: 2= 2:1 with x = a sin (2t + ) and y = b sin t

For = 0, : Figure of eight

For = :3,4

4

Double parabola

For = :3,2

2

Single parabola


18/35




18

GRAVITATION

(i) Newtons law of gravitation:

(a) F = G m1m2/r2; (b) a = 6.67 x 10

11K.m

2/(kg)

2; (c)

r

dr2

F

dF=

(ii) Acceleration due to gravity (a) g = GM/R2; (b) Weight W = mg

(iii) Variation of g:(a) due to shape ; gequator< gpole(b) due to rotation of earth: (i) gpole= GM/R

2 (No effect)

(ii) gequator= RR

GM 22

(iii) gequator< gpole

(iv) 2R = 0.034 m/s2(v) If 17 0 or T = (T0/17) = (24/17)h = 1.4 h, then object would

float on equator

(c) At a height h above earths surface g = g Rhif,g

h21


19/35




19

(xii) Orbital velocity of satellite:

( ) ( )( )

;hR2

Rvvb;

r

GMva e00 +

== (c) v0ve/2 (if h


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20

SURFACE TENSION

(i) (a)!

F

Length

ForceT == ; (b)

A

W

areaSurface

energySurfaceT ==

(ii) Combination of n drops into one big drop: (a) R = n1/3

r

(b) Ei= n(4r2T), E

= 4R2T, (E

/E

i) = n

1/3,

''(

)

**+

,=

1/3i n

11

E

E

(c) E = 4R2T (n1/31) = 4R3T '(

)*+

,

R

1

r

1

(iii) Increase in temperature: =s

T3

'

(

)*+

,

R

1

r

1 or '

(

)*+

,

R1

r

1

sJ

T3

(iv) Shape of liquid surface:

(a) Plane surface (as for water silver) if Fadhesive>2

Fcohesive

(b) Concave surface (as for water glass) if Fadhesive>2

Fcohesive

(c) Convex surface (as for mercuryglass) if Fadhesive Fc/2 ;(b) obtuse: if Fa Tsa

(vi) Excess pressure:

(a) General formula: Pexcess= ''(

)**+

,+

21 R

1

R

1T

(b) For a liquid drop: Pexcess= 2T/R(c) For an air bubble in liquid: Pexcess= 2T/R(d) For a soap bubble: Pexcess= 4T/R(e) Pressure inside an air bubble at a depth h in a liquid: Pin= Patm+ hdg + (2T/R)

(vii) Forces between two plates with thin water film separating them:

(a) P = T ;R

1

r

1'(

)*+

,

(b) ;R

1

r

1ATF '

(

)*+

,=

(c) If separation between plates is d, then P = 2T/d and F = 2AT/d

(viii) Double bubble: Radius of Curvature of common film Rcommon =rR

rR

(ix) Capillary rise:

(a) ;rdg

cosT2h

=

(b)rdg

2Th= (For water = 00)


21/35




21

(c) If weight of water in meniseus is taken into account then T =

'(

)*+

,+

cos2

3

rhrdg

(d) Capillary depression,( )

rdg

cosT2h

(x) Combination of two soap bubbles:(a) If V is the increase in volume and S is the increase in surface area, then 3P0V + 4TS =

0 where P0is the atmospheric pressure

(b) If the bubbles combine in environment of zero outside pressure isothermally, then S = 0 orR3= ( )2221 RR +

ELASTICITY

(i) Stress: (a) Stress = [Deforming force/crosssectional area];

(b) Tensile or longitudinal stress = (F/r2);(c) Tangential or shearing stress = (F/A);(d) Hydrostatic stress = P

(ii) Strain: (a) Tensile or longitudinal strain = (L/L);(b) Shearing strain = ;(c) Volume strain = (V/V)

(iii) Hooks law:

(a) For stretching: Stress = Y x Strain or( )LAFL

Y

=

(b) For shear: Stress = x Strain or = F/A

(c) For volume elasticity: Stress = B x Strain or B = ( )V/V

P

(iv) Compressibility: K = (1/B)

(v) Elongation of a wire due to its own weight: L = YgL

2

1YA

MgL2

1 2=

(vi) Bulk modulus of an idea gas: Bisothermal = P and Badiabatic = P (where = Cp/Cv)

(vii) Stress due to heating or cooling of a clamped rod

Thermal stress = Y(t) and force = YA (t)

(viii) Torsion of a cylinder:

(a) r = !(where = angle of twist and = angle of shear);(b) restoring torque = c(c) restoring Couple per unit twist, c = r4/2!(for solid cylinder)

and C = (r24 r1

4)/2!(for hollow cylinder)

(ix) Work done in stretching:

(a) W =2

1x stress x strain x volume = Y

2

1(strain)

2x volume =

( )volumex

Y

stress

2

12

(b) Potential energy stored, U = W =2

1x stress x strain x volume

(c) Potential energy stored per unit volume, u =2

1x stress x strain

(x) Loaded beam:


22/35




22

(a) depression, = ( )rrectangulaYbd4

W3

3!

(b) Depression, = ( )lcylindricarY12

W2

3

!

(xi) Positions ratio:(a) Lateral strain =

r

r

D

D =

(b) Longitudinal strain = (L/L)

(c) Poissons ratio =L/L

rr

strainallongitudin

strainlateral

/

=

(d) Theoretically, 1 < < 0.5 but experimentally 0.2 0.4

(xii) Relations between Y, , B and :(a) Y = 3B (12) ;(b) Y = 2(1+ );

(c)

+=3

1

B9

1

Y

1

(xiii) Interatomic force constant: k = Yr0(r0= equilibrium inter atomic separation)

KINETIC THEORY OF GASES

(i) Boyles law: PV = constant or P1V1= P2V2

(i) Chares law: (V/T) = constant or (V1/T1) = (V2/T2)

(ii) Pressure temperature law: (P1/T1) = (P2/T2)

(iii) Avogadros principle: At constant temperature and pressure, Volume of gas,

V number of moles, Where = N/Na [N = number of molecules in the sample

and NA= Avogadros number = 6.02 x 1023

/mole]

M

Msample= [Msample= mass of gas sample and M = molecular weight]

(iv) Kinetic Theory:

(a) Momentum delivered to the wall perpendicular to the xaxis, P = 2m vx(b) Time taken between two successive collisions on the same wall by the same molecule: t =

(2L/vx)

(c) The frequency of collision:coll. = (x/2L)(d) Total force exerted on the wall by collision of various molecules: F = (MN/L)

(e) The pressure on the wall : P = 2rms2rms

22x v

3

1v

V

mN

3

1v

V3

mNv

V

mN==><

(v) RMS speed:

(a) rms= (v12+ v2

2+ + v 2N /N);

(b) rms= (3P/) ;(c) rms= (3KT/m);

(d) rms= (3RT/M) ; (e)( )

( ) 12

1

2

2rms

1rms

M

M

m

m==

(vi) Kinetic interpretation of temperature:


23/35




23

(a) (1/2) Mv2

rms = (3/2) RT ;(b) (1/2) mv

2rms = (3/2) KT

(c) Kinetic energy of one molecule = (3/2) KT ;(d) kinetic energy of one mole of gas = (3/2) RT(e) Kinetic energy of one gram of gas (3/2) (RT/M)

(ix) Maxwell molecular speed distribution:

(a) n (v) = 4N KT2/mv-23/2

2ev

KT2m '

()*

+,

(b) The average speed:M

RT60.1

M

RT8

m

KT8v =

=

=

(c) The rms speed: vrms=M

RT73.1

M

RT3

m

kT3==

(d) The most probable speed:p=M

RT41.1

M

RT2

m

KT2==

(e) Speed relations: (I) vp< v < vrms

(II) vp: v : vrms= (2) : (8/) : (3) = 1.41 : 1.60 : 1.73

(x) Internal energy:(a) Einternal= (3/2)RT (for one mole)

(b) Einternal= (3/2 RT (for mole)

(c) Pressure exerted by a gas P = E3

2

V

E

3

2=

(xi) Degrees of freedom:(a) Ideal gas: 3 (all translational)(b) Monoatomic gas : 3 (all translational)(c) Diatomic gas: 5 (three translational plus two rotational)(d) Polyatomic gas (linear molecule e.g. CO2) : 7 (three translational plus two rotational plus two

vibrational)(e) Polyatomic gas (nonlinear molecule, e.g., NH3, H2O etc): 6 (three translational plus three

rotational)

(f) Internal energy of a gas: Einternal = (f/2) RT. (where f = number of degrees of freedom)

(xii) Daltons law: The pressure exerted by a mixture of perfect gases is the sum of the pressuresexerted by the individual gases occupying the same volume alone i.e., P = P1+ P2+ .

(xiii) Van der Walls gas equation:

(a) ( ) =''

(

)

**

+

, + Rb-V

VaP

2

2

(b) ( ) RTbVV

aP m2m

2

=''

(

)

**

+

, + (where Vm= V/= volume per mole);

(c) b = 30 cm3/mole

(d) Critical values: Pc= ;Rb27

a8

T,b3V,b27

aCC2 ==

(e) 375.08

3

RT

VP

C

CC ==

(xiv) Mean free path: =nd2

1

2,

Where n= (N/V) = number of gas molecules per unit volume andd = diameter of molecules of the gas


24/35




24

FLUID MECHANICS

(i) The viscous force between two layers of area A having velocity gradient (dv/dx) is given by: F =

A (dv/dx), where is called coefficient of viscosity(i) In SIsystem, is measured IPoiseiulle (P!) 1P!= 1Nsm2= 1 decapoise. In egs system, the unit

of is g/cm/sec and is called POISE(ii) When a spherical body is allowed to fall through viscous medium, its velocity increases, till the sum

of viscous drag and upthrust becomes equal to the weight of the body. After that the body moveswith a constant velocity called terminal velocity.

(iii) According to STOKEs Law, the viscous drag on a spherical body moving in a fluid is given by: F =

6r v, where r is the radius and v is the velocity of the body.

(iv) The terminal velocity is given by: vT=( )

gr

9

2 2

where is the density of the material of the body and is the density of liquid

(v) Rate of flow of liquid through a capillary tube of radius r and length !

R

p

r/8

p

8

prV

4

4

=

=

=

!!

where p is the pressure difference between two ends of the capillary and R is the fluid resistance(=8 !/r4)

(vi) The matter which possess the property of flowing is called as FLUID (For example, gases andliquids)

(vii) Pressure exerted by a column of liquid of height h is : P = hg (= density of the liquid)(viii) Pressure at a point within the liquid, P = P0+ hg, where P0is atmospheric pressure and h is the

depth of point w.r.t. free surface of liquid

(ix) Apparent weight of the body immersed in a liquid Mg = Mg Vg(x) If W be the weight of a body and U be the upthrust force of the liquid on the body then

(a) the body sinks in the liquid of W > U

(b) the body floats just completely immersed if W = U

(c) the body floats with a part immersed in the liquid if W < U

(xi) solidofdensity

solidofdensity

solidofvolumetotal

solidaofpartimmersedofVolume

=

(xii) Equation of Continuity: a1v1= a2v2

(xiii) Bernouillis theorem: (P/) + gh +2

1v

2= constant

(xiv) Accelerated fluid containers : tan =g

a x

(xv) Volume of liquid flowing per second through a tube: R=a1v1= a2v2 ( )2221 aagh2

(xvi) Velocity of efflux of liquid from a hole:

v = (2gh), where h is the depth of a hole from the free surface of liquid

I do not ask to walk smooth paths, nor bear an easy load.

I pray for strength and fortitude to climb rock-strewn road.

Give me such courage I can scale the hardest peaks alone,

And transform every stumbling block into a stepping-stone.

Gail Brook Burkett

ax

Fig. 4


25/35




25

HEAT AND THERMODYNAMICS

(i) L2 L1= L1(T2 T1); A2 A1= A2(T2 T1); V2 V1= V1(T2 T1)where, L1, A1, V1 are the length, area and volume at temperature T1; and L2, A2, V2 are that at

temperature T2. represents the coefficient of linear expansion, the coefficient of superficialexpansion and the coefficient of cubical expansion.

(ii) If dtbe the density at t0C and d0be that at 0

0C, then: dt= d0(1T)

(iii) : : = 1 : 2 : 3(iv) If r, abe the coefficients of real and apparent expansions of a liquid and gbe the coefficient of the

cubical expansion for the containing vessel (say glass), then

r= a+ g

(v) The pressure of the gases varies with temperature as : Pt= P0(1+ T), where = (1/273) per0C

(vi) If temperature on Celsius scale is C, that on Fahrenheit scale is F, on Kelvin scale is K, and onReaumer scale is R, then

(a)4

R

5

273K

9

32F

5

C=

=

= (b) 32C

5

9F +=

(c) ( )32F95

C =

(d) K = C + 273 (e) ( )459.4F9

5K +=

(vii) (a) Triple point of water = 273.16 K(b) Absolute zero = 0 K = 273.15

0C

(c) For a gas thermometer, T = (273.15) ( )KelvinP

P

triple

(d) For a resistance thermometer, R= R0[1+ ]

(viii) If mechanical work W produces the same temperature change as heat H, then we can write:W = JH, where J is called mechanical equivalent of heat

(ix) The heat absorbed or given out by a body of mass m, when the temperature changes by T is: Q= mcT, where c is a constant for a substance, called as SPECIFIC HEAT.

(x) HEAT CAPACITY of a body of mass m is defined as : Q = mc(xi) WATER EQUIVALENT of a body is numerically equal to the product of its mass and specific heat

i.e., W = mc(xii) When the state of matter changes, the heat absorbed or evolved is given by: Q = mL, where L is

called LATENT HEAT(xiii) In case of gases, there are two types of specific heats i.e., cpand cv[cp= specific heat at constant

pressure and Cv= specific heat at constant volume]. Molar specific heats of a gas are: Cp= Mcpand Cv= Mcv, where M = molecular weight of the gas.

(xiv) Cp> Cvand according to Mayers formula Cp Cv= R

(xv) For all thermodynamic processes, equation of state for an ideal gas: PV = RT

(a) For ISOBARIC process: P = Constant ;TV =Constant

(b) For ISOCHORIC (Isometric) process: V = Constant;T

P=Constant

(c) For ISOTHERMAL process T = Constant ; PV= Constant

(d) For ADIABATIC process: PV= Constant ; TV

1=Constant

and P(1) T= Constant

(xvi) Slope on PV diagram


26/35




26

(a) For isobaric process: zero(b) For isochoric process: infinite(c) For isothermal process: slope = (P/V)

(d) For adiabatic process: slope = (P/V)(e) Slope of adiabatic curve > slope of isothermal curve.

(xvii) Work done(a) For isobaric process: W = P (V2 V1)(b) For isochoric process: W = 0

(c) For isothermal process: W=RT loge(V2/V1)RT x 2.303 x log10 (V2/V1)P1V1x 2.303 x log10 (V2/V1)

RT x 2.303 x log10 (P1/P2)

(d) For adiabatic process:( )( )

( )( )1

=

1

= 221121

VPVP

TTRW

(e) In expansion from same initial state to same final volume

Wadiabatic< Wisothermal< Wisobaric

(f) In compression from same initial state to same final volume:

Wadiabatic< Wisothermal< Wisobaric

(xviii) Heat added or removed:

(a) For isobaric process: Q = CpT(b) For isochoric process = Q = CvT(c) For isothermal process = Q = W = Rt loge(V2/V1)(d) For adiabatic process: Q = 0

(xix) Change in internal energy

(a) For isobaric process = U = CvT(b) For isochoric process = U = CvT

(c) For isothermal process = U = 0

(d) For adiabatic process: U = W =( )( )1

TTR 12

(xx) Elasticities of gases(a) Isothermal bulk modulus = BI= P

(b) Adiabatic bulk modulus BA= P

(xxi) For a CYCLIC process, work done W = area enclosed in the cycle on PV diagram.Further, U = 0 (as state of the system remains unchanged)So, Q = W

(xxii) Internal energy and specific heats of an ideal gas (Monoatomic gas)

(a) U =2

3RT (for one mole);

(b)2

3U= RT (for moles)

(c) U =2

3 RT (for moles);

(d) Cv= R2

3

!

U

1='

(

)*+

,


27/35




27

(e) Cp= Cv + R =2

3R + R =

2

5R

(f) = 67.13

5R

2

3R

2

5

C

C

v

p=='

(

)*+

,=''

(

)

**

+

,

(xxiii) Internal energy and specific heats of a diatomic gas

(a)2

5U= RT (for moles);

(b) U =2

5 RT (for moles)

(c) Cv= ;R2

5

T

U1='

(

)*+

,

(d) Cp= Cv+ R =2

5R + R =

2

7R

(e) 4.15

7

2

R5

2

R7

C

C

v

p=='

(

)*+

,='

'(

)**+

,=

(xxiv) Mixture of gases: = 1+ 2

21

221121

NN

mNmNMMM

+

+=

+

++=

21

21

21

21

21

21

+

+=

+

+= 2121

ppp

vvv

CCCand

CCC

(xxv) First law of thermodynamics

(a) Q = U + W or U = Q W(b) Both Q, W depends on path, but U does not depend on the path(c) For isothermal process: Q = W = RT log | V2/V1|, U = 0, T = Constant, PV = Constant

and Ciso=

(d) For adiabatic process: W =( )( )

,1

TTR 12

Q = 0, U = Cv(T2T1), Q = 0,

PV= constant, Cad= 0 and

+==2

1C

C

v

p

(where is the degree of freedom)

(e) For isochoric process: W = 0, Q = U = CvT, V = constant, and Cv= (R/1)(f) For isobaric process: Q = CpT, U = CvT., W = RT, P = constant and

Cp= (R/1)

(g) For cyclic process: U = 0, Q = W(h) For free expansion: U = 0, Q = 0, W = 0(i) For polytropic process: W = [R(T2T1)/1n], Q = C (T2T1), PV

n= constant and

n1

RRC

+

1=

(xxvi) Second law of thermodynamics(a) There are no perfect engines(b) There are no perfect refrigerators


28/35




28

(c) Efficiency of carnot engine: = 11

2

1

2

1

2

T

T

Q

Q,

Q

Q=

(d) Coefficient of performance of a refrigerator:

=21

2

21

22

TT

T

QQ

Q

W

Q

orrefrigeratondoneWork

reservoircoldfromabsorbedHeat

=

==

For a perfect refrigerator, W = 0 or Q1= Q2 or =

(xxvii)The amount of heat transmitted is given by: Q = KA tx

, where K is coefficient of thermal

conductivity, A is the area of cross section, is the difference in temperature, t is the time of heatflow and x is separation between two ends

(xxviii) Thermal resistance of a conductor of length d = RTh=AK

d

(xxix) Flow of heat through a composite conductor:

(a) Temperature of interface, =( ) ( )

( ) ( )22112211

d/Kd/K

d/Kd/K

+

+ 21

(b) Rate of flow of heat through the composite conductor: H =( )

( ) ( )2211 K/dK/dA

t

Q

+

= 21

(c) Thermal resistance of the composite conductor

( ) ( )2Th1Th2

2

1

1TH RR

AK

d

AK

dR +=+=

(d) Equivalent thermal conductivity, K =( ) ( )2211

21

K/dK/d

dd

+

+

(xxx) (a) Radiation absorption coefficient: a = Q0/Q0

(b) Reflection coefficient: r = Qr/Q0(c) Transmission coefficient: t = Qt/Q0(d) Emissive power: e or E = Q/A .t [t = time]

(e) Spectral emissive power: e=( )dAtQ

and e = ee

(f) Emissivity: = e/E ; 0 1(g) Absorptive power: a = Qa/Q0

(h) Kirchhoffs law: (e/a)1 = (e/a)B= = E

(i) Stefans law: (a) E=T4(where =5.67x108Wm2 K4)For a black body: E = (T4T0

4)

For a body: e = (T4T04)

(j) Rate of loss of heat: )(Adt

dQ 40

4 =

For spherical objects:( )

( ) 22

21

2

1

r

r

dt/dQ

dt/dQ=

(k) Rate of fall of temperature: ( ) ( )404404

=

=

sV

A

ms

A

dt

d

( )( ) 1

2

1

2

2

1

2

1

r

r

V

Vx

A

A

dtd

dtd==

/

/

-0


29/35




29

(For spherical bodies)

(l) Newtons law of cooling:dt

d= K (0) or (0) e

KT

(m) Weins displacement law: mT = b (where b = 2.9 x 103

m K)

(n) Weins radiation law: Ed=

A'(

)*+

,

5 (T) d= '

(

)*+

,

5

A

e

a/T

d

(o) Solar Constant: S =2

ES

S

R

R

''(

)**+

, T4 or T =2/1

S

ES4/1

R

R

S''(

)**+

,'(

)*+

,


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30

WAVES

1. Velocity: v = nand n = (1/T)

2. Velocity of transverse waves in a string: v =dr

T

m

T2

=

3. Velocity of longitudinal waves:(a) In rods: v = (Y/) (Y Youngs modulus, = density)(b) In liquids: v = (B/) (B = Bulk modulus)(c) In gases: v = (P/) (Laplace formula)

4. Effect of temperature:

(a) v = v0(T/273) or v = v0+ 0.61t(b) (vsound/vrms) = (/3)

5. Wave equation: (a) y = a sin2

(vtx)

(b) y = a sin 2 '(

)*+

,

x

T

t

(c) y = a sin (t kx), where wave velocity v = =

nk

6. Particle velocity: (a) vparticle= (y/t)(b) maximum particle velocity, (vparticle)max = a

7. Strain in medium (a) strain = (y/x) = ka cos (t kx)(b) Maximum strain = (y/dx)max= ka(c) (vparticle/strain) = (/k) = wave velocityi.e., vparticle= wave velocity x strain in the medium

8. Wave equation:''

(

)

**

+

,

=

2

22

2

2

x

yv

t

y

9. Intensity of sound waves:(a) I= (E/At)

(b) If is the density of the medium; v the velocity of the wave; n the frequency and a theamplitude then I= 22v n2a2 i.e. In2a2

(c) Intensity level is decibel: 10 log (I/I0). Where, I0=Threshold of hearing = 1012

Watt/m2

10. Principle of superposition: y = y1+ y2

11. Resultant amplitude: a = (a12+ a2

2+ 2a1a

2cos )

12. Resultant intensity: I= I1+ I2+ 2(I1I2cos )(a) For constructive interference: = 2n, amax= a1+ a2and Imax= (I1+ I2)

2

(b) For destructive interference: = (2n1) , amin= a2a2and Imin= (I1=I2)2

13. (a) Beat frequency = n1 n2and beat period T = (T1T2/T2T1)(b) If there are N forks in successive order each giving x beat/sec with nearest neighbour, then

nlast= nfirst+ (N1)x

14. Stationary waves: The equation of stationary wave,(a) When the wave is reflected from a free boundary, is:

y = + 2a cos tsinkxcosa2T

t2sin

x2=


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31

(b) When the wave is reflected from a rigid boundary, is:

Y + 2a sinT

t2cos

x2

=2a sin kx cos t

15. Vibrations of a stretched string:

(a) For fundamental tone: n1=

m

T1

(b) For p th harmonic : np=m

Tp

(c) The ratio of successive harmonic frequencies: n1: n2: n3:.. = 1 : 2 : 3 :

(d) Sonometer:m

T

2

ln

!= (m = r2d)

(e) Meldes experiment: (i) Transverse mode: n =m

T

2

p

!

(ii) Longitudinal mode: n =m

T

2

p2

!

16. Vibrations of closed organ pipe

(a) For fundamental tone: n1= '(

)*+

,

L4

v

(b) For first overtone (third harmonic): n2= 3n1(c) Only odd harmonics are found in the vibrations of a closed organ pipe

and n1: n2: n3: ..=1 : 3 : 5 :

17. Vibrations of open organ pipe:(a) For fundamental tone: n1= (v/2L)(b) For first overtone (second harmonic) : n2= 2n1(c) Both even and odd harmonics are found in the vibrations of an open organ pipe and

n1: n2: n3: =1 : 2 : 3 : .

18. End correction: (a) Closed pipe : L = Lpipe+ 0.3d(b) Open pipe: L = Lpipe+ 0.6 d

where d = diameter = 2r

19. Resonance column: (a) !1+e =4

; (b) !2+ e =

4

3

(c) e =2

3 12 !! ; (d) n =( )

( )1212

2or2

v!!

!!=

20. Kundts tube:rod

air

rod

air

v

v

=

21. Longitudinal vibration of rods(a) Both ends open and clamped in middle:

(i) Fundamental frequency, n1= (v/2!)(ii) Frequency of first overtone, n2= 3n1(iii)Ratio of frequencies, n1: n2: n3: = 1 3: 5 : ..

(b) One end clamped

(i) Fundamental frequency, n1= (v/4!)(ii) Frequency of first overtone, n2= 3n1


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32

(iii) Ratio of frequencies, n1 : n2: n3: = 1 : 3 : 5 :

22. Frequency of a turning fork: n E

t

2!

Where t = thickness, != length of prong, E = Elastic constant and = density

23. Doppler Effect for Sound(a) Observer stationary and source moving:

(i) Source approaching: n =vv

v

sx n and =

v

vv s x

(ii) Source receding: n =svv

v

+x n and =

v

vv s+ x

(b) Source stationary and observer moving:

(i) Observer approaching the source: n =v

vv 0+ x n and =

(ii) Observer receding away from source: n =vvv 0 x n and =

(c) Source and observer both moving:

(i) S and O moving towards each other: n =s

0

vv

vv

+x n

(ii) S and O moving away from each other: n =s

0

vv

vv

+

x n

(iii) S and O in same direction, S behind O : n =s

0

vv

vv

x n

(iv) S and O in same direction, S ahead of O: n=s

0

vvvv

++ x n

(d) Effect of motion of medium:sm

0m

vvv

vvv'n

=

(e) Change in frequency: (i) Moving source passes a stationary observer: n=2s

2

s

vv

vv2

x n

For vs


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33

(b) Source between observer and wall

n = nxvv

v

s+ (for direct waves)

n =svv

v

x n (for reflected waves)

(ii) Source moving away from wall(a) Observer between source and wall

n = nxvv

v

s+ (for direct waves)

n =svv

v

+x n (for reflected waves)

(b) Source between observer and wall

n = nxvv

v

s (for direct waves)

n =svv

v

+x n (for reflected waves)

(g) Moving Target:(i) S and O stationary at the same place and target approaching with speed u

n = nxuv

uv'(

)*+

,

+

or n = nxv

u21 '

(

)*+

,+ (for u


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34

(c) Doppler Broadening = 2= 2 '(

)*+

,

c

v

(d) Transverse Doppler effect:

For light, n = nxc

v

2

11nx

c

v1

2

2

2

2

''

(

)

**

+

,= (for v


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35

STUDY TIPS

Combination of Subjects

Study a combination of subjects during a day i. e. after studying 23 hrs of mathematicsshift to any theoretical subject for 2 horrs. When we study a subject like math, aparticular part of the brain is working more than rest of the brain. When we shift to atheoretical subject, practically the other part of the brain would become active and thepart studying maths will go for rest.

Revision

Always refresh your memory by revising the matter learned. At the end of the day youmust revise whatever youve learnt during that day (or revise the previous days workbefore starting studies the next day). On an average brain is able to retain the newlylearned information 80% only for 12 hours, after that the forgetting cycle begins. Afterthis revision, now the brain is able to hold the matter for 7 days. So next revision shouldbe after 7 days (sundays could be kept for just revision). This ways you will get rid of the

problem of forgetting what you study and save a lot of time in restudying that topic.

Use All Your Senses

Whatever you read, try to convert that into picture and visualize it. Our eye memory ismany times stronger than our ear memory since the nerves connecting brain to eye aremany times stronger than nerves connecting brain to ear. So instead of trying to mug upby repeating it loudly try to see it while reapeating (loudly or in your mind). This isapplicable in theoritical subjects. Try to use all your senses while learning a subjectmatter. On an average we remember 25% of what we read, 35% of what we hear, 50%of what we say, 75% of what we see, 95% of what we read, hear, say and see.

Breathing and Relaxation

Take special care of your breathing. Deep breaths are very important for relaxing yourmind and hence in your concentration. Pranayam can do wonders to your concentration,relaxation and sharpening your mined (by supplying oxygen to it). Aerobic exercises likeskipping, jogging, swimming and cycling are also very helpful.

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