physics review

Physical World

1. Physics deals with the study of the basic laws of nature and their manifestation in different phenomena.

2. Gravitational Force

It is the force of mutual attraction between any two objects by virtue of their masses. It is a universal force. It plays a key role in the large-scale phenomena occurring in the

universe, such as formation and evolution of stars, galaxies and galactic clusters.

3. Electromagnetic Force

Electromagnetic force is the force between protons is 1036 times the gravitational force between them, for any fixed distance.

It is mainly the electromagnetic force that governs the structure of atoms and molecules

4. Strong Nuclear Force

The strong nuclear force binds protons and neutrons in a nucleus. The strong nuclear force is the strongest of all fundamental forces, it is charge independent and acts equally between a proton and a proton, a neutron and a neutron, and a proton and a neutron. Its

range is, extremely small, of about nuclear dimensions (1015m). It is responsible for the stability of nuclei.

5. Weak Nuclear Force

The weak nuclear force appears only in certain nuclear processes such as the decay, the nucleus emits an electron and an uncharged particle called neutrino. The weak nuclear force is not as weak as the gravitational force, but much weaker than the strong nuclear and electromagnetic forces. The range of weak nuclear force is exceedingly

small, of the order of 1016 m.

6. Nature of Physics Laws The physical quantities that remain unchanged in a process are called conserved

quantities. Some of the general conservation laws in nature include the laws of conservation of mass, energy, linear momentum, angular momentum, charge etc. some

conservation laws are true for one fundamental force but not for the other. Conservation laws have a deep connection with symmetries of nature. Symmetries of space and time,

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and other types of symmetries play a central role in modern theories of fundamental forces in nature.

7. Sir C.V. Raman (1888-1970) The Raman Effect deals with scattering of light by molecules of a medium when they are

excited to vibration energy levels.

8. Satyendra Nath Bose (1894 - 1974) Bose gave a new derivation of Plancks law, treating radiation as a gas of photons and

employing new statistical methods of counting of photon states. An important consequence of Bose-Einstein statistics is that a gas of molecules below a certain temperature will undergo a phase transition to a state where a large fraction of atoms populate the same lowest energy state.



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Units and Measurements

1. The numerical value (n) of a physical quantity is inversely proportional to the unit (u) in which it is expressed.

1n

u (Or) n1u1 = n2u2.

2. The quantities that are independent of other quantities are called fundamental quantities. The

units of these fundamental quantities are called fundamental units.

3. The quantities that are derived from fundamental quantities are called derived quantities. The units of these derived quantities are called derived units.

4. The basic systems of units :

5. Fundamental quantities in SI system :

Physical quantity Unit Symbol

Length Metre m

Mass Kilogram kg

Time Second s

Electric current Ampere A

Thermodynamic temperature Kelvin K

Intensity of light Candela cd

Quantity of substance Mole mol

Fundamental Quantity System of units

C.G.S. M.K.S. F.P.S.

Length Centimeter Metre Foot

Mass Gram Kilogram Pound

Time Second Second Second



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Supplementary quantities

Plane angle Radian rad

Solid angle Steradian sr

13. Metre: A metre is equal to 1650763.73 times the wavelength of the light emitted in vacuum due to electronic transition from 2p10 state to 5d5 state in Krypton86. But in 1983, 17th General Assembly of weights and measures, adopted a new definition for the metre in terms of velocity of light. According to this definition, metre is defined as the distance travelled by

light in vacuum during a time interval of 1/299, 792, 458 of a second.

14. Kilogram: The mass of a cylinder of platinumiridium alloy kept in the International Bureau

of weights and measures preserved at Serves near Paris is called one kilogram.

15. Second: The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of caesium133 atom is called one

second.

16. Ampere : The current which when flowing in each of two parallel conductors of infinite length and negligible crosssection and placed one metre apart in vacuum, causes each conductor to experience a force of 2x107 Newton per metre of length is known as one

ampere.

17. Kelvin: The fraction of 1/273.16 of the thermodynamic temperature of the triple point of water is called Kelvin.

18. Candela: The luminous intensity in the perpendicular direction of a surface of a black body of area 1/600000 m2 at the temperature of solidifying platinum under a pressure of 101325 Nm2 is known as one candela.

19. Mole: The amount of a substance of a system which contains as many elementary entities as there are atoms in 12x103 kg of carbon12 is known as one mole.

20. Radian: The angle made by an arc of the circle equivalent to its radius at the centre is known as radian. 1 radian = 57o17l45ll.

21. Steradian: The angle subtended at the centre by one square metre area of the surface of a sphere of radius one metre is known as steradian.



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22. Dimensions of a physical quantity are the powers to which the fundamental units are raised

to obtain one unit of that quantity.

23. The expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity is called the dimensional formula of that quantity.

24. Dimensional Constants: The physical constants which have dimensions are called dimensional constants. Eg: Gravitational constant (G), Plancks constant (h), Universal gas constant (R) etc.

25. Dimensionless quantities: The physical quantities which do not have dimensions are called

dimensionless quantities

a) Dimensionless quantities without units. Eg: numbers, , sin, tan.etc., b) Dimensionless quantities with units. Eg : Angular displacement radian, Joules constant joule/calorie, etc.,

26. Dimensional variables: The physical quantities which have dimensions and do not have fixed value are called dimensional variables. Eg: Velocities, acceleration, force, work, etc.

27. Dimensionless variables: The physical quantities which do not have dimensions and do not have fixed value dimensions are called dimensionless variables. Eg: Specific gravity,

refractive index, coefficient of friction, etc.

28. Uses of dimensional formulae: These are used to a) verify the correctness of a physical equation, b) derive relationship between physical quantities and c) to convert the units of a physical quantity from one system to another system.

29. Principle of homogeneity: In any correct equation representing the relation between physical quantities, the dimensions of all the terms must be the same on both sides.

Quantities having same dimensions can only be added or subtracted or equated. 30. Limitations of dimensional analysis

1. Dimensionless quantities and proportionality Constants cannot be determined by this

method.

2. This method is not applicable to trigonometric, logarithmic and exponential functions.

3. In the case of physical quantities which depend upon more than three physical quantities,

this method will be difficult.

4. If the constant of proportionality also possesses dimensions, this system cannot be used.



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If one side of equation contains addition or subtraction of physical quantities, this method

cannot be used

Dimensional formulae for some physical quantities

Physical quantity Unit Dimensional

formula

Boltzmanns constant JK1 ML2T2 1

Bulk modulus Nm2, Pa M1L1T2

Coefficient of linear or areal or volume

expansion oC1 or K1 1

Surface tension Nm1 or Jm2 MT2

Thermal conductivity Wm1K1 MLT3 1

Coefficient of viscosity (F = dxdvA ) poise ML1T1

Compressibility Pa1, m2N2 M1LT2

Electric capacitance CV1, farad M1L2T4I2

Electric conductance Ohm1 or mho or siemen

M1L2T3I2

Electric conductivity siemen/metre or Sm1

M1L3T3I2

Electric charge or quantity of electric charge coulomb IT

Electric current ampere I

Electric dipole moment Cm LTI

Intensity of electric field NC1, Vm1 MLT3I1

Electric resistance ohm ML2T3I2

Emf (or) electric potential volt ML2T3I1 Energy density Jm3 ML1T2

coefficient of self induction Henry (H) ML2T2I2



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Intensity of gravitational field Nkg1 L1T2

Intensity of magnetization Am1 L1I

Joules constant or mechanical equivalent of

heat Jcal1 MoLoTo

Latent heat Jkg1 MoL2T2

Magnetic dipole moment Am2 L2I

Magnetic flux Weber (Wb) ML2T2I1 Magnetic induction NI1m1 or T MT2I1

Magnetic pole strength Am LI

Modulus of elasticity Nm2, Pa ML1T2

Moment of inertia kgm2 ML2

Momentum kgms1 MLT1

Permeability of free space Hm1 or NA2 MLT2I2

Permittivity of free space Fm1 or C2N1m2 M1L3T4I2

Plancks constant Js ML2T1

Poissons ratio MoLoTo

Pressure coefficient or volume coefficient oC1 or 1 1

Radioactivity disintegrations per second

MoLoT1

Refractive index MoLoTo

Resistivity or specific resistance m ML3T3I2

Specific conductance or conductivity siemen/metre or

Sm1 M1L3T3I2

Specific heat Jkg1 1 MoL2T2 1

Stefans constant Wm2 4 MLoT3 4

Universal gravitational constant Nm2kg2 M1L3T2



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Quantities having same dimensions a) Work, energy, torque, moment of force, energy b) Angular momentum, Plancks constant, rotational impulse c) Stress, pressure, modulus of elasticity, energy density. d) Force constant, surface tension, surface energy. e) Angular velocity, frequency, velocity gradient f) Gravitational potential, latent heat. g) Thermal capacity, entropy, universal gas constant and Boltzmann are constant.



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Measurements and Errors

Accuracy: Closeness of the measured value to the true value is called accuracy Precision: Closeness of the measurements done with an instrument to one another called Precision

E.g.: The time period of a seconds pendulum is T = 2 sec Clock A: 2.10 sec, 2.01 sec. 1.98 sec (Accurate) Clock B: 2.56 sec, 2.57 sec 2.57 sec (Precise)

Error: The difference between the measured value and true value of a physical quantity is called Error

Type of errors: 1) Systematic Errors 2) Random Errors and 3) Gross Errors 1) Systematic Errors: These may be either positive or negative. a) Constant or Instrumental errors: These are due to i) imperfect design and ii) zero error b) Imperfection in experimental arrangement: In the calorimeter experiment, the loss of heat

due to radiation, the effect on weighing due to buoyancy of air cannot be avoided. c) Environmental Errors like changes in temperature, pressure wind velocity etc.

d) Personal or Observational errors are due to the improper setting of the apparatus, carelessness in taking observations

2) Random Errors: These are due to fluctuations in temperature, voltage supply etc . Accurate value can be obtained by taking a number of readings and finding the arithmetic mean

of all the readings. 3) Gross Errors: These due to the carelessness of the observer in taking measurements towards the

sources of error.

In tangent galvanometer experiment, the coil should be placed in magnetic meridian position and other magnetic materials should be kept away. Neglecting these precaution result in gross errors

No corrections can be applied to these errors. Care should be taken to avoid these errors

Estimation of errors a) Absolute Errors ( )a : The magnitude of the difference between the true value of a physical

quantity and the individual measured value is called absolute error of that measurement

Absolute error = True value-measured value



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Or i mean i =

Absolute error is always positive. It has the same units as that of the quantity measured

b) Mean absolute Error ( )mean : The arithmetic mean of all the absolute errors is called mean absolute error (or) final absolute error

Mean absolute error

1 2 3 ..... nmean

n

+ + + + =

1

1 ni

in

=

=

Mean absolute error is always positive and has the same units as that of the measured physical quantity.

c) Relative Error: The ratio of mean absolute error to the mean value of the quantity measured is called relative error.

Relative error meanmean

=

Relative error has no units.

d) Percentage Error ( ) : When the relative error is multiplied by 100, it is called percentage

error 100 %meanmean

=

Combination of errors

a) Error of a sum or a difference i) If x = a + b Let a and b be the absolute errors in a and b respectively. Let the error in x be x

x = a + b

Maximum possible value of x a b = +

Relative error, x a bx a b

+ =

+

Percentage error, % 100 %x a bx a b

+ = +

ii) If x = a b Maximum possible value of x a b = +

Relative error, x a bx a b

+ =

Percentage error, % 100 %x a bx a b

+ =



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b) Errors of multiplication or Division i) If x = a b

Maximum relative error, x a bx a b

= +

Percentage error, % 100%x a bx a b

= +

ii) If axb

=

Maximum relative error % 100%x a bx a b

= +

c) Errors of a measured quantity that involves product of powers of observed quantities: i) If nx a=

Maximum relative error, % 100 %x anx a

=

ii) If p q

r

a bx

c=

Maximum relative error, x a b cp q rx a b c

= + +

Percentage error, % 100%x a b cp q rx a b c

= + +

Significant figure: Significant figures in a measurement are defined as the number of digits that are known reliably plus the uncertain digit

Rules for determining the number of significant figures 1. All the non-zero digits in a given number are significant without any regard to the location of

the decimal point if any E.g.: 4205, 42.05, 4.205, 420.5 all have 4 significant digits. 2. All zeros occurring between two non-zero digits are significant without any regard to the

location of decimal point if any E.g.: 2.002, 20.02, 200.2 all have 4 significant digits. 3. All the zeros to the right of the decimal point but to the left of the first non zero digit are not

significant

E.g.: 0.003 in these number significant digits are 1. 4. All zeros to the right of the last non zero digit in a number after the decimal point are

significant

E.g.: 0.4200 has 4 significant figures



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5. All zeros to the right of the last non zero digit in a number having no decimal point are not significant

E.g.: 4200 has 2 significant figures

But if the zeros are obtained from actual measurement, then the number of significant figures in 4200 are 4.

Rounding off 1. The preceding digit is raised by one if the immediate insignificant digit to be dropped is more

than 5 E.g.: When 4228 is rounded off to three significant figures, it becomes 4230 2. The preceding digit is to be left unchanged if the immediate insignificant digit to be dropped is

less than 5 E.g.: If 4228 is rounded off to two significant figures it becomes 4200 3. If the immediate insignificant digit to be dropped is 5 then there will be two different cases a) If the preceding digit is even, it is to be unchanged and 5 is dropped Eg: If 4.728 is to be rounded off to two decimal places, it becomes 4. 72 b) If the preceding digit is odd, it is to be raised by 1 E.g.: If 4.7358 is to be rounded off two decimal places it becomes 4.74

Rules for arithmetic operations with significant figures 1. Addition and subtraction: For addition and subtraction, the rule in terms of decimal places i) After completing addition or subtraction, round off the final result to the least number of

decimal places (n) Eg 1): Find the value of 2.2 + 5.08 + 3.125 + 5.3755

Ans: 15.78 is rounded off to 15.8 Eg (2): Find the value of 44.8 21.235

Ans: 23.565 is rounded off to 23.6 2. Multiplication and division

In multiplication and division, the rule is in terms of significant figures i) In a given set of numbers, notice the number with the least number of significant figures ( )n and the round off the other number to ( )1n + significant figures. Complete the arithmetic operation

ii) After completing multiplication or division round off the final result to the least number of significant figures ( )n


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E.g.: (1): Find the value of 1.2 2.54 3.257 1.2 2.54 3.26 9.93468 =

Final result is rounded off to 9.9

Eg: 2) Find 9.27 41

9.27 0.226097541

=

Final result is rounded off to 0.23


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Horizontal Motion

1. An object is said to be at rest, if the position of the object does not change with time with respect to its surroundings.

2. An object is said to be in motion, if its position changes with time with respect to its surroundings.

3. Rest and motion are relative.

Ex: A person travelling in a bus is at rest w.r.t. the co-passenger and he is in motion w.r.t. the person on the road.

4. Distance and Displacement a) The difference between the final and initial positions of a particle is known as

displacement.

Displacement x = xf xi

b) Displacement of a particle is the shortest distance between its initial and final position and directed form initial position to final position.

c) The length of the actual path covered by a particle in a time interval is called distance.

d) Distance is a scalar quantity and displacement is a vector quantity.

5. Speed a) Speed of a body is the rate at which it describes its path. Its SI unit is ms 1.It is a

scalar quantity.

Speed =taken timetravelled distance

.

b) A body is said to move with uniform speed, if it has equal distances in equal intervals of time, however small the intervals may be.

c) A body is said to move with non uniform speed, if it has unequal distances in equal intervals of time or equal distances in unequal intervals of time, however small the intervals may be.



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d) Average speed = Total distance Total time

e) Instantaneous speed =dtds

tsLt

0t=

.

f) If a particle covers the 1st half of the total distance with a speed v1 and the second half with a speed v2.

Average speed =21

21vv

vv2+

.

g) If a particle covers 1st 1/3rd of a distance with a speed v1, 2nd 1/3rd of the distance with speed v2and 3rd 1/3rd of the distance with speed v3.

Average speed = 1 2 31 2 3

3v v vv v v+ +

.

6. Velocity a) The rate of change of displacement of a body is called velocity. Its SI unit is ms1.It

is a vector quantity.

b) A body is said to move with uniform velocity, if it has equal displacements in equal intervals of time, however small these intervals may be.

c) If the direction or magnitude or both of the velocity of a body change, then the body is said to be moving with non-uniform velocity.

d) The velocity of a particle at any instant of time or at any point of its path is called

instantaneous velocity. V

=dtds

tsLt

0t=

7. Average velocity

a) Average speed = Total distance Total time

b) If a particle under goes a displacement s1 along a straight line t1 and a displacement s2 in time t2 in the same direction, then

Average velocity=21

21ttss

+

+

c) If a particle undergoes a displacement s1 along a straight line with velocity v1 and a displacement s2 with velocity v2 in the same direction, then



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Average velocity =1221

2121vsvs

vv)ss(+

+

d) If a particle travels first half of the displacement along a straight line with velocity v1 and the next half of the displacement with velocity v2 in the same direction, then


21vv

vv2+

e) If a particle travels for a time t1 with velocity v1 and for a time t2 with velocity v2 in the same direction, then


2211tt

tvtv+

+

f) If a particle travels first half of the time with velocity v1 and the next half of the time with velocity v2 in the same direction, then

Average velocity =2

vv 21 +

8. Acceleration a. The acceleration is defined as the time rate of change of velocity. b. The acceleration and velocity of a body need not be in the same direction. eg : A

body thrown vertically upwards. c. If equal changes of velocity take place in equal intervals of time, however small

these intervals may be, then the body is said to be in uniform acceleration. d. Negative acceleration is called retardation or deceleration. e. The acceleration of a particle at any instant or at any point is called instantaneous

acceleration.

a

=0 tt

v dvLtdt

=

f. A body can have zero velocity and non-zero acceleration. Eg: for a body projected vertically up, at the highest point velocity is zero, but acceleration is g.

g. If a body has a uniform speed, it may have acceleration. Eg : uniform circular motion h. If a body has uniform velocity, it has no acceleration.



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i. Acceleration of free fall is called acceleration due to gravity (g) and it is equal to 980 cms 2 or 9.8 ms 2.

9. The equations of motion for uniform acceleration 1) v = u + at 2) s = ut +

21

at2

3) v2 u2 = 2as 4) sn = u + 2

a (2n 1)

5) s =

+

2vu t

10. One dimensional motion (uniform acceleration) a) If a body starting from rest travels a distance Sm in mth second and Sn is in nth

second, then a =

n mS Sn m

.

b) If a particle travels along a straight line with uniform acceleration and travels distances Sn and Sn+1 in two successive seconds, the acceleration of the particle is

a = Sn+1 - Sn

c) If a particle travels along a straight line travels distances S1 and S2 in two successive intervals of n seconds each, the acceleration of the particle is

a = 2 12S S

n

d) If a body starting from rest, attains a velocity 'v' after a displacement 'x', then its velocity becomes 'nv' after a further displacement (n2 - 1)x.

e) If a bullet loses (1/n)th of its velocity while passing through a plank, then the

number of such planks required to just stop the bullet is = 1n2

n2

f) The first compartment of a train crosses a pole with a speed u

and the last compartment of the train crosses the pole with a speed v , the speed with which

the middle compartment of the train crosses the pole with a speed 2 2

2u vV +=



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g) Starting from rest a body travels with an acceleration '' for some time and then with deceleration '' and finally comes to rest. If the total time of journey is 't', then the maximum velocity and displacement are given by

2max

12

V t and s t

= = + +

Also, average velocity =

2Vmax

h) A body is projected vertically up from a topless car relative to the car which is moving horizontally relative to earth.

i. If the velocity of the car is constant, ball will be caught by the thrower.

ii. If the velocity of the car is constant, path of ball relative to the ground is a

parabola and relative to the car is straight up and then straight down.

iii. If the car accelerates, ball falls back relative to the car.

iv. If acceleration or retardation of the car is constant path relative to car is a

straight line and relative to ground is a parabola.



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Motion under Gravity

1. Freely falling body

a) The equations of motion i) V = u + at ii) s = ut + at2

V = gt h = gt2

V t h t2

2

1

2

1tt

VV

= 22

21

2

1

tt

hh

=

iii) V2 = u2 + 2as iv) sn = )12(2 + na

u

V2 = 2gh hn = 2g (2n-1)

V2 h hn (2n 1)

2

1

2

1hh

VV

=

b) The average velocity during fall (V) = 2

gh

c) The ratio of distance traveled in 1st, 2nd, 3rd.. n seconds is 1:3:5(2n-1) d) The ratio of distances traveled in first, first two, first three seconds is

1:4:9..n2

e) The ratio of time taken to travel first, 2nd, 3rdnth unit of distances is 23:12:1 ( )1 nn

f) The ratio of times taken to travel first, first two, first three first n units of distances is :3:2:1 .. n

g) If x is the distance traveled in the n th second, then the distance traveled in the (n +1)th

second is xn

n

+

1212

(or) x + g.

h) If x is the distance traveled in the n th second, then distance traveled in the (n-1) th

second is 2 32 1n

xn

(or) x - g



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i) The ratio of distances covered in the nth second and the distance traveled in n seconds is 22

1212n

n

nns

s n ==

j) + =n 1 nS s g

k) If a body travels n

1 th of the total distance in the last second the total time of fall

[ ])1( += nnnT l) If a particle takes x seconds less and acquires a velocity y ms-1 more at one place than at

another in falling through the same distance. If g1 and g2 are accelerations due to

gravity at these two places, then x: y is ( )21/1 gg . m) The acceleration of a body in a medium is given by gI = g

bm

dd1

=

b

m

dd1g

Where dm = density of the medium and db = density of the body

n) If a body is dropped into a well of depth h the time taken to hear the sound from start

(v is the velocity of sound) T is given by v

hghT += 2

2. Body thrown vertically upwards

a) The equations of motion a) v = u + at b) s = ut + at2 v = u gt h = ut gt2

c) V2 = u2 + 2as d) sn = u + )12(2 na

ghu 2= hn = u - )12(2 ng

b) Maximum height reached = H = g

u

2

2

c) Time of ascent = Time of descent = gu

Time of flight = 2ug

d) Maximum height H = 2 21

2 2 8gT gT

g

=

e) The velocity of the body at the half of the maximum height is gh (or) 2

2u



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f) A body projected vertically up from the top of a tower of height h reaches the ground

in a time t , then 221 gtuth += and h = g2uv 22

g) A body is projected up with a velocity u and another body is also projected up from the same point with same velocity but after t sec. Then they will meet after a time

2t

guT +=

h) A body projected up from the top of a tower with a velocity u reaches the ground in a time t1. Another body projected down with same velocity reaches the ground in time t2

i) The time difference (t1 t2) = gu2

ii) Time take by the freely falling body to reach the ground is 21tt

iii) Height of the tower is h = 21tgt21

iv) Velocity of projection is u = )tt(2g

21

i) If air resistance is considered, time of ascent < time of descent.



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Graphs

1. Displacement-time graph 1) Slope of straight line gives velocity 2) Smooth curves represents uniform acceleration 3) Zig zag curve represents non-uniform acceleration

Displacement Velocity Acceleration

1. At rest

O t

c

x

x = c

O t

v

O t

a

2. Motion with

constant

velocity

O t

x

x

= vot + xo

xo

O t

v

vo

O t

a

3. Motion with constant

acceleration

O t

x x

= vot +(1/2)aot2

O t

v

v = aot

O t

a

ao

4. Motion with

constant

deceleration.

O t

x x

= vot - (1/2)aot2

O t

v

vo

a0

t

a

O

2. Velocity-time graph



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1) Slope gives the acceleration. 2) Area under the graph gives the distance travelled 3) Curve represents non-uniform acceleration. 4) Straight line represents uniform acceleration.



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Vector Addition

1. Scalar: A physical quantity having only magnitude but no direction is called a scalar.

eg: Time, mass, distance, speed, electric charge, etc.

2. Vector: A physical quantity having both magnitude and direction and which obeys the laws of vector addition is called a vector quantity.

eg: Displacement, velocity, acceleration, intensity of electric field, etc.

3. Surface area can be treated both as a scalar and a vector. A is magnitude of surface area which is a scalar. If n is a unit vector normal to the surface, we can write A n as a vector.

4. Electric current and velocity of light are not vectors even though they have direction since they do not obey the laws of addition.

5. A vector quantity which has direction by its nature is called a polar vector. Ex: velocity. 6. A vector quantity which has direction by a convention is called a pseudo (or) axial (or)

non-polar vector. The direction of pseudo vector can be known from right hand thumb rule. Ex: Angular velocity.

7. Equal vectors: Vectors having same magnitude and which have same direction are called equal vectors. Their corresponding components are equal.

8. Negative vectors: A vector which has the same magnitude as that of another and which is opposite in direction is called a negative vector.

9. Null Vector (Zero Vectors): A vector whose magnitude is zero and which has no specific direction is called a null vector.

e.g. 1) The cross product of two parallel vectors is a null vector. 2) The difference of two equal vectors is a null vector. 10. Unit vector: It is a vector whose magnitude is unity. A unit vector parallel to a given

vector.

If A

is a vector, the unit vector in the direction of A

is written as | |AAA

=

. kandj,i are units

vectors along x, y and z axis.

11. Position vector: The position of a particle is described by a position vector which is drawn from the origin of a reference frame.The position vector of a

particle P in space is given by OP r xi yj zk= = + +

.

O

P



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Its magnitude is given by 2 2 2r x y z= + +

Unit vector of r is given by 2 2 2

r xi yj zkr

r x y z

+ += =

+ +

Addition of Vectors

12. Resultant can be found by using

a) Triangle law of vectors b) Parallelogram law of vectors c) Polygon law of vectors

13. Triangle law: If two vectors are represented in magnitude and direction by the two sides of a triangle taken in order, then the third side taken in the reverse order represents their sum or resultant both in magnitude and direction.

14. Parallelogram law

If two vectors Q and P

are represented by the two sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that

point.

R = ++ cosPQ2QP 22

Tan+

=

cosQPsinQ

and +

=

cosPQsinPtan

15. If the resultant R of P and Q makes an angle with P and with Q and if P Q> then < .

16. For two vectors P and Q , maxR P Q= + and minR P Q=

17. If two vectors P and Q have equal magnitudes x, then 2 cos2

R x =

18. Vectors addition obeys

a) Commutative law: ABBA +=+ b) Associative law: ( ) ( ) CBACBA ++=++ c) Distributive law: ( ) BmAmBAm +=+ where m is a scalar.

19. Polygon law: If a number of vectors are represented by the sides of a polygon taken in the same order, the resultant is represented by the closing side of the polygon taken in

the reverse order.

P

RQ



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20. Resolution of a vector

Consider a vector A represented along OA

in two co-ordinate system, which makes an

angle with X-axis.

x yA A i A j= + and 2 2x yA A A= +

x

AxCos A A CosA

= =

yy

ASin A A Sin

A = =

If the vector makes an angle with X-axis, with Y axis and with Z-axis Then x y zA A i A j A k= + + and 2 2 2x y zA A A A= + +

; yxAACos Cos

A A = = = = m and zACos m

A = =

2 2 2Cos Cos Cos + + = 1 (or) 2 2 2 1m n+ + = And 2 2 2 2Sin Sin Sin + + = (Law of cosines)

21. If 1 1 1, ,m n and 2 2 2m n the direction cosines of two vectors and is the angle between

them then cos = 1 2 1 2 1 2m m n n+ + .

22. Component of a vector is a vector.

23. If vectors kAjAiAA zyx ++=

and x y z B B i B j B k= + +

are parallel, then z

z

y

y

x

x

BA

BA

BA

== and

A

=KB

where K is a scalar.

Equilibrium

24. Equilibrium is the state of a body in which there is no acceleration i.e., net force acting on

a body is zero.

25. The forces whose lines of action pass through a common point are called concurrent

forces.

26. Resultant force is the single force which produces the same effect as a given system of forces acting simultaneously.

27. A force which when acting along with a given system of forces produces equilibrium is

called the equilibrant.

Ax

Ay

xA i

yA jA

X

Y

A

O



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28. Resultant and equilibrant have equal magnitude and opposite direction. They act along the same line and they are themselves in equilibrium.

29. Triangle law of forces: If a body is in equilibrium under the action of three coplanar forces, then these forces can be represented in magnitude as well as

direction by the three sides of a triangle taken in order. |R|r

|Q|q

|P|p

==

Where p, q, r are sides of a triangle. R,Q,P

are coplanar vectors.

30. Lamis theorem: When three coplanar forces R and Q ,P

keep a body in

equilibrium, then

== sinR

sinQ

sinP

.

31. If 0A....AAA n321 =++++

and A1 = A2 = A3 = An, then the adjacent vectors

are inclined to each other at an angle N2pi

or N

360.

32. N forces each of magnitude F are acting on a point and angle between any two adjacent

forces is , then resultant force Fresultant = NFsin2

sin( / 2)

.

33. Body Pulled Horizontally The horizontal force required to pull a suspended body

through an angle with the vertical is given by

sinT F = and cosT mg =

( )22 2T F mg= + FTan

mg =

F mg Tan =

Motion of the boat in a river

34. Let VB be the velocity of the boat and RV the velocity of the river.

1. The time taken by the

boat to go from A to B and B to A in still water 2B

dTV

=

2. If the river is flowing

Q

R

P

R r pq

P

Q

A B

mg

FT

l

x

l-x

22



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downB R

dt

V V=

+ and up

B R

dt

V V=

2 22 B

d uB R

d VT t tV V

= + =

3. ( )Bd V Sin t= Time taken by the boat to cross the river is

B

dt

V Sin=

If 90 ,o = then t is minimum .i.e. the boat can cross the river in a shortest time if it

moves along AB.

35. Shortest Path

R

B

VSinV

= Or 1 RB

VSinV

=

And 190 RB

VSinV

+

with stream

Resultant velocity = 2 2B RV V

Time taken to cross the river 2 2

B R

dt

V V=

36. Shortest time

Resultant velocity = 2 2B RV V+

Time taken to cross the river is B

dt

V=

Also R RB B

V VxTan x dV d V

= = =

37. Subtraction of two vectors

a) If Q and P are two vectors, then P Q is defined as P + )Q( where Q is the negative vector ofQ

.

If QPR

= , then += PQCos2QPR 22

In the parallelogram OMLN, the diagonal OL represents BA

+ and the diagonal NM

represents BA

BV cos

BV sin BV

B

d

A

2 2B RV VVB

A

BRVCRV

RV

d

BV

RVBx

C

A

2 2netV B RV V= +

d



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b) subtraction of vectors does not obey commutative law ABBA c) subtraction of vectors does not obey Associative law

C)BA()CB(A

d) Subtraction of vectors obeys distributive law m BmAm)BA( = .

38. For two equal vectors 22

R x Sin =

N L

AO

B

M



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Relative Velocity

1. Relative velocity: When the distance between two bodies is altering either in magnitude

or direction or both, then each is said to have a relative velocity with respect to the other.

Relative velocity is vector difference of velocities.

a. The relative velocity of body 'A' w.r.t. 'B' is given by BAR VVV

=

b. The relative velocity of body 'B' w.r.t. 'A' is given by ABR VVV

=

c. ABBA VV andVV

are equal in magnitude but opposite in direction

d. +== cos.VV.2VVVVV BA2B2ABAR

e. For two bodies moving in the same direction, relative velocity is equal to the

difference of velocities. ( = 0.cos 0 = 1)

RV

= VA VB

f. For two bodies moving in opposite direction, relative velocity is equal to the sum of

their velocities. ( =180;cos180 = 1)

RV

=VA + VB

g. If they move at right angle to each other, then the relative velocity = 2221 vv + .

2. Rain is falling vertically downwards with a velocity RV

and a person

is travelling with a velocity .VP

Then the relative velocity of rain

with respect to the person is PR VVV

= .

Relative velocity = 2P2R VV|V| +=

.

3. The direction of relative velocity (or) the angle with the vertical at which an umbrella is to

be held is given by Tan =R

PVV

.

-VP

VR

VP

V



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Product of Vectors

1. Dot Product

a) Scalar product (or) dot product is defined as the product of the magnitudes of two vectors and the cosine of the angle between them. The dot product of two vectors a

and

b

is given by . cosa b ab =

b) Scalar product is commutative i.e., a.bb.a =

c) Scalar product is distributive i.e., c.ab.a)cb.(a +=+

d) If a

and b

are parallel vectors, then a

.b

= ab.

e) Ifa andb

are perpendicular to each other, thena

.b

= 0.

f) If a and b

are anti-parallel vectors, then a

.b

= ab.

g) Component of a along b .cos a bab

=

h) Component of b along a ( ) .cos a bba

=

i) Vector component of a along cos bb ab

=

j) Vector component of b along cos aa ba

=

k) In the case of unit vectors,

1k kj ji i === . . . and 0i kk jj i === . . . .

l) If x y z A A i A j A k= + +

x y z and B B i B j B k= + + , then x x y y z zA B A B A B A B= + +.

.

2. Applications of Dot Product

a) The dot product of force and displacement is called work done W = S.F . b) The dot product of force and velocity is called power P = V.F .

3. Cross Product

a) Cross product (or) vector product of two vectors is a vector which is the product of the magnitudes of the two vectors and the sine of the angle between them.

sinA B AB = . n

a

b

( )a b



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Where n is a u nit vector along A B .

Eg: angular momentum ( = rL ), torque ( Fr = ), angular velocity ( rV = ) etc. b) The direction of A B can be known from right hand thumb rule (or) Cork screw rule.

c) 0i i j j k k = = = i j k = j i k = j k i = k j i = k i j = i k j =

d) If kBjBiBB and kAjAiAA zyxzyx ++=++=

, then B A

=

zyx

zyxBBBAAAkji

=

k)BABA(j)BABA(i)BABA( xyyxxzzxyzzy +

e) If B x A = 0 and B and A are not null vectors, then they are parallel to each other. B x A

= AxB

)B(m AB )A(m)B Am( ; C AB A)CB( A

==

+=+

f) ABBA (Anti-commutative)

g) The area of the triangle formed by B andA BA21

is sides adjacent as

.

h) The area of the parallelogram formed by B andA as adjacent sides is .BA

i) If B andA are diagonals of a parallelogram, then area of parallelogram= 1 ( )2

A B .

4. Applications of Cross Product

a) Torque is the cross product of radius vector and force vector, Fr = b) Angular momentum is the cross product of radius vector and linear momentum, prL =



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Motion in a Plane Oblique Projection

1. A body which has uniform velocity in the horizontal direction and uniform acceleration in the vertical direction

is called a projectile. 2. The path of a projectile is called trajectory and it is a

parabola.

3. For a projectile, the horizontal component of velocity (ux = u cos ) remains constant throughout its motion.

4. The vertical component (uy = u sin ) is subjected to acceleration due to gravity. 5. Equations of a projectile

a) Maximum height reached = g2

sinu 22

b) Time of flight =gsinu2

Time of ascent = time of descent =g

sinu

c) Range =g

2sinu2

d) tan=R

H4 max and tan=

R2gT2

6. At the highest point of the projectile a) Velocity is u cos (minimum). b) Vertical component of velocity is zero.

c) KE of the body is 22 cos21

mu .

d) PE of the body is 22 sin21

mu .

e) Angle between the velocity and acceleration is 90.



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f) The direction of motion of the body is horizontal. 7. Velocity after time "t"

a) Vertical component of velocity Vy = u sin gt

b) Velocity of a projectile after t seconds v = 22 )gtsinu()cosu( + .

c) The angle made by a projectile after t seconds, then

=

cosu

gtsinutan .

8. If projected from level ground a) Velocity of the projectile when it moves perpendicular to its initial velocity is U

cot.

b) Time taken for the velocity to become perpendicular to the initial velocity is sing

u

9. Position of the projectile after time 't' a) If x and y represent the horizontal and vertical displacements with respect the

point of projectiont seconds after projection x = (u cos) t

y = (u sin) t 2gt21

b) Equation of trajectory is

Y = ( ) 22 2tan 2 cosg

x xu

10. If y = Ax Bx2, then

a) The angle of projection = tan-1A

b) Maximum height H = B

A4

2

c) Range BAR =

d) Time of flight Bg

T 2=

e) Velocity of projection B

Agu

2)1( 2 +

=



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11. At the half of the maximum height

a) Vertical component of velocity is 2

sinu

b) Horizontal component of velocity is u cos

c) Velocity of the body is 2/12

2cos1

+ u

12. If the angle of projection is and (90 - ) (Complementary angles) a) Range is same

b) Sum of maximum heights is g

u

2

2

c) Ratio of max heights is tan2:1 d) Ratio of times of flight is tan : 1 e) If h1 and h2 are the maximum heights, then 214 hhR =

f) Range = 2121 TgT where T1 and T2 are the times of flights.

13. If a man throws a body to a maximum distance R then he can project the body to vertical height R/2.

14. If a body is projected down at an angle with the horizontal from the top of a tower then

h = (u sin) t + 2gt21



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Law of Motion

1. Newtons first law of motion: Every body continues to be in the state of rest or of uniform

motion unless it is compelled by an external unbalanced force to change that state.

2. The first law of motion gives the concepts of force and inertia.

3. Inertia is the inability of a body to change its state of rest or of uniform motion along a straight line in the absence of any external force

4. Inertia is of three types i) Inertia of rest ii) Inertia of motion and iii) Inertia of direction

5. Inertia of rest: The inability of a body to change its state of rest by itself is called inertia of rest.

Eg: When a bus at rest starts suddenly passengers fall back.

6. Inertia of motion: The inability of a body to change its uniform motion by itself is called as inertia of motion. Eg: When a bus in uniform motion suddenly stops, the passengers fall forward.

7. Inertia of direction: The inability of a body to change its direction of motion by itself is called inertia of direction. Eg: When a bus takes a turn passengers will be pulled outwards.

8. Force: Force is that which changes or tries to change the state of rest or of uniform motion of a body along a straight line.

9. Momentum: Momentum is the product of mass and velocity ( vmP = ). SI unit is kg ms 1. It is a vector quantity.

10. Newtons second law of motion: The rate of change of momentum of a body is directly proportional to the impressed force and it takes place in the direction of force.

11. Newtons second law gives the quantitative definition of force and defines the unit force.

12. ( )dp d m vFdt dt

= = OR dv dmF m vdt dt

= +

(a) If m = constant , dvF mdt

= = ma



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(b) If v = constant , dmF vdt

= ( For a variable mass system )

13. A unit force: A unit force is one which when acting on unit mass produces unit acceleration in its direction. Unit of force : newton

Gravitational unit of force :1kgwt= 9.8 Newton

14. Rocket Propulsion

Velocity of a rocket at any instant of time is given by 00 logr em

v u vm

= +

m0 = mass of the rocket at t = 0

v0 = velocity of the rocket at t = 0

m = mass of the rocket at any instant of time

v = velocity of the rocket at any instant of time

vr = velocity of the exhaust gases relative to the rocket

a) If the rocjet is to moveup with constant speed over comming its weight then

Thrust on the rocket = dm udt

= Mg

b) If the rocket moves up with constant acceleration a. then

Thrust on the rocket = dm udt

= Mg + Ma

15. A liquid of density d flowing through a pipe of length l and cross section A with a velocity V strikes a vertical wall normally

a) If the liquid comes to rest after striking the wall then 2F Av d= b) If the liquid moves back with same velocity then 22F Av d= c) Power P = 3P Av d= d) If the rate of water ejected is n times the initial rate then, force become n2 times and

power becomes n3 times.

c) If water reflects with a velocity 1v then, ( )'F A v d v v= + e) Pressure exerted on the wall Fp

A=



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b) In the above case if water strikes the surface at angle with the normal and reflects with the same speed, force exerted on the wall is 2Av2dCos.

16. If a machine gun fires n bullets each of mass m with a velocity u in a time t , then the force

required to hold the gun is mnuFt

=

17. If a plate of mass M is suspended in air by firing the bullets on to it

a) If bullets comes to rest after striking the plate then m n u M gT

=

b) If the bullets gets reflected back with the same Velocity perpendicular to the plate, then 2 m n u M g

T=

18. A body of density Bd moves down in a liquid of density ld then the acceleration of the body is

given by B lB

d da g

d

=

19. Linear Momentum

a) If the initial velocity of a body is u and final velocity is v then the change in momentum is given by ( )P m v u =

2 2 2 cosP m u v uv = + Where is the angle between u and v .

If v = u, 2 sin2

P mu =

b) If a ball of mass m strikes a wall normally and bounces back with same velocity, then the change in momentum is

( ) ( )P mv i mv i = 2P m v i =

The magnitude of the change in momentum is 2P mu =

c) If a ball of mass m strikes a wall at angle of incidence with a velocity u and bounces back with same velocity at the same angle

i) Along the wall, ( ) ( )sin sinP mu j mu j = = 0 ii) Perpendicular to the wall,



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( )cos cos ( )P mu i mu i = ( )2 cosmu i= The magnitude of the change in momentum is 2 cosP mu = +

d) If a ball of mass m strikes a wall at an angle with the wall with a velocity u and bounces back with same velocity at the same angle then

i) Along the wall , ( ) ( )cos cos 0P mu j mu i = = ii) Perpendicular to the wall,

( ) ( )sin sinP mu i mu i = ( )2 sinmu j= The magnitude of the change in momentum is 2 sinP mu =

iii) A body of mass m is released from a height h, the momentum of the body on reaching the ground is 2m gh

20. Impulse

i. Very large force acting for a short interval of time is called impulsive force. Eg:

Blow of a hammer on the head of a nail.

ii. The impulse of a force is defined as the product of the average

force and the time interval for which it acts.

Impulse J = FAV t = m v

- mu

iii. Impulse due to a variable force is given by the area under Ft

graph.

iv. If a force F1 acts on a body at rest for a time t1 and after that another force F2 brings

it to rest again in a time t2, then F1t1 = F2t2.

v. While catching a fast moving cricket ball the hands are lowered, there by increasing

the time of catch which thus decreases the force on hands.

vi. A person jumping on to sand experiences less force than a person jumping on to a hard floor, because sand stops the person in more time.

21. The gravitational force that acts on a body is called its weight (W = mg). It is a vector always pointing in a vertically downward direction.

22. A bird is in a wire cage hanging from a spring balance when the bird starts flying in the cage, the reading of the balance decreases.



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23. In the above case, if the bird is in a closed cage or air - tight cage and it hovers in the cage, the reading of the spring balance does not change.

24. In the above case for a closed cage if the bird accelerates upward reading of the balance is

R = Wbird + ma, where m is the mass of the bird and a, its acceleration.

25. Apparent weight of a person in a lift

Consider a person of mass m in a stationary lift whose weight is W = mg

a) If the lift moves up with an acceleration a apparent wt

'W mg ma= + ' 1 aW Wg

= +

b) If the lift moves down with an acceleration a 'W mg ma= OR

' 1 aW Wg

=

c) If the lift is freely falling under gravity i.e. If a = g, then ' 0w =

d) If the lift is moving up or down with uniform velocity, i.e. if a = 0 then 'W W mg= =

26. Newtons third law: For every action there is an equal and opposite reaction.

27. Newtons first and third laws are only special cases of second law.

28. Limitations of Newtons law of motion

a) It is applicable only for speeds V

Dynamics-Connected Bodies

1. Atwoods machine a) For body A, 1 1m g T m a =

For body B, 2 2T m g m a =

Acceleration and tension of the system respectively are

( )( )

1 2

1 2

m m ga

m m

=

+ And ( )

1 2

1 2

2m m gTm m

=

+

b) ( )( )( )( )

( )( )

( )( )

1 1 2 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2CM

m a m a m m a m m m ma g

m m m m m m m m

= = =

+ + + +

2

1 2

1 2

m ma g

m m

= +

c) The reaction at the pulley, R = 2T ( )1 2

1 2

4m m gRm m

=

+

2. If the mass of the pulley is also taken into consideration, then

For body A, 1 1 1m g T m a = --- (1) For body B, 2 2 2T m g m a = --- (2)

For the pulley, ( )1 2T T R I = , where R is the radius of the pulley.

( ) 21 2 12aT T R MRR

=

1 212

T T Ma = --- (3)

From equations (1), (2) and (3)

( )1 21 2 2

m m ga

mm m

=

+ +

, 1 2

1

1 2

22

2

Mm m g

T Mm m

+ =

+ +

and 2 12

1 2

22

2

Mm m g

T Mm m

+ =

+ +

3. One object having horizontal motion and other having vertical motion:

a) For the body at A, 2 2m g T m a =



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For the body at B, 1T m a=

( )2

1 2

m ga

m m =

+ and ( )

1 2

1 2

m m gTm m

=

+

Thrust on the pulley is 2T

b) If the coefficient of friction between table and the mass is ' ' then For the body at A, 2 2m g T m a =

For the body at B, 1 1T m g m a =

( )( )

2 1

1 2

m m ga

m m

=

+ And ( )( )

1 2

1 2

1mm gT

m m

+=

+

4. For the body at A, 1 1 1m g T m a =

For the body at A, 2 2 2T m g m a =

For the body at C, 1 2T T Ma =

( )( )

1 2

1 2

m m ga

m m M

=

+ + and

[ ]1 21

1 2

2m m M gT

m m M+

=

+ + and [ ]2 12

1 2

2m m M gT

m m M+

=

+ +

5. Two masses m1 and m2 connected by a string pass over a pulley. m2 is suspended and

m1slides up over a frictionless inclined plane of angle

1 1T m g Sin m a =

2 1 2m g T m a =

Acceleration, a = 21

12mm

g)sinmm(+

and

Tension in the string T = m2g m2a = )mm(g]sin1[mm

21

21+

+

6. 1 1m g Sin T m a =

2 2T m g Sin m a =

Tension (T) = 21

21mm

g)sin(sinmm+

+

T

m2

TR

mgsi

n1

m g1

m1

m cos1



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Acceleration (a) = 21

21 )sinsin(mm

mmg+

Force on the pulley 9022

F T Cos =

7. The acceleration in the following case

a) 2 11 2

m ma g

m m

=

+ 3 3m ggm

= =

b) T= pulling force F = 2mg T mg = ma

mg = ma

a = g

8. Consider the following system

a) 2 11 2

m ma g

m m

= +

22 3

M M ggM M

= = +

b) Tension in the string AB is T Mg = Ma

43MgT Mg Ma= + =

c) Tension in the string BC is Mg T1 = Ma

T1 = Mg Ma

Or T1 = 2

3Mg

9. Two weights w1 and w2 are suspended as shown. When the pulley is pulled up with an acceleration g, the tension in the string is

( ) ( )1 2

1 2

2m mT g gm m

= ++

Or ( ) ( )1 2 1 2

1 2 1 2

4 4m m g w wTm m w w

= =

+ +



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10. Bodies in contact: a. Consider two bodies m and M which are in contact and placed on a horizontal smooth

surface. Let a force f is applied on the system as shown. Let R be the contact force between the two bodies

1) faM m

=

+

2) f R Ma = and R ma=

b.

1) faM m

=

+

2) f R ma = and R Ma= 11. A uniform rope of length L is placed on a horizontal smooth

surface and pulled with a force F at one of its end. Let m be the mass per unit length of the rope. The tension in the rope at a distance l force the end where force is applied is given by

a) FamL

=

b) L lT FL

=

12. A block of mass M is pulled by a rope of mass m by a force P on a smooth horizontal plane.

P m M


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a) Acceleration of the block a = mM

p+

b) Force exerted by the rope on the block

)mM(MpF+

=

13. Consider the following system

For the 1st body, 1 1F T m a =

For the second body, 1 2 2T T m a =

For the third body, T2 = m 3 a Acceleration of the system

1 2 3

Fa

m m m=

+ + And ( )2 3

11 2 3

m m FT

m m m

+=

+ +

( )3

21 2 3

m FTm m m

=

+ +

If the force (F) acts on 3m , then

( )( )

1 22

1 2 3

m m FT

m m m

+=

+ +And

( )1

11 2 3

m FTm m m

=

+ +

14. Masses m1, m2, m3 are inter connected by light string and are pulled with a string with

tension T3 on a smooth table.

a) Acceleration of the system

a = )mmm(T

321

3++

b) Tension in the string

T1 = m1 a = 321

31mmm

Tm++

T1 T2 T3 m1 m2 m3



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( ) ( )321

321212

mmm

TmmammT

++

+=+=

T3 = (m1 + m2 + m3) a



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Horizontal Plane

1. Friction in due to the interlocking of irregularities between the surfaces in contact with each other.

2. Frictional force is a non-conservative force.

3. Frictional force is electromagnetic in nature.

4. Frictional force acts along the tangent drawn to the surface in contact

5. Advantages of friction

i) Safe walking on the floor is possible due to the friction between the floor and the feet.

ii) Nails and screws are driven in the walls due to friction. iii) Friction helps the fingers to hold a drinking water tumbler. iv) Vehicles move on the roads without sliding due to friction and they can be

stopped due to friction.

6. Disadvantages of friction

i) Friction results in the power loss in engines. ii) The wear and tear of the machine increases due to the friction.

7. Methods of reducing friction

i) Friction between two surfaces of contact can be reduced by polishing the surfaces. ii) A lubricant is a substance which forms a thin layer between two surfaces in contact and reduces the friction.

iii) Ball bearings reduce the friction because rolling friction is minimum. iv) Automobiles and aero planes are stream lined to reduce the air friction.

8. Methods of increasing the friction



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i) Tyres of a vehicle have irregular projections to increase the friction. ii) Belts on the wheels of grinding machines are waxed frequently to increase the friction.

iii) Sand in poured on the railway tracks during rainy season to increase the friction. iv) Wheels of a train (or) rails are smoothened to increase the friction. v) Vigorous polishing increase the friction due to the increase in the intermolecular forces.

9. Types of Friction

a) Static Friction

i. The force of friction between the bodies in contact which have no relative motion is called Static friction.

ii. Static friction is a self adjusting force (i.e.) it is equal to the applied force until the body just begins to move.

iii. When the body is ready to slide the static friction becomes maximum and it is called limiting friction.

iv. Limiting friction (Fs) is independent of the area of contact of the surfaces.

v. sss whereNF = is called the coefficient of static friction. It depends upon

the nature of the surfaces in contact and their state of roughness.

vi.s between two given surfaces is independent of the normal force between the two surfaces.

vii.s > 0, it can also be greater than one, but in most of the cases it is less than one

viii. The angle between normal reaction and the resultant of normal reaction and friction is

called the angle of friction .



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Tan =

b) Kinetic friction

The frictional force that exists between the bodies which are in relative motion with each other in called kinetic (or) sliding friction.

It is constant and in independent of velocity of the body provided the velocity is low.

The force of kinetic friction is independent of the area of the surfaces in

contact and is proportional to the normal reaction Fk N.

i. Fk = k N

ii. Where k is coefficient of kinetic friction

Whenever a body is in motion

Net force = Applied force - friction force

ma = F - f K

Force of kinetic friction may be greater than the static friction but it is

1. Always less than the

limiting friction.

2. OA static friction

3. A Maximum Static friction

4. BC kinetic friction

C) Rolling friction

i. Rolling friction comes into play when a body rolls on a surface.



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ii. Rolling friction arises due to the deformation of the two surfaces in contact

with each other.

iii. Greater the deformation greater is the rolling frictional force.

iv. The rolling frictional force is inversely proportional to the radius of the

rolling body.

v. If R is the coefficient of rolling friction

vi. R < k < s for a given pair of surfaces.

10. Block on a rough fixed horizontal surface

a) If a force required to just move the body is continuously applied then the acceleration given by

a = (s - k) g b) If the block slides with an acceleration under the influence of an external force F,

the acceleration of the block is a = m

fF k

11. Motion on a rough horizontal plane

(a) Pulled with a horizontal force F: (i) Body moving with uniform velocity F = k mg.

(ii) Body moving with uniform acceleration F = m ( k g + a).

(b) Pulled with a force F inclined at an angle with the horizontal and the body moving with uniform velocity.

F= sincos sin cos( )

k

k

mg mg

=

+ Where is the angle of friction

between the two surfaces.

c) The minimum possible force among all directions required to just move the body is mg sin (or)

2s

s

1

mg

+

where is the angle of friction.



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c) Pushed with a force F inclined at an angle with the horizontal and the body moving with uniform velocity:

F =

sincos

mgk

k.Hence pulling is easier than pushing.

12. A uniform chain of length L lies on a table. If the coefficient of friction is , then the

maximum length of the chain which can overhang from the edge of the table without

sliding down is 1

L+

.

13. If a block is pushed with are initial velocity u and released and if the block comes to rest after traveling some distance s, then

212 k

mu mg= (Or) 2

2 ku

sg

=

Also, v gt=

14. If a vehicle is moving on a curved un-banked road 2

kmv

mgr

= (or)

kg

v rg andr

= =

15. When a bicycle is pedaled (accelerated) the direction of the frictional force on the front and rear wheels are

a) Front wheel - Opposite to the direction of motion (b) Rear wheel - In the direction of motion 16. When a bicycle is in uniform motion, then the direction of frictional forces is

a) Front wheel - Opposite to the direction of motion

b) Rear wheel - Opposite to the direction of motion

17. A body of mass m is at the back side of an open truck. If the trolley moles forwards with an acceleration a, then a pseudo force (ma) acts an the body

(1) If sma mg< , the block does not slide, then, Frictional force f = ma



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(2) If sma mg= . Then the block just slides. Then the frictional force sf mg=

(3) If sma mg> . The block moles over the truck in the backward direction with acceleration ka g=

18. If a body is pushed with a force P towards a vertical wall then

1) If the block is at rest 0xF N P= = (or) N = P

0y sF f mg= = (Or) sf mg=

( )s sf mg or P mg

The block is at rest if s

mgP

mins

mgP

=

19. A vehicle is moving on a horizontal surface. A block of mass 'm' is stuck on the front part of the vehicle. The coefficient of

friction between the truck and the block is ''. The minimum acceleration with which the truck should travel, so that the body may not slide down is a = g/ .

20. If a book is pressed between two hands then 2mg F= where F is the pressing force

applied by each hand.



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Friction

Inclined Plane

1. For a block of mass m on an inclined plane of inclination with the horizontal,

a) If Applied force < frictional force

Or m g sin f < Or sin cossmg mg <

Or sTan < Or Tan Tan < < Then the block doesnt

slide. Since the static friction is rest adjusting,

Frictional force = sinmg

b) If Applied force = frictional force

sinmg f = (or), if = , then the body tends to move (or) ready to move. The angle of inclination in this condition is called the angle of repose. Angle of repose in independent of the weight of the body

Frictional force = sin cossmg mg =

s Tan =

c) If Applied force > frictional force

Or sin smg N > Or sin cossmg mg > or sTan > Or >

Now, the frictional force = k kf N=

cosk kf mg =

The resultant force on the block is given by

( )sin coskma mg = Or ( )sin coska g = This is the acceleration of the body sliding down the inclined plane.

2. Sliding down the inclined plane

a) If , the block slides down with an acceleration given by a = g [sin - k cos].

mg mg cos

mg sin f

N



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b) If , and the block slides down from the top of the inclined plane. Velocity at the bottom of the plane is

V = 2 (sin cos )kgl .

c) In the above case time of descent is t = )cos(singL2

k .

3. Moving up the inclined plane

a) If a block is projected up a rough inclined plane, the acceleration of the block is a = g [sin + k cos].

b) Force opposing the motion of the block is F = mg sin + k mg cos.

c) The distance traveled by the block up the plane before the velocity becomes zero is

S = )cos(sing2u

k

2

+.

d) The time of ascent is t =k

u

g(sin cos ) + . In the above case the block will come down

sliding only if .

e) In the above case if time of decent is n times the time of ascent, then

= tan

+

1n1n

2

2.

f) Force needed to be applied parallel to the plane to move the block up with constant velocity is

F = mg (sin + k cos). g) Force needed to be applied parallel to the plane to move the block up with an

acceleration a is

F = mg (sin + k cos) + ma. h) If block has a tendency to slide, the force to be applied on the block parallel and up the

plane to prevent the block from sliding is F = mg( sin - s cos). 4. Smooth inclined plane

a) Normal reaction (N) = mg cos a) Acceleration of sliding block a = g sin

mg mg cos

mg sin

N



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b) If l is the length of the inclined plane and h is the height. The time taken to slide down starting from rest from the top is

t = 221 2sin sin

hg g

= .

d) Velocity of the block at the bottom of the inclined plane is V = gh2singl2 = same as the speed attained if block falls freely from the top of the

inclined plane.

e) Distance traveled up the plane before its velocity becomes zero is

S =sing2

u2.

5. If the time taken by a block to slide downs a rough inclined plane of angle is an n time

that on identical smooth inclined plane. Then of the rough plane is 211 Tann

=

.

6. Sand is piled up on a horizontal ground in the form of a regular cone of a fixed base of

radius R. The maximum volume of the cone without the sand collapsing is3

max 313

hV pi

= .

7. An inclined plane of inclination is upper half smooth and lower half rough. A body starts sliding from the top from rest and comes to rest at the bottom. If the coefficient of

friction of the lower half is , then 2 tan = .



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Friction

Block on block

Case I: (lower block pulled and there is no friction between lower block and the horizontal surface.) a) When the lower block is pulled upper block is accelerated by

the force of friction acting upon it.

b) The maximum acceleration of the system of two blocks for them to move together without slipping is a = s g, where s is the coefficient of static friction between the two blocks.

c) If a < s g blocks move together and applied force is F = (mB + mu ) a. d) If a < s frictional force between the two blocks f = mu a. e) The maximum applied force for which both blocks move together is Fmax = s g (mu + mB). f) If F > Fmax blocks slip relative to each other and have different accelerations. The

acceleration of the upper block is k g and lower block is

a =uB mm

F+

.

Case - II (Upper block pulled and there is no friction between lower block and the horizontal surface) a) When the upper block is pulled, lower block is accelerated by

the force of friction acting upon it.

b) The maximum acceleration of the system of two blocks for them to move together without slipping is amax = g

m

m

B

us (s = coefficient of static

friction between the two blocks) c) If a < amax frictional force between the two blocks is f = MB a.

m

mu f f

F

mu F

f

mB

f



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d) If a < amax' then applied force on the upper block is F = (mB + mu) a.

e) The maximum force for which both blocks move together is Fmax = s gm

m

B

u (mu +

mB). f) If F>Fmax blocks slide relative to each other and hence have different accelerations.

The accele-ration of the lower block is gm

m

B

uk and the acceleration of the upper

block isu

ukm

)gmF( .



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Rotatory Motion

Horizontal Circular Motion

1. In translatory motion, every point in the body follows the path of its preceding one with same velocity including the centre of mass.

2. In rotatory motion, every point move with different velocity with respect to the axis of rotation. The particle on the axis of rotation will have zero velocity.

3. The angle described by the radius vector in a given interval of time is called the angular displacement.

4. Angular displacement is a vector passing through the centre and directed along the perpendicular to the plane of the circle whose direction is determined by right hand

screw rule (It is a pseudo vector). 5. Angular displacement is measured in radians.

6. The rate of change of angular displacement is called angular velocity (). 1

radst

= .

7. Angular velocity is a vector lying in the direction of angular displacement.

8. Linear velocity r )V( = .

9. Rate of change of angular velocity is called angular acceleration (). Unit is rads 2.

timevelocityangularinchange

= .

10. Linear acceleration = radius angular acceleration. r a = .

11. Resultant acceleration a = where ar = radial acceleration and aT = tangential

acceleration.

12. Comparison of linear and angular quantities.

Translatory motion Rotatory motion



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v = u + at

21

2s ut at= +

2 22v u as= +

2

u vs

+=

Mass (m)

F = ma

Impulse = Ft

Linear momentum p = mv

Work = FS

Power = FV

1 1 2 2mv m v=

2

2

pE

m=

21

2KE mv=

= +2 1

t

2

1

1

2t t = +

= +2 22 1

2

1 2

2

+=

Moment of inertia (I)

Torque = I

Angular impulse = t

Angular momentum L = I

=W

=P

1 1 2 2I I =

2

2

LE

I=

Rotational KE = 212I

13. If a particle makes n rotations per second 2 n pi= .

14. Angular velocity is a pseudo vector (or) axial vector. v r= and . . 0v r r = = .



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15. Rate of change of angular velocity is called angular acceleration ( ). Unit is rad s 2.

timevelocityangularinchange

= The angular acceleration is a pseudo (or) axial vector.

16. The direction angular acceleration ( ) is along the change in angular velocity. a r= and . . 0a r a = = .

17. The direction will be same as that of if it is increasing and opposite to that of if it is decreasing.

18. Uniform circular motion

a. Tangential acceleration is due to change in the speed and normal acceleration is due to the change in the direction.

b. Tangential acceleration = =Tdv

a rdt

. This is along the tangent drawn along the

circular path.

c. For vertical circular motion the tangential acceleration is given by

= = sinTa r g

d. Radial (or) normal (or) centripetal acceleration pi= = =2

2 2 24N

va r n r

r

e. In uniform circular motion: ( =constant) i) Tangential acceleration is zero ( 0ta = ) ii) Normal acceleration aN = constant 19. In non uniform circular motion

Net acceleration = +2 2N Ta a a

(or) ( )2

22v

a rr

= +

and = +N Ta a i a j

20. aN = 0 and aT = 0 uniform linear motion.

21. aN = 0 and 0Ta - accelerated (or) non uniform linear motion. 22. 0Na and aT = 0 uniform circular motion.



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23. 0Na and 0Ta - non uniform circular motion.

24. The force which makes a body move round a circular path with uniform speed is

called the centripetal force. This is always directed towards the centre of the circle.

Centripetal force= pi= =2

2 2 24

mvmr n mr

r.

25. A body moving round a circular path with uniform speed experiences an inertial or pseudo force which tends to make it go away from the centre. This force is called the

centrifugal force and this is due to the inertia of the body.

26. Centrifugal force = centripetal force (but these are not action and reaction). 27. No work is done by centripetal force.

28. The kinetic energy of the body revolving round in a circular path with uniform speed

is E. If F is the required centripetal force, then r

E2F =

29. Uses of centrifugal forces and centrifugal machines.

i) Cream is separated from milk (cream separator) ii) Sugar crystals are separated from molasses. iii) Precipitate is separated from solution. iv) Steam is regulated by Watts governer. v) Water is pumped from a well (Electrical pump). vi) Hematocentrifuge, Grinder, Washing machine, etc.

30. The angle through which a cyclist should lean while taking sharp turnings is given by

the relation

=

rgvTan

21

.

31. Safe speed on an unbanked road when a vehicle takes a turn of radius r is v = rg

where = coefficient of friction.

32. The maximum speed that is possible on curved unbanked track is given by g = v2h/ar

Where h = height of centre of gravity and a = half the distance between wheels.



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33. Angle of banking

At curves the outer edge of the road is slightly above the lower edge. The angle made by the tilled road with the horizontal is called angle of banking.

2

sin mvNr

= and cosN mg =

2vTanr g

=

For small angles,

tan sin = 2 2h v v lh

l r g r g = =

34. Conical Pendulum: Let Tbe the tension in the string.

a. 2

sin mvTr

= And cosT mg =

2 2v rTanrg g

= =

b. rTanh

= 2

r r gh g h

= =

c. Time period 2 hTg

pi = (Or) 2 2

2 l rTg

pi

=

And frequency 12

gn

hpi= .

d. ( )2 2sin sinT mr m l = = .



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Dynamics

WorkPowerEnergy

1. Conservative force a) A force is said to be conservative if the work done by it is independent of path followed by

the body. b) Work done by a conservative force for a closed path is zero. c) Work done by a conservative force depends only on the initial and final positions of the

body. d) Work done by a conservative force is the product of Force and displacement. e) During a round trip the body attains the same initial K.E.

Ex. Gravitational force, Electrostatic force etc.

2. Non - Conservative force a) A force is said to be non- conservative if the work done by it depends on the path followed

by the body. b) Work done by a non-conservative force for a closed path is not zero. c) During a round trip the body attains a different K.E. as that of initial. d) Work done by a non-conservative force is the product of Force and distance. e) Due to a non-conservative there may be a loss of mechanical energy but the total energy is

constant.

Ex. Frictional force

3. Work

a. Work is said to be done when the point of application of force has some displacement in the direction of the force.

b) The amount of work done is given by the dot product of force and displacement. W = = cosFss.F

c) Work is independent of the time taken and is a scalar. d) If the force and displacement are perpendicular to each other, then the work done is zero. e) A person rowing a boat upstream is at rest with respect to an observer on the shore.

According to the observer the person does not perform any work. However, the person



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performs work against the flow of water. If he stops rowing the boat, the boat moves in the direction of flow of water and work is performed by the force due to flow, as there is displacement in the direction of flow.

f) If the work is done by a uniformly varying force such as restoring force in a spring, then the work done is equal to the product of average force and displacement.

g) If the force is varying nonuniformly, then the work done = ds.F = cos.ds.F . h) The area of Fs graph gives the work done. i) SI unit of work is joule. j) Joule is the work done when a force of one Newton displaces a body through one metre in

the direction of force.

k) CGS unit of work is erg; 1 J = 107 ergs. l) If the force or its comp

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