Unit 1: Physical World and Measurement

Important Results and Formulae:1. Fundamental Quantities:

Fundamental quantity Units Symbol (a) Length Metre

(b) Mass Kilogram (c) Time Second

(d e)

Electric current Ampere A

Thermodynamic temperature Luminous intensityy

Kelvin K

( Candela Cd

Amount of Substance Mole Mol

2 Dimensional Formulae (for topics related to class XI only) Symbol Quantity

Displacement Formula S.I. Unit D.F.

Metre or m M°LT

xb (Metre) or (Metre or

Area M°L?T°

Volume Ixbxh M°L

MLT- Velocity m/s

MLT M°LT?

Momentum kgm/s

Acceleration m/s a At

Force Newton or N MLT Ma

Impulse N.sec MLT-1 FXt

ML'T?

MLT-2

Work W F.d N.m

Energy KE or U K.E. = mo Joule or ]

ML'T-3

P Power W watt or W

Density d= mass/volume kg/m MLT

P=F/A Pascal or Pa ML Pressure Torque ML'T2

M°LOTO

T=rxF N.m.

radian or rad Angular displacement radius

rad/sec M°LT? Angular velocity

rad/sec ML T Angular acceleration

a At

ML'T MLT1

Moment of Inertia | kg-m I m

Angular kgm Jor L J mor

momentum S

Frequency voff hertz of Hz M°LT

Stress F/A N/m ML-'T?

A. AA AV

TAV Strain M°L°T°

Youngs modulus

(Bulk modulus) F/A N/7 ML-IT2

Surface tension MLOT2

Force constant F= km N/m MLOT2

(Spring)

Coefficient of kg/ms(poise in

C.G.S) ML T viscosity

Gm,n 2

N-m

Gravitational M'LT?

constant Fr G=

m2 Gravitational Vg PE

Constant M°L?T2 kg

Kelvin or K_

Joule or Calorie M°L°T°9*1

ML2T-2 Temperature

Heat 2 Qm x Sx At

Joule kg. Kelvin

Specific heat S Q=mxSx At M°L?T2¢-1

Joule kg

Latent heat Q= mL M°LT2

Coefficient Q= KA(0-4 Joule m secK of thermal K MLT9-1

conductivity Oule

mol.K Universal gas PV nRT

MLPT9 R constant

Relative Error: Relative error = Aamean/mean

3. Four Fundamental Forces i) Gravitational force

Percentage Error: (ii) Electromagnetic force

(ii) Nuclear force (iv) Weak force

oa =(Aa mean/mean)x 100%

5. Combination of Errors

) Ifr = (a + b), then Ar = t [Aa + Ab]

Ci) Ifx= (a -b), then Ax =+ [An - Ab] Fg: Fw: Fe : Fs = 1: 103 106 1038

4. Absolute, Relative and Percentage Error

suppose the values obtained in several measurements are a1, @2 Bgr.

GHi) Ifx =ax b, then +

amean(@1 +02+03.-.+ a)/n Absolute Error: (iv) Ifx=a/b), then =t|

M mean"1, M2 = umean 2

(V) Ifr = 40, ***** *****

**** ****

M,=mean"n then -Mean Absolute Error:

Amean=(|Aa,|+| Aa2| +|4,lt| Aa, |)/n

Unit 2: Kinematics

Important Results and Formulae: 1. Speed = Distance travelled / time taken

Velocity = displacement / Time Acceleration = Change in velocity/ time taken

2. Kinematic Equations

B b,i+j+ k are two vectors

.a ( a)i +(.a)l + (2 a)k 6. Vector Joining two points

If P, (z1, y, z,) and P2 (T V z) are any two points, then

For accelerated For deceleration motion motion PP = (-z) i + Va-vi) + (a-z)

UU-at

s=ut+ a s=ut- a 7. Resolution of Vectors

|=+2as =u2-2as A = A cos e

A, A sin 8

A = A+A |5, =u+(2n-1) |S,=u+(2-1)

3. The unit vector in the direction of tan 6 Y

A a is given byand is represented by

4. Addition of Vectors

If a a,i +ai +ak and b

are two vectors

a +b ( +b) í + (a,+ bi) + (a,+b,) k 5. Scalar Multiplication

ai+j+a,k and If A?

8. Triangle Law of Vectors

Let the vectors a andb be so positioned that initial point of one coincides with terminal

B sin6 tan A+B cos8e 11. Scalar or Dot Product

The scalar or dot product of two given vectors

a+b having an angle 0 between them is

point of the other, a = AB, b = BC. Then,

the vector is a + b is respresented by the third side of AABC i.e, defined as

a.b =1aI cos e f a =ai +a,i +a, k and b = bi +b,j +

b are two vectors a.b = , b, + a2 b, + a b

Properties of Scalar Product:

7- (a) a.b is a scalar quantity.

a (b) When 8=0, a - b =| all b AB+BC = AC

(c)When 0= * a.b =| a || b |cos=0 or AC-AB = BC

When a L b, a . b = 0

and AC-BC = AB (d) When either a = 0 or b= 0 a b = 0.

This is known as triangle law of a addition,

9. Parallelogram law of addition: If the two

vectors a and b are represented by the two

adjacent sides OA and OB of a parallelogram

OACB, then their sum a +b is represented

in magnitude and direction by the diagonal AC of parallelogram through their common point

te ii-ij=kk =1

12. Vector or Cross Product

The cross product of two vectors a and b

having angle 6 between them is given as

axb =| al| b Isin 0 i, O.

Where is a unit vector perpendicular to the

plane containing a and b

ai +a +ak and b =b,i +, i +bs fi

k are two vectors

a xb =

ie, OA +OB = OC

10. Magnitude and Direction of Resultant in Triangle Law and Parallelogram Law

(ba-bc) i + (a -c,e)I +(a,h,-ab,k

Properties of Vector Product: (a)a x b = | a || b |sin . i

(1) If a =0 or b = 0,a x b =0.

i) a llb, a x b =0(or @ = 0) (b) ax b is a vector ie,, a x b =- bxa

axb # b xa

R= VA+B+2ABcos e axb is not commutative.

positive a

(c) When =,a x b = |a|| b]xh

orl a x bl= | a|| b1.

(d) ixi =ixi =hxk =0

and i x i= , i xk =i,kxi =i

ixi--k, E xi =-i, i xk =-i.

at rest positive acceleration

negative a a0

(e) Sin e= axb

( If a and b represent adjacent sides of a negative acceleration Zero acceleration

parallelogram, then its area| a x b|. Velocity time Graph

(g) If a, b represent the adjacent sides of a

triangle, then its area= |axb| 13. Angle Between Two Vectors

If a =a,i +42} +ay k and b = b, i + bi b3 k are two vectors

Motion in positive direction with positive acceleration

Motion in positive direction with negative acceleration a.b

cos =

14. Projectile Motion Path followed by the projectile is parabola.

Velocity of projectile at any instant t,

V= [(u*-24gtsin 8 + g"]/2 Horizontal range

R Sin20 Motion in negative direction

with negative acceleration

Motion of an object with negative acceleration that

changes direction at time For maximum range e= 45,

Rnax 4"/g Flight time T = 2usin /s

Height

Relative Velocity

IfvVVa=V0 H=sin'e

48 r (m 3

40 For maximum height 6 = 90°

max =u"/2g

Important Graphs: Position time Graph

T245 6

with positive velocity with negative velocity

120 Ifva Vg=V4 is negative

140 120

x (m100

100 x (mg0

0 If V=V is positive +0

201 50

3 () B 40 20

20

40

Unit 3: Laws of Motion

4. Lami's Theorem > Important Results and Formulae:

1. Newton's Laws:

(1) First Law: It gives the concept of inertia. That is mass is the measure of the inertia

of the body.

(ii) Second Law:

Fa p dt

F mdo dt

2

F ma

Pa (ii) Third Law:

-Fa 2. Linear Momentum

F3 p mv

Sina Sinß Sin y Conservation of momentum

5. Types of Frictional Forces

1. Static frictional force Kinetic frictional force

p Constant

Rolling frictional force

6. Angle of friction (0) and angle of repose (4)

tan 6 = H

p initial= p final

3. Impulse

tan 4 I =FvgX ie.,

I =p2-P1

7. Circular Motion

rc

Angular Displacement radius Centripetal Acceleration

r

a= rw Angular Velocity

Centripetal Force Relation Between Angular Velocity and Linear

Velocity F. = miw

W= 8. Condition for Banking of Road

Without friction = rw

Time Period of Uniform Circular motion tane

With Friction

T47

W =RgH+ tan 8

Dmax 1-H, tan® ) Different Cases for Application of Second and Third Law (Without Friction)

Case Equation

N

N= mg

mg

N 7mg F ma

ng

N

N mg

F F-F= ma

1 sine

Fi

F2 N+F, sin 6 = mg

Fi sos -F2= ma

N= mg cos 6

gsine mg sin 6 ma

ngcose

*************

N=mg cos F-F2-mg sin mgsin

ma

mgcos mg

N, N

N = m,8

N, m F-T= m14 T r

m28

N

N m8 mi8 T = m,a

m2g-T= ma

Im28

Different Cases for Application of Second and Third Law (Without Friction):

Case Equations

N

N mg

F=f=4N

N Fsine

N+Fsin = mg

Fcos 0-= ma

Fcos -HN= ma Fcost

HaN

mg

max)H

A N= mg cos 8

mg sin =f.=4, N mgsine

mg Cosg

N= mg Cos mgsine

mg cos 6-= ma

mg sin 6-HN= ma

mgcos 68

N= mg cos

F-f mg sin 8 = ma

F-HN-mg sin 6 = ma

ngsine

mgcose

****

Unit 4: Work, Energy and Power 6. Work Energy theorem: Important Results and Formulae:

1. Different Formulae for Work Done: W (Fcos6) x s W = Fscos

W = Fs

W- mo- mo

W =K-K,

W :AK dW F.ds 7. Power

AW w-F. Total work done avg Total time taken

Pinst Lim Pavg W JF.Ddt

Pinst Lim AW

2. Different cases of Work Done At

Positive Work Done: PnstF.v = 0

8. Coefficient of Restitution: e=1 for perfectly elastic collision e=0 for perfectly inelastic collision

ec1 for perfectly elastic and inelastic collision

W = Fs cos

V = Fs cos 0

W= Fs Zero Work Done:

9. Elastic Collision in 1-D 0 Velocities of both bodies after collision

W = Fs cos 0

2m,42 W = Fs cos 90

W =0 (7 +ma) (m1 + ma)

2m,4 (7 +ma)

Negative Work Done: 0 (m +ma)

W = Fs cos W= Fs cos 180

10. Inelastic Collision on 1-D

W=-Fs (m +m2) 3. Kinetic Energy:

Important Graphs and Figures: Parabolic plots of the potential energy V and kinetic energy K of a block attached to a spring.

K.E. =mv 2

K Zm

4. Potential Energy: P.E. = mgh

5. Restoring Force and Elastic Potential energy FR kx

-E -K+V

E.P.E. kx? m

Unit 5 Motion of System of Particles and Rigid Body Important Results and Formulae

1. Centre of Mass:

m +2+ ma t.+

(vi)M.L. of a solid sphere about its diameter,

IMR M

2. Torque (vii) M.I. of spherical shell about its diameter,

-TxF 1-MR 3. Angular Momentum: Comparison between Linear and Rotational

Motion:

L XPy-YPL=Tp Sn o

4. Relation between Torque and Angular

Linear Motion Rotational Motion Na

Angular displacement (s)displacement (6)_

Angular velocity,

Distance/ Momentum

2. Linear velocity,

dt as d6

dt 5. Law of Conservation of Angular Momentum dt

Linear Angular 12 = a constant acceleration, acceleration =

6. Moment of Inertia: I= m,+ m +m, "+.+m, dr

dr a a

t dt

Mass (m) Moment of inertia 7. Radius of gyravation

+ K= Linear Angular momentum, momentum,

8. Theorem of perpendicular axes:

=+1y 9. Theorem of parallel axes:

+Mh

P mv L=I b. Forece, Torque, t= la

F ma

Also, force, Also, torque, 10. K.E. of rotation= lo

iL F

dt T

11. Moment of inertia and torque 1t

T 12. MOI of some bodies

Translational KE, Rotational KE,

) MIL of a rod about an axis through its C.m. and perpendicular to rod, Kale

Work done, IML Work done,

W Fs W t (ii) M.I of a circular ring about an axis through

its centre and perpendicular to its plane,

I= MR? (ii) M.. of a circular disc about an axis through

its centre and perpendicular to its plane,

10. Power, Power, P=Fo P=t

Equation of

translator motion

|11. (i) =u + at

Equations of rotational motion

IMR )0 + at iv)M.I. of a right circular solid cylinder about

its symmetry axis, Gi) s=ut+at)=ot+ar I=MR

Cii) - u = 2as (ii) o - a = 2a0 (v) M.I. ofa right circular hollow cylinder about

its axis = MRE

Unit 6: Gravitation

>Important Results and Formulae 6. Gravitational potential energy GM

1. Kepler's Law U = - Xm 1. Law of orbitsAll planets move in

eliptical orbits with the Sun situated at one of the foci of the ellipse. Law of areas: The line that joins any

planet to the sun sweeps equal areas in

equal intervals of time.

3. Law of periods :The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the elipse traced out by the planet.

7. Escape speed

2GM 2gR V. R 8. Satellite:

Orbital Velocity

V =RR+h T =

2. Coulomb's Law Time Period

CR+h T 2 GM F Gh

Where G= 667 x 10-11Nm2kg? 3. Acceleration due to gravity (g):

GM G R2 T-2Rh

8

Height of Satellite 4. Variation of acceleration due to gravity: (a) Effect of altitude, g' =g(1-2h/R) (b) Effect of depth g' = 8(1-d/R) (c) Effect of rotation of earth:

8 g- Rat cos

21/3

Total energy of satellite -GMm

E = P.E. + KE =

2(R+h) 5. Gravitational potential

W_GM V r