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