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Mohammed
Asif
Name :
Roll No. :
Topic : Electric Field &Potential
Ph : 939132!"#
$%!"
2. MEA'REMENT (F P)*+A, -'ANT+T+E
1. If a physical quantity P. is given by
P = x + y
m nn m
AbsoluteerrorinP P X y
x y x yPRelativeerror !errorinP 1""
P x y x y
#. IfP x y P x y$
x yP P!errorinp 1""
P x y P
%. IfP x y orx & y .
m x n yPP x y
= +
+ + = = + +
= = + +
= =
=
= +
i.e. ! error in P = m. ! error in x + n. ! error in y.
'. a b cP ( ) * . =
If an arethepercentageerrorsinthemeasurementxof() an*
thenpercentage errorinP a b c .
= + +
,ignificant figurei- all non/ero igits are ,ignificant 0x. #%.2 3o. of s. f 4 'ii- 5ero in bet6een t6o non/ero igits are ,ignificant 0x. #%"'" 3o. of s. f 4 2iii- All /eros to the left of the first non/ero are not ,ignificant 0x. ""#%2 ".""#%2 3o. of s. f 4 'iv- All /eros to the right of the last non/ero igit in a measure value are ,ignificant
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0x. #%2"" m 3o. of s. f 4 2
(ean eviation 1 # % n+ + + ...+
@eviation -n
+ + += =
,tanar eviation# # #
1 # n ... @6hennislarge.-n
+ + + =
Probable error = r = ".2>' = *ruevaluex X r Probableerrorr n
1
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3. E/T(R
M 0 +: Manitde and direction of a ector:
$ $= + +
= + +
$
# # #
1. IfA ai b B c8
*hen@a-magnitue A a b c .
@b- Cnit vector in the irection of =uur AA isgivenbyA .
A
= = =
+ + =
+ + =
ur
ur
r
# # #
# # #
@c-If an aretheanglesmaebyA6ithxy an/axesthen
a b cDos Dos Dos .
A A A
AlsoDos Dos Dos 1.
@-IfAis aunit vector then a b c 1.@e-EectorparalleltoAanhavingmagnitueequal
toanothervector
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Q
P
R
r
M 0 +++: Paralleloram la4 of ectors
2 2
2 2
If R P Q(sumof the vectors)
Magnitudeof ther esul t an t R P Q 2PQCos .
Q sin P sinDirection tan tan
P Q cos Q P cos
If R P Q (difference) R P Q 2PQCos
Q sin P sintan tan
P Q cos Q P cos
If P Q then
= +
= + +
= =
+ +
= = +
= =
+ +
> = =
Q
R
P
M 0 +: Resoltion of ectors:
If a vector P ma8es an angle 6ith hori/ontal then its hori/ontal componentPx = Pcos $ Eertical component = Py = P sin
2 2
x yP p p= +
= = +$ $x x yy
Ptan P P i P BP
Py
Px
P
M 0 : Pol5on la4 of ectors:
n equal force each of magnitue 7 are acting simultaneously on a particle.
0ach force ma8es an angle 6ith one then
n
%2"a- If then theresul tan tforceis/ero.=
n
%2"b- If 1thentheresultantforceis7. =
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Area of parallelogram A < A an < are the t6o aBacent sies of aparallelogram
1Area of triangle A < A an < are the t6o aBacent sies of atriangle.
#
=
= r rur ur
$ $x y / x y /
f- Cnit vector prepen+icular to vectors A an+ < is
A T(N? ,A> (F M(T+(N@
1. 7orce 7 @m - m ma 6hen mass is constant
t t m
7 @m - if m is changes in the case of a variable masst t
= = =
= =
,ystem li8e a roc8et.#. a- Fhen a lift moves up6ars 6ith acceleration a @or- o6n6ars 6ith a
eceleration a then apparent 6eight of a boy in the lift 6 = m @g + a-.
b- Fhen a lift moves up6ars 6ith eceleration a @or- o6n6ars 6ith anacceleration a then apparent 6eight of a boy in the lift 6 = m @g a-. c- Fhen the lift is stationary or moving 6ith uniform velocity the 61= mg.
- Percentage change in 6eighta
1"".g
=
e- If h is the epth of 6ater in a buc8et place in the lift then pressure at thebottom of the buc8et is P = h @g a-.
%. Fhen a machine gun fires NnG bullets in a time t then the force require to holthe gun is
mn
7 t
= .
'. a- Fhen a metallic plate of mass m is hel in mi air by firing NnG shots&secon 6ith avelocity u if the bullets stop ea after stri8ing the plate then mnu = mg.
b- If the shot comes bac8 6ith a velocity E then nm @v+u- = mg.. If 6ater from a narro6 pipe hits a 6all hori/ontally 6ith a velocity v an stops ea
force acting on the 6all#
# H7 A H 6hereH avA
= = = =
@volume flo6 rate-.A = area of cross section of the pipe
= ensity of 6ater.If 6ater bounces bac8 6ith same spee 7 = # Av#
2. If a ball hits a 6all normally 6ith a velocity v an rebouns 6ith same velocity changein momentum = # mv.
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7orce acting on the 6all#m
t
= 6here NtG is the time of collision.
>. a- A boy of mass m moving 6ith a velocity v hits 6all an rebouns 6ith samevelocity then change in momentum perpenicular to the 6all p #mvsin = force
on the 6all#m sin
7 @ is the angle ma+e 6ith the 6all-t
=
b- Dhange in momentum parallel to 6all = ".J. 0xtra force require to 8eep the conveyor belt moving 6ith a velocity E 6hen
,an @any mass- is falling on it the rate ofm m
is given by 7 E.t t
=
?. If hot gases are coming out of the roc8et at the rate of
mt
6ith a velocity E
then
a- *hrust on the roc8et =
m7 .
t
b- If the roc8et moves up 6ith uniform velocity then 7 = (g(mass of the roc8et.
c- If the roc8et moves up 6ith uniform acceleration a then 7 = (@g + a-.If gravity is neglecte then 7 = (a
- Acceleration of the roc8et after t sec a
+m
+t@neglecting gravity-.
+m( t
+t
=
,A> (F /(NERAT+(N (F ,+NEAR M(MENT'M AN6 /(,,++(N
M 0 +: ,a4 of conseration of momentm:
A boy of mass m1 moving 6ith a velocity u1collies 6ith another boy of mass m#moving 6ith a velocity u# if v1an v#are their final velocities then Accoring to la6 ofconservation of momentumm1u1 + m#u# = m1v1 + m#v#If the t6o boies stic8 together after collision then common velocity
=
+1 1 # #
1 #
m u m uE
m m
1. If boies are moving in mutually perpenicular irections before collision then
common Eelocity( ) ( )+
=+
# #
1 1 # #
1 #
mu m uE .
m m
#. In case of obliquic collision i.e. after collision if the irection of motion of boies
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ma8es an angle 6ith initial irectiona- 1 1 # #mu m u+ = @ 1 1 # # xi xf@m v m v -cos @P P -+ =
b- 1 1 # # yi yfm v m v . @P P -= =
%. If a bullet of m come out 6ith a velocity from a gun of mass ( the Recoil velocity
of gunm
E .(
=
bullet
gun
0 ( .0 m
= Eelocity of bullet relative to gun = v + u.
'. If a stationary shell brea8s into t6o fragments they 6ill move in oppositeirections 6ith velocities in the inverse ratio of their masses.
1 #
# 1
E m.
E m=
. In the above case the inetic energy of the t6o fragments is inversely proportionalto their masses.
1 #
# 1
0 m0 m=
2. *he total energy release in the explosion # #1 1 # #1 1
m m# #
= +
>. If a shell at rest exploes into three fragments having masses in the ratio m 1 m#(%if first t6o fragments travel in mutually perpenicular irection 6ith velocitiesv1an v#then the velocity of the thir fragment is given by
( ) ( )
# #
1 1 # #%
%
mu m uE .
m
+=
J. If a shell of mass m moving 6ith a velocity E exploes into t6o fragments havingmasses m1 an m# an if 1 #E anE are the velocities of the fragments then
1 #1 #m m m = +
M 0 ++: Elastic collision:
1. 7or one imensional elastic collision v#4 v1= u14 u#.
1 # # #1 1
1 # 1 #
1# 1
# #1 # 1 #
m m #m uu
m m m m
#mum m
um m m m
= + + +
= + + + ,pecial cases
a- Fhen m1= m#$v# = u1v1 = u#.b- If u# = " @i.e. secon boy is at rest-then
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If m1= m# v# = u1v1 = "If m# OO m1 v1 = u1v# = "If m1 OO m# v1 = u1v# = # u1.
#. In case of elastic collision if the secon boy is at rest *he fraction of .0 transferre or lost by the first boy
( )
1 #
##1 #
'm m 'n
.@m m - n 1= =+ + Fhere n = m1 & m#
%. 7raction of .0 retaine by the first boy is
# #
# 1 f
1 # i
0m m 1 n
m m 0 1 n
= =
+ + Fhere n = m1 & m#
If m1= m#fraction of 0 transferre is maximum an that is 1"" !.M 0 +++: +n elastic collision:
1. a- 1 1 # #mu m u+ = 1 1 # #m v m v+
b-# 1
1 #
e
u u
=
7inal velocity can be foun by using above t6o equations
#. If u# = " i.e. if the secon boy is at rest ratio of final velocities1
#
1 e.
1 e
= +
( ) ( )1 11 #u u
1 e 1 e .# #
= = +
# # # #1 1 # # 1 1 # #
1 1 1 1%. )oss of 0 mu m u m m
# # # # = + +
( )
( ) ( )# #1 #
1 #1 #
1 m mu u 1 e
# m m=
+
'. A bloc8 of mass ( is suspene freely. A bullet of mass m is fire in to the bloc86ith a velocity . If the bullet gets embee in to the bloc8 then
a- Eelocity of bloc8 after collisionm
M m
=
+
b- ;eight rise by the bloc8
=#
#;
g
c- (aximum angle mae by the bloc8 6ith vertical cos#
#gl
=
- )oss of .0( )
#
#(m
( m =
+
M 0 +: /oefficient of Restittion:
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1. Doefficient of restitution# 1
1 #
.eu u
=
a- Fhen a boy falls from a height h an rises to a height h1after first collision
6ith the groun then 1h
eh
=
b- ;eight rise after n bounces hn = e#n. h
c- Eelocity after n bounces vn = en. E = en
#gh
#. If a boy falls from a height h the total istance travele before coming to rest.#
#.
ex h
e
1 += 1
%. If a boy falls from a height h the total time ta8en to come to rest.
# 1
.1
h et
g e
+ =
'. If a boy hits the groun 6ith a velocity v1an rebouns 6ith a velocity v#then
## #1 # 1
p I0e
p I0
= = =
. Percentage loss of velocity after n bounces = @1 en- X 1"".2. Percentage loss of momentum after n bounces = @1 en- X 1"".
>. Percentage loss of 0 after n bounces = @1 4 e#n- X 1"".J. A boy hits groun 6ith a velocity u ma8ing an angle 1 6ith vertical an rebounces
6ith a velocity v at an angle # then
#
1
coscos
eu
=
If it is a smooth surface then1
#
tan .tan
e
=
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. >(R 0 P(>ER ENERB*
M 0 +: >orC:
1. a- If a force N7G acts on a boy ma8ing an angle 6ith hori/ontal an isplaces the
boy through a istance N,G 6or8 one "F 7., 7,cos .6hen ?" F ".= = = =
b-F Dos @If 7 an , are same-.#c-F Dos
@If 7 an t are same-.#. For8 one against gravity F = mgh.
%. For8 one in imparting velocity to a boy #1
F m#
=
'. For8 one against friction on a level surface. F 7x, mgs. = = coefficient of friction.
. For8 one in compressing or elongating a spring#
#1 7 1F 8x 7x.# #8 #
= = =
4 spring constant x 4 elongation2. For8 one in moving a boy up on to a smooth incline plane is F = @mg sin - ,.>. For8 one to compress or expan a gas F Px E= at constant pressure.J. a- For8 one in pulling the bob of a simple penulum asie through an angle from
the vertical is
( ) #F mgl 1 cos #mglsin .# = =
m 4 mass of the bob l length of penulumb- For8 one in isplacing the penulum from an angle 1 to an angle # is
( )1 1 #F mg cos cos=
?. For8 one in pulling asie a bar through an angle from the vertical is
( ) #1
F mgl 1 cos mglsin# #
= =
m 4 mass of the ro l length of the ro
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1". *he 6or8 one in lifting a boy of mass NmG an ensity NsG in a liqui of ensity N1Gthrough a height NhG uner gravity is
1
s
F mgh 1 .
=
11. For8 one by external resultant force on a boy is equal to change in 0 of theboy
( )# #1
F m u 0.#
= =
1#. For8 one in lifting 6ater from a 6ell from a 6ell of epth h is "m
F m gh#
= +
m(ass of 6ater m"(ass of rope1%. A ro of mass m an length l lies hori/ontally on a NfloorG. *he 6or8 one in rotating
the ro through an angle about one ege in a vertical plane is givenF = K mgl sin .
1'. In the above point the 6or8 one in bringing the ro to vertical position is
F = K mgl.1. A uniform chain of length l an mass m is place on a friction less table such that
1&nth of itslength hanging over the ege. *he 6or8 one in pulling the hanging part
on to the table is #.#
mglF
n=
M 0 ++: Po4er:
1. Average po6er P = F&t.#. Instantaneous po6er = 7 x E = . .7 E%. *he po6er of machine gun firing NnG bullets each of mass NmG in one secon 6ith
velocity NvG is #1. .#
P n m=
'. A motor sens a liqui 6ith a velocity NEG in a tube of cross section NAG an NG is the
ensity of the liqui then the po6er of the motor is %1
.#
P A+E =
. A motor lifts m 8g of 6ater to a height h in t sec. *hen the po6er of the motor ismgh
Pt
= .
If is the efficiency of the motor then .mgh
Pt
=
If the 6ater comes at of the pipe 6ith a velocity then#1
# .mgh m
Pt
+=
2. 0xtra po6er require to 8eep a conveyor belt moving 6ith constant velocity if
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gravel is falling the belt at a rate of m&t is#.
+mP
+t
=
>. A vehicle moves 6ith a constant velocity E on a rough hori/ontal roa. *he frictionalforce acting on the vehicle is f then the po6er o f the engine P= f x v.
M 0 +++: Ener5:
1. 0 = K mv#.#. P0 = mgh.%. P0 of a liqui in a capillary tube P0 = @- @ah- g h = agh# & #.
4 ensity of liquia 4 area of cross section of the capillary tube.h 4 height of the liqui in the capillary tube.
'. P0 of a bloc8 place on the groun =h
lbh g. .#
. P.0 in a spring = K 8x# .
M 0 +: ,a4 of conseration of ener5:1. A boy is roppe freely from a height h uring collision 6ith groun it looses x !
of energy an then rises to a height NhG then( ) 11"" x mgh mgh $
1""
=
@1""x- remaining energy$1 1"" xh h.
1""
#. In the above question if the boy is thro6n vertically o6n6ars 6ith a velocity Ethen
# 11"" x 1mgh m mgh
1"" #
+ =
%. A boy of mass m is roppe from a height h on to a spring of spring constant .If x is the compression prouce in the spring then
( ) #1
mg h x 8x#
+ =
If x Q Q Q h mgh = K 8x#
'. A boy of mass m moving 6ith a velocity E collies 6ith a spring of spring constant an comes to rest. If x is the compression prouce in the spring then
K mv# = K 8x#
M 0 : >orC 0 Ener5 theorem:1. For8 one = Dhange in 0$
7 x , = K m @E# u# -.#. A boy of mass m is roppe from a height h it reaches the groun 6ith a velocity
( )E E #gh< 6or8 one by air resistance F = mgh 4 K mv# .
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%. A 8nife ege of mass m is roppe from a height h on to a table. It penetrates aistance x into the table. If the resistance offere by the table is 7 then7 x x = mg @h+x-. if x Q Q Q h then 7x= mgh
M 0 +: Relation 8et4een E and Momentm:
( )
( )
#
#
P 11. 0 0 if P is same .
#m m
0 P if m is constant .
=
( )
( )
0#. P #m8 P m If0 is same
P 0 If m is same .
=
%. *6o boies of ifferent mass are moving 6ith same momentum if same brea8
force is applie on them. *hen
1 #
# 1
10 Accoringly6.e.theoremF. 0.
m1 , m
a-7x, 0 " 0 , 0m , m
=
= =
)ighter boy 6ill travel more istance bet6een coming target.b- In pulse = change in momentum
7 x t = P 4 " ,in 7 an P are same t 6ill7 x t = P t1 t#= 1 1.
,o both ta8e same time to come to rest.'. *6o boies of ifferent mass are moving 6ith same 0. If same brea8 force isapplie on them then
1 #
P m
a-7 x, 0 ,ince7 an 0$are same
,6illbesame, , 1 1.
=
=
,o both travel same istance before coming to rest.
b- 7 x t = Pi 4 Pf it P m
i # #t t m m=
;eavier boy ta8es more time to come to rest.
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". /ENTRE (F MA
M 0 +: /oordinates of center of mass:
1. Particles istribute in space. If @x1y#/1- @x#y#/#- 4 are the position coorinatesof particles of masses m1 m#4 the position coorinates of their center of mass
are
1 1 # # n ncm
1 # n
1 1 # # n ncm
1 # n
1 1 # # n ncm
1 # n
m x m x ..... m xx
m m ..... m
m y m y ..... m yy
m m ..... m
m / m / ..... m //
m m ..... m
+ + +=
+ + +
+ + +=
+ + +
+ + +=
+ + +
#. In vector notation. If r1 r# r%.. Are the position vectors of particles of
masses m1 m# m% *hen the position vector of their center of mass is1 1 # # n n
1 # n
m r m r ..... m rr
m m ..... m
+ + +=
+ + +
M 0 ++: elocit5 /enter of Mass:
%. a- Eelocity of center of mass
If 1 # % nEE E .....E are the velocities of particle of masses.
m1 m# m% mnthe velocity of their center of mass is
1 1 # # % % n n 1 # % ncm
1 # % n 1 # % n
m m m ..... m P P P ...PE
m m m ..... m m m m .....m
+ + + + + += =
+ + + + +
cm 1 # nb- (E P P ... P= + + + i.e. total momentum of the system is the prouct of massof the 6hole system an the velocity of the center of mass.
c- If E1an E#are the magnitues of velocities of t6o particles of masses m1an m#then the magnitue of velocity of c.m.
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M 0 +: Motion on the hori ?%?1%#22>. 666.asifiitphysics.vriti.com #
D. FR+/T+(N
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F r i c t i o n
N
m g
F s i n
F c o s
( )
6sinmg7 . angleoffriction
cos sin cos
= =
+
7 6ill be minimum if = .
min #
mg7 6sin .
1
= =
+
J. In case of pushing3 = mg + 7 sin .
7 = Lmg + 7 sin M87cos fa .
m
=
(inimum pushing force require to isplace the boy
( )
6sinmg7 .
cos sin cos
=
+ "if ?"+ 7 becomes infinity.
;ence ?" ?"+ < < N
m g
F s i n
F c o s
F
?. Acceleration 87 f
am
= @If force is applie hori/ontally-.
1". Acceleration of a freely falling boyf
a g .m=
11. eceleration of a vertically proBecte boyf
a g .m
= +
@ffriction force ue to air-.
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1#. 7or a vertically proBecte boy au mu
t .g f&m mg f
= =+ +
@If air resistance is ta8en into account-#u
;f
# gm
= +
Eelocity before reaching grounmg f
umg f
=
+
1%. (inimum force require to isplace a boy is applie an the same force is
continue then the acceleration of the boy ( )s 8a g.=
1'. A chain of uniform length N)G is place on a rough hori/ontal table. *hecoefficient of friction bet6een the chain an table is then the maximumfractional length of chain that can be hung freely from the ege of the table is.x
) 1
=
+
(inimum fraction of length of chain that can be on the table is1
.1 +
1. bloc8 slips or slies on the floor. 7rictional force on the bloc8
8mg.= *he acceleration @a- of the bloc8 relative to lorry is1
8a a g=
12. Dar moving on an umban8e circular roaa- Fhen a car goes aroun an unban8e circular roa the static frictional force
bet6een the 6heels an the roa provies the necessary centripetal force.
b- (aximum spee 6ith 6hich curve can be negotiate 6ithout s8iing ismax sE rg=
s = coefficient of static frictional bet6een 6heels an the roa.c- *he maximum angular velocity is
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smax
gF
r
=
1>. a- If a bloc8 having initial velocity u slies on a rough hori/ontal surface ancomes to rest the acceleration of the bloc8 is
8a g=
b- istance travele by the bloc8 before coming to rest is#
#
8
u, , C# g
=
c- *ime ta8en by the bloc8 to come to rest is
8
ut t C
g=
7od5 on ertical srface.
1J. Fhen a boy of mass m is presse against a vertical surface 6ith a force P thena- 3ormal reaction 3 =P.
b- )imiting frictional force )f P.=
c- 7rictional force bet6een the boy an surface P or mg 6hich ever is less.- (inimum force require to slie the boy up6ars.
7 = mg + P
e- (inimum force require to prevent the boy from sliing o6n 7 = mg P
P N
m g
f
1?. A boo8 of mass m is presse bet6een t6o hans by applying a hori/ontal force P
6ith each han the boo8 oes not slie ifmg
.#P
=
#". A vehicle is moving on a hori/ontal surface. A bloc8 of mass NmG is stuc8 on thefront part of the vehicle. *he coefficient of friction bet6een the truc8 an thebloc8 is . *he minimum acceleration 6ith 6hich the truc8 shoul travel sothat the boy oes not slie o6n is
ga .=
Lmg = maM
mooth inclined plane:
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#1. ?%?1%#22>. 666.asifiitphysics.vriti.com #?
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- If the boy is sliing o6n 6ith uniform velocity then
8 8a " f mgsin tan .= = =
e- 3et contact force acting on the boy = # #f 3 .+
mg
sin
m g mg
cos
N
f
#%. If the bloc8 slie o6n from the top of the incline plane. Eelocity at thebottom of the plane is
( ) ( )8 8E #gl sin cos #gh 1 cot= =
#'. In the above case time of escent is
( )8#)t g sin cos=
#. *he time ta8en by a boy to slie o6n on a rough incline plane is NnG times thetime ta8en by it to slie o6n on a smooth incline plane of same inclination anlength then coefficient of friction is
#
1tan 1 .
n =
#2. If a bloc8 is proBecte up a rough incline plane the acceleration of the bloc8 is
8a g sin cos = +
#>. a- *he istance travele by the bloc8 up the plane before the velocity becomes/ero is
( )
#
8
u,
#g sin cos=
+
b- *he time of ascent is ( )8
ut .
g sin cos=
+
c- If the boy slies o6n 6ith uniform velocity on the same plane then#
8
C Csin cos then, t
'gsin #gsin = = =
#J. In the above case if time of ecent is n times the time of ascent then#
#
n 1tan .
n 1
= +
#?. a- 7orce neee to be applie parallel to the plane to move the bloc8 up 6ith
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constant velocity is
( )8 87 mgsin mgcos mg sin cos .= + = +
b- 7orce neee to be applie parallel to the plane to move the bloc8 up 6ith anacceleration a is
87 mgsin mgcos ma.= + =
%". If bloc8 has a tenency to slie the force to be applie on the bloc8 parallel anup the plane to prevent the bloc8 from sliing is
( )s s7 mgsin mgcos mg sin cos .= =
%1. If a force P is applie on the boy up the plane then the acceleration of theboy is sin .
( )8P mg sin cosa .m
+ =
%#. An air craft of mass m travels through a istance , on a run6ay an ta8e off6ith a velocity E in t sec. *hen.
a- 7orce applie by the engine#m
7 mg#,
= +
b- For8 one by the engine #1
F mgs m#
= +
c- Po6er of the engine#1mgs m
#P .t
+ =
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M ++: Motion of a 8od5 in a ertical circle:
Dase I If a boy of m is tie to a string of length 1 an rotate in a vertical circle
6ith uniform spee ( )v. v g1 .>
1". *ension in the string 6hen it ma8es an angle 6ith vertical is given by#m
* mgcos .
l
=
a- *ension at the highest point @ = 1J"" -
( )#
#minh
m* mg ml mg * .
l
= = =
b- *ension 6hen the string is in hori/ontal position ( )"?"=#
#m* mll
= =
c- *ension at the lo6est point ( )"" . =
( )#
#
) max
m* mg ml mg * .
l
= + = + =
- ifferent bet6een maximum an minimum tensions.
max min* * #mg. =
e-#
max
min
n 1*If nthen .
* n 1 1g
+ = =
f- *ime perio ( )# l l
* # Ifv g1 .g
= = =
g- *otal energy at the highest point#
; ;h
10 0 P0 mv #mgr.#= + = +
h- *otal energy at the lo6est point
i- # #) ) )1 1
0 0 P0 mv " mv .# #
= + = + =
B- ; ) ; ) ; )0 0 0 "$ 0 0 0 #mgr$ P0 P0 P0 #mgr. = = = = = =
/ase 0++:
11. If the boy rotate 6ith non uniform spee. If u is the velocity at the highestpoint an v is the velocity at the lo6est point then.
a- #u 'gl. = +
b-#
; min
mu* mg * u gl
l = = >
#
max
m*) mg *
l
= + =
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c-#
max#
min
u gl*.
* u gl
+=
-#
max#
min
*u n 1If nthen .
* n
= =
/ase 0+++:
1#. If the boy is revolving 6ith critical speea- (inimum or critical spee of the boy 6hen the string ma8es an angle 6ith
vertical is given by
( )gr % #cos . = +
At the lo6est position ( )"1 gr " = =Q
At the hori/ontal position ( )"1 %gr ?" = =Q
At the highest point ( )"% gr 1J" = =Q
1 # %
1 # %
v v v % 1
0 0 0 % 1.
=
=b- *ension in the string 6hen it ma8es an angle 6ith vertical * = %mg @1 + cos-.
At the lo6est position ( )"1 max* 2mg * " .= = =Q
Fhen the string is in hori/ontal position ( )"# max* %mg * ?" .= = =Q
At the highest position ( )"% min* " * 1J" .= = =Q
max min* * 2mg. =
c- *otal energy at the highest position.
; ; ;
mgr 0 0 P0 #mgr mgr.
# #
= + = + =
*otal energy at the lo6est position
) ) )
0 0 P0 mgr " mgr.
# #= + = + =
) ; ; ) ; )0 0 0 #mgr$ 0 0 0 "$ P0 P0 P0 #mgr. = = = = = =
/ase 0+:
1%. ,imple penulum is given a hori/ontal velocity u at the lo6est position @meanPosition- then
a- Ifu #gr< the boy oscillates about A.
b- If #gr u gr< < the boy leaves the 6ithout completing the circle.c- Ifu gr= the boy completes the circle.
- ;eight at 6hich velocity u = ". is h = u#g.
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e- ;eight at 6hich tension#u rg
* "ish .%g
+=
f- Angle 6ith vertical at 6hich velocity#u
v ".isDos 1 .#gr
= =
g- Angle 6ith vertical at 6hich the tension #* ".isDos #&% u &%gr.= =
h- *ension in the string at an angular isplacement
6ith vertical is#* mu &r mg@# %cos -.=
1'. ,afe spee of a car going on a convex brige to travel in contact 6ith the brige
is v gr.=
1. A ball of mass NmG is allo6e to slie o6n from rest from the top of a incline ofheight NhG. 7or the ball to loop in a loop of raius NrG
a- (inimum height of incline h = r.b- NhG is inepenent of mass of the ball.
12. A ball of mass N(G is suspene vertically by a string of length NlG. A bullet of mass
NmG is fire hori/ontally 6ith a velocity NuG on to the ball stic8s to it. 7or thesystem to complete the vertical circle the minimum value of NuG is given by
( mu gl.
m+
=
1>. If the bob of the simple penulum is given an initial isplacement then
a- Eelocity at the lo6est position ( )#gl 1 cos . =
b- *ension at the lo6est position * mg % #cos .= 1J. A boy is place on the top of a hemispherical bo6l an it is given a hori/ontal
velocity v then
a- *hrust on the bo6l 6hen the raius vector turns through an angle
( )#m
3 mg # %cosr
= +
*he boy looser contact at# #
Dos .%rg %
= +
b- If v = " then 3 = mg L# 4 % cosM.
*he boy looser contact at ( )1Dos #&% or h #r&%. = =
1?. 0quation of motion#
# 1 1
# # 1 ## 1
1
a-6 6 t c- 6 t t#6 6
b-6 6 # - t#
= + = + + =
Also 6 = # n @if n is r.p.s-F= # n&2" @if n is in rpm-.
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#". *orque )# 1l l
I .t t
= = =
#1. For8 one 6 = x .
##. Po6er P = x6.
#%. 0 = K I6#.
#'. For8 one ( )
# #
# 1
16 I 6 6 .
#=
#. If a boy of mass m is revolving in a circular path of raius r 6ith a velocity vthen angular momentum of the boy about origin is
) = mvr = mr# 6 = I6 6here I = mr#
#2. A boy of mass m is moving 6ith a velocity v along a straight line p representeby ax + by + c = " then the angular momentum of the boy about origin
# #
c) m .
a b=
+#>. Angular momentum of a proBectile about point of proBection 6hen it is at its
highest point.# #u sin
) mucos ; mucos .#g
= =
#J.#)
0#I
=
#?. (oment of inertial # #I mr (8 .= =
Parallel axes theorem #I Ig (x .= +
Perpenicular axes theorem x y /I I I .+ =
%". 7or a uniform circular ring.# %
# %
I (R aR +
a area of cross section
+ +ensity.
i.e.I R if m is same$ I + if R is same. I R if + is same
%1. 7or a uniform circular isc.
( )#
# '
#
mI (R R t t thic8ness
t
i.e. I R if m is same.
I R if is same.
%#. 7or a sphere& %
# #& %
mI (R R t
.
#i.e. I R if m is same.
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I R if is same.
%%. Rolling of a boy 6ithout slipping on a hori/ontal surface.a- *he boy possesses both transnational an rotational 8inetic energy.b- *otal energy of the boy
( ) ( )# # # # # # #*1 1 1
0 mv I .0 mv 1 &r .0 1 &r .# # #
= + = + = +
* *ransalational 0 of the boy.c- 7raction of energy associate 6ith translation motion ( )# #1& 1 &r .= +
- 7raction of energy associate 6ith rotational motion ( )# #R&0 1& 1 r &8 .= +
%'. Rolling of a boy o6n an incline 6ithout slipping.a- Acceleration of the boy
( )# #gsin
a inclinationoftheplane.1 &r
=
+b- Eelocity acquire by boy on reaching the bottom
# ##ghE .1 8 &r= +h height of the incline.c- *ime ta8en to reach the bottom
#
#
8#h 1
rt 1&sin .
g
+
=
%. Fhen a boy rolls o6n 6ithout sliing on an incline plane 6ith an inclinationan height NhG.
,.3o. g sin 1"gh&>
#. isc K r# #&% g sin 'gh&%
%. ,oli cyliner K r# #&% g sin 'gh&%
'. ;ollo6 sphere #&% r# %& g sin 2gh&
. Ring r# K g sin gh
2. ;ollo6 cyliner r# K g sin gh
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M(MENT (F +NERT+A (F (ME REB',AR 7(6+E
,.3o. & (r#
2. ;allo6sphere
a- Any iameterb- Any tangent
#&% (r#
&% (r#
>. *hinuniform ro
a- Passing through its centre an normal to lengthb- Passing through one en an normal to its length
(1#&1#(1#&%
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1%. BRA+TAT+(N
M 0 +: epler?s ,a4s:
1. If * is the time perio of plant an R is average istance of the planet from ,un*hen
( )#
# # # %'* R or * RT(
=
#. If v1 is the velocity of the planet 6hen its istance from ,un is an v#is thevelocity 6hen the istance is #then accoring to la6 of conservation of angularmomentum.
# #1 1 # # 1 # #+ v + v + 6 + 6 .= =
M 0 ++: Ne4ton?s ,a4 of raitation:
%. Travitational force bet6een t6o boies of manes m1 an m# separate byistance
is given by. 1 ##Tm m
7 .
=
'. Fhen t6o similar spheres each of raius r are in contact then the gravitation
force bet6een then 7 'r . . Travitational force acting on a boy of mass m place at a istance x istance x
from the centre @x Q R- is given by %T(m
7 xR
=
M 0 +++: Relation 8et4een and B:
2. #T( '
g . g TR R %
= = ensity of earth
M 0 +: ariation of 4ith heiht:
>. Acceleration ue to gravity at a height h above the surface of the earth is given
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- *ime perio of the satellite%
"
# R R %* # # #
6 g T( ' T
= = = =
M 0 +++: Escape elocit5:
11. 0scape velocity#
#c #Tm J#gR TRR % = = =
c "v # v=
M 0 +: Ener5 of a satellite:
1#.T(m T(m
P0 0r #r
= = @r = raius of the orbit$ r = R + h-.
*otal energyT(m
0 P0 0 .#r
= + =
1%. For8 one in shifting a satellite from orbit of r1to another orbit of raius r#isgiven by
# 11 #
T(m 1 16 0 0
# r r
= =
1'. For8 one by lifting a boy to a height h @h is comparable to R" = change in P0-
h T(m h6 mgR
R h R R h = = + +
1. If a boy is proBecte 6ith a velocity v from the surface of the earth then theheight raise by the boy can be foun by
#e e
1 hm mgR . v comparable to v . If v v $ h
# r h
= = = = +
If v O vethen velocity at infinity can be foun using la6 of conservation ofenergy i.e.
# #1 #
1 T(m 1m m .
# R #
+ =
v1 = Eelocity of proBection.v#= Eelocity at infinity.
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11. .).M
M 0 +: 6isplacement# elocit5 and Acceleration:1. isplacement after t sec y = r sin ( )6t . If particle starts from mean
position the "$y rsin6t. = =
If particle starts from extreme position ?"$y rcos6t. = =
#. Eelocity after t secons E r 6 cos 6t.=
Eelocity 6hen the isplacement is # #y 6 r y . =
At mean position v = r6 @maximum-At extreme position v = " @minimum-
%. Acceleration a = r6#sin 6t
i.e. a = 6#y.at 0.P a = r6#@maximum-.At (.P a = " @minimum-.
'. 7orce 7 = ma = mr6# sin 6t.= m6#y
at (.P = 7= " at 0.P 7 = mr6#
. *ime perioy#
* # .6 a
= =
2. 7requency
1#1 6 a
n .* # y
= = =
( )
( )
>. In y r sin 6t
6t is calle phase
is calle phase constant.
=
M 0 ++: Ener5 of the particles:
( )
( )
( )
# # #
# #
# #max min
J. 0 1m6 r y
1At (.P 0 m6 r maximum
#
At0.P 0 " minimum0 0 1
Average 0 m6 r# '
=
=
=+
= =
?. P0= K m6#y#
At (P P0 = " @minimum-
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Average P0 = U m6# r#At 0P P0 = K m6# r#@minimum-.
1". *otal energy 0 = K m6# r#
# ## # # # #
max max #
1 #m r0 P0 0 m6 r #m n r .
# *
= = = = =
M 0 +++: Time period of a simple pendlm:
11. *ime perio of a simple penuluml* # .g
=
1* l *
g
1# # #
1 1 1 #
g* l *
* l * g = =
1#. 7or a simple penulum in a lift moving up 6ith an acceleration or coming o6n
Fith a ecelerationl
* #
g a
=
+1%. In a lift moving up 6ith eceleration or coming o6n Fith a acceleration time
perio of simple penuluml
* #g a
=
In a freely falling lift effg = ". ,o the penulum oes not oscillate. *he timeperio is infinity.
1'. 7or a penulum of infinite length @or- for a boy roppe in tunnel along the
iameter the time perioR
* # J'.2min.g
= =
1. *ime perio of a secons penulum is #s.
*ime perio remains same if1 #
1 #
l l.
g g=
Its length is #g
l .=
A secons penulum is ta8en from one place to another place then the Dhange in
the length of the seconsG penulum # 1#g g
l .
=
,o that time perio remains constant
12. If the simple penulum is in a cart moving hori/ontally 6ith an acceleration a
then time perio # #1
* #g a
= +
If the cart is moving along a circular path of raius r 6ith a spee v then a = v#&r
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If the cart is sliing o6n an incline plane incline at angle 6ith the
hori/ontall
* # .gcos
=
1>. Fhen t6o simple penulum of lengths l1 an l#are get into vibration in thesame irection at the same instant 6ith same phase.Again they 6ill be in same phase after he shorter penulum has complete noscillations. *o fin the value of n
n *s= @n 1- *1an * 1
( ) 1
s s
n * nor .
n 1 * n 1 l = =
l
l
, = shorter$ l= longer1J. *6o penulum of lengths l1an l#@l1O l#- start vibrating from the mean position in
the same phase. *hey 6ill be again in the same phase at the mean position after
larger penulum completes n oscillations an the shorter one completes @n + 1-oscillations. *hen
( )
1 1 # #
## 1
#
n l n l .
n ln l n 1 l .
n 1 l
=
= + =+
1?. *6o simple penulum of time perio *1 an *# start vibrating from the meanposition in same phase. *he phase ifference bet6een them after t sec. is givenby
t
# 1# 1 # 1
# # 1 16 t 6 t # t .* * * *
= = =
*hey 6ill be in same phase after a time1 #
1 #
**t .
* *=
M 0 +: Time period of a loaded sprin:
#". *ime perio of a loae spring(
#8
= 6here 8 = 7&x.
*a8ing the mass of the spring m into consierationm
(%* # .
8
+ =
*ime perio of the springx
* #g
= 6here x is elongation uner the loa.
#1. If a spring is cut in n equal parts the spring constant of each part is 81= n8.*hen time perio of each part uner the same loa is given by
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1 1** 3 n3.n
= =
##. If a spring of spring constant NG an length N l G is cuts in to t6o springs oflengths N l1G an N l#G then the spring constants of the t6o parts is
( ) ( )1 # 1 #1 #
1 #
an .
+ += =
l l l l
l l
#%. Fhen t6o spring of force constants 1 an # are connecte in series. *hen theeffective force constant is
1 #
1 #
.
=
+
#'. Fhen t6o spring of force constants 81 an 8# are connecte in parallel. *heeffective spring constant is = 81 + 8# .
#. * 1an *#are the time perios of t6o springs uner same loa.If the springs are connecte in series an the same loa is attache.
*ime perio # #1 #* * * .= +
If the springs are in parallel *ime perio1 #
# #1 #
*** .
* *=
+
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12. E,AT+/+T*
Re storing 7orce 7
1. ,tress .Area A= =
#. a- )ongituinal strainl
l
= percentage change in length = longituinal
strain x 1"".b- ,hear strain
)ateral +isplacement bet6een t6o layers l
Perpen+icular +istance bet6een t6o layers l
= = =
c- Eolume strain .
=
,hear strain = # x longituinal strain volume strain = % x longituinal strain.M 0 +: *on?s modls:
%. :oungGs moulus #)ongituinal stress mgl7l
: .)ongituinal strain Ae r e
= = =
7 Applie force$ A Area of cross section) Sriginal length$ e 0longationR Raius of the 6ire.
'. 0longation
#1 1 # #
## # 1 1
7l e1 7 l r :e .
Ay e# 7 l r :
= =
If 7 an : are same
#
# '
l l me .
r m r If youngGs moulus = stress then strain = 1 an final length
= # x initial lengthm mass of the 6ire.
( )
( )
( )
# '1 1 #
# '# # 1
1 1
# #
1 #
# 1
e l rif m is same
e l r
e mif r is same
e me m
if l is samee m
= =
=
=
Fhen t6o 6ires are connecte in series an same force is applie thena- ,tress is same in both 6ires
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b- Ratio of the strains = :#&:1c- *otal elongation e = e1+ e#
Fhen t6o 6ires are connecte in parallel an same force is applie thena- ,train is same in both the 6iresb- Ratios of the stresses = :1&:#c- *otal force 7 = 71+ 7#
. Fhen a rubber cor is suspene vertically from a support it elongates uner itso6n 6eight.
#l gelangation e .
#:=
2.
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@ = ensity of material-1". Fhen a pressure p is applie on a substance its ensity changes. 3e6 ensity
G = 8& @8 p -11. Dompressibility *he reciprocal of bul8 moulus is calle Dompressibility
1D
8=
7or incompressible substances D = " 8 = M 0 +: Poisson?s ration & relation 8et4een *# n and C:
( )lateral contraction strain
1#. Poissions Ratiolongituinal elongation strain
rtransverse strain ! ecrease in raiusr
llongituinal strain ! increase in lengthl
=
=
= =
i- PoissonGs ratio has no unit an no imensionsii- *heoretical limits of 1 to".
iii- Practical limit of " to".
iv- If ". = the substance is perfectly incompressible.1%. Relation among elastic constants : n
( )
( )( )
? 1 %i- ii- y #n 1
y 8 n
%8 #niii- y %8 1 # iv-
# n %8
= + = +
= =
+
1'. Percentage change in volume of a 6ire ue to elongation@!E- = @!1- +#@!r-! change in volume = ! change in length + # @! change in raius-
M 0 : train Ener5
1. For8 one ins stretching 6ire = strain energy store in the 6ire= K x 7orce x elongation = K 7 x e
#
#
1 :Ae :Ae6 ,.0 7
# l l
1 7l 7le# AE lA
= = =
= =
Q
Q
12. If t6o material of same material are stretche by same amount i.e. e is same
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( )
# '
#
' #1 1 1 #
' ## # # 1
1 # 1 1
# 1 # #
A r r m6
l l m l
6 r 6 lIf m is same
6 r 6 l
6 m 6 mif r is same if 1 is same.
6 m 6 m
= =
= =
1>. If t6o 6ires of same material are stretche by same force then#
# '
l l l m6
A r m r
1J. 0lastic strain energy = K stress x strain x volume,train energy per unit volume = K stress x strain
( )( )
##stress :
strain .#: #
= =
1?. If l1an l#are the lengths of a 6ire uner tensions *1 an *# the actual length
of the 6ire1 # # 1
# 1
l* l *
.* *
= *hermal stress evelope in a boy ue to change in temperature : t=
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13. 'RFA/E TEN+(N
M 0 +: Force de to .+:
1. ,urface tension 7orce 7* .)ength l= =
# 1
6or8 one 6increase in Area A A
= =
( )# 16 * A A =
#. a- 7orce in aition to 6eight require to lift a thin 6ire of length l from liquisurface is 7 = #l x * = #*l.b- 7orce require to lift a thin8 circular plates of raius R from liqui surface is7 # Rx* # R*.= =
c- 7orce in aition to 6eight require to pull a circular ring from 6ater surfaceis ( )1 #7 # * r r= +
r1an r#are internal an external raius.If it is a thin ring then7 ' r* = .- 7orce require to pull a thin8 rectangular plate of length 1 an breath b is
from liqui surface is ( )7 #1x* #bx* #* 1 b .= + = +
e- 7orce require to pull a thin8 rectangular frame from liqui surface is
( ) ( )7 # x #* 1 b '* 1 b .= + = +
f- 7orce require to pull a capillary tube from 6ater surface is 7 ' R*.=
%. If a small rop of 6ater is squee/e bet6een t6o plates the force require toseparate the plates
#*7 A.
=
A= Area of 6ater la6yer. = *hic8ness of 6ater layer.'. A metallic 6ire of ensity floats hori/ontal in 6ater. *he maximum raius of
the 6ire so that the 6ire may not sin8 6ill be @surface tension of 6ater = *-.
#*a-
g
. A metallic 6ire of ensity is laying hori/ontal on the surface of 6ater. *hemaximum length of 6ire so that it may not sin8 6ill be@surface tension of 6ater = *-
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# *a-
g
M 0 ++: rface ener5:
2. ,urface tension6or8 one 6
* 6 * Aincrease in area A
= = = =
*his 6or8 6ill be store as surface energy0 *. A= .>. a- For8 one in forming a liqui rop of raius #R ' R *.= b- For8 one in forming an air bubble of raius r is #6 J r *.= c- For8 one in increasing the raius of a liqui rop from R1to R#
( ) ( )# ## 1 # 16 * A A ' * R R= =
- For8 one in forming of a soap film of si/e l x b is = #* lbe- For8 one in increasing the raius of a soap bubble from R1to R#
( ) ( )# ## 1 # 16 * A A J * R R= =
J. Fhen a big rop of raius R is split in to n ientical rops each of raius r.
a- R = n1&% r.b- Increase in surface area ( ) ( )# # # 1 %A n ' r ' r ' R n 1 .= = =
c- For8 one in splitting the rop ( )# 1&%' R * n 1 .=
- If this process ta8es place uner aiabatic conitions the temperature of V
the rop ecreases by%* 1 1
t .s r R
= ?. Fhen n ientical small rops each of raius r combine to from a big rop of
raius R then
a- R = n1&%
r.b- ecrease in surface area ( ) ( )# # # 1&%n' r ' R ' R n 1 .= =
c- 0nergy release in this process ( )# 1&%' R * n 1 .=
- If this process ta8es place uner aiabatic conitions then the temperature
of the rop increases by%* 1 1
t .s r R
= ( )# 1&%L' R * n 1 ms *M =
e- If the energy converts into 0 then ( )# 1&% #' R * n 1 1mv . =
2* 1 1E .
r R
=
1". *6o rops of raii r1an r#combine in vacuum uner isothermal conitions. *hen
if r is the raius of the rop forme % % %1 #r r r .= +
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If P"is outsie pressure total pressure insie =P"+ excess pressure
1. If t6o bubbles of raii r1an r# are in contact the raius of curvature of the
interface is1 #
# 1
rrr .r r
=
12. An air bubble of raius r is at a epth h is 6ater.#*
Pr
=
Insie pressure i a#*
P P hg .r
= + +
1>.
1&% 1
1 # # #
1 1 1 1
P r E A.
P r E A
= = =
1J. A bubble of raius r1 is insie another bubble of raius r#. *he raius of a singlebubble 6hose excess pressure is equal to ifference in pressure bet6een insieof inner bubble an out sie the outer bubbles is
1 # 1 #
1 1 1 '* '* '* R r r R r r
= + = +
1?. A long capillary tube of raius r is fille 6ith 6ater an place vertically. *he
height of 6ater column remaining the capillary tube'*
.rg
=
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M 0 +: iscos force:
1. Eiscous force 7 A Eelocity graient.x x
=
7orce require to rag a plate place on an oil layer of thic8ness t 6ith a
velocity v is given by 7 A
t
=
Area of the plate.
M 0 ++: Poiseille?s e;ation:
2. Eolume of liqui flo6ing through a capillary tube in unit time is given by
( ) '' '1 1 1 # #'
# # # 1 1
hg rPr E h r lE .
J l J l E h r l
= = =
P Pressure ifference across the tube. P = hg.r Raius of the capillary$ 1 )ength of the tube.
'rE
l
@if P an are same-.
Also 'P J l
E 6here RR r
= =
it is 8no6n as flui resistance.
%. Fhen t6o capillary tubes are connecte in series then
1 #
1 #
E E E$
P P P
= =
= +
1P Pressure ifference across the first tube.#P Pressure ifference across the secon tube.
'
1 1 #
# # 1
P l r.
P l r
=
If the t6o tubes are replace by a single tube of raius r then
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1 #' '
1 #'
1 # # ' '1 #'
l l lr r r
1 1 1if 1 1 1 then
r r r
= +
= = = +
'. Fhen t6o capillary tubes are connecte in parallel
1 #.
1 #
P P
E E E
=
= +
'1 1 #
'# # 1
E r l.
v r l
=
If the t6o tubes are replace by a single tube of raius r
'rl
=' '
1 #
1 #
r r.
l l
@If r1= r#= r-
*hen1 #
1 1 1
l l l
= +
M 0 +++: toCes formlas:
. Fhen a spherical boy of raius r is moving through a flui 6ith a velocity Ethen the viscous force acting on boy is given by 7 2 rv.=
1&% 1&%7 r E m
After some time the boy attains constant velocity calle terminalvelocity in this case.
( )*
# #&%
*
mg 2 rv if buoyancy is neglecte
m
E r E .r
=
If n ientical rops moving 6ith terminal velocity v. Dombine to form a signal mpthen terminal velocity of the big rop E = n#&% v
M 0 +: /ontinit5 e;ation:
2. #1 1 # # 1 1 # #1
E a v a v r v r v .A
= =
Eolume of liqui flo6ing through a tube in unit time @or- volume flux q = av.
M 0 : 7o5ant force:>. If a boy of volume E ensity sis immerse in a liqui of ensity 1then
a-
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1 1 s 1 1 sE g E E E . = =
c- Apparent 6eight of the boy 61= mg 4 7b.
( )
( )
1s 1
s 1 1 s 1
1 s
i- If+ + $ 6 ".
ii-If+ + $6 vg + +
mg 1 + & + .
=
> =
=
3ote Fith increase in temperature tbecreases hence 61
increases.M 0 +: 7ernolli?s theorem:
J.# #
1 1 # #1 #
P E P Egh gh
# #+ + = + + @all are in ,I units-
#P 3&m also P hg. =
If the liqui is flo6ing hori/ontally then# #
1 1 # #
1
P E P E # #
+ = +
Dhange in 0 per ay unit mass# #
# 1 1 #E E P P .#
= =
?. If P1an P#are pressures at the top an bottom of the 6ings of the aero plane
then the ynamic lift on the aero plane ( )# 17 P P A$A Area of the 6ings.= +# #
# 1# 1
E EP P
#
=
Fhere E1 an E#are the velocities at the top an bottom of
the 6ings an is ensity.
M 0 : Torricelli?s theorem:
1". A vessel fille 6ith a liqui up to a height ; has a small hole near the bottom. Ifh is epth of the hole then.
a- Eelocity of efflux from the orifice is v #gh=
b- Eolume efflux H = aE
c- *ime ta8en by the 6ater to touch the base level t( )# ; h
tg
=
- ;ori/ontal range ( )R # h ; h=
e- R is maximum if h = ;Rmax= ;
f- *ime ta8en for emptying the tan8A #;
t ha g
=
g- *ime after 6hich the level falls from h1to h#is ( )1 #A #t h ha g=
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1!. EPAN+(N (F (,+6
1. )inear expansion 1 1 t. =
( ) ( )# 1
# 1# 1
l l
1 1 1 tl1 t t
= = +
If same amount of heat given thenH l H
t then lms ms
= =
( )
( ) ( )
( ) ( )
#
# 1# 1
1 # 1
# 1# 1
1 # 1
l 1l r raius of the ro .
m rAral expansion A A t.
A AA A 1 t
A t t
Eolume expansion E E t.
E E 1 tt t
=
= = +
=
= = +
! change in length tx1""=
! change in area # tx1""=
! change in Eolume % tx1""=
#. )oss or gain of a penulum cloc8 ue to a change in temperature per ay is
( )1
*# *1 J2'""sec.#
%. If t6o ros of ifferent material have equal lengths of 1 an at temperature *1as the temperature is increase to *# ifferent in their final length is
( )( )# 1 # 11 1 * * . =
'. 7or an anisotropic substance if x y / an are the coefficients of linear
expansion in three perpenicular irections then x y / . = + +
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. Fhen a bimetallic strip is heate it bens. *hen the raius of curvature of thebimetallic strip
( ) ( )# # 11
R .
* *=
Fhere is the thic8ness of each strip.
( )# 1* * is change in temperature
1 # are coefficients of linear expansion of the t6o metals.
2. " " " "
in per 7 in per in per 7 in per D.? '
= =
>. *hermal stress evelope in a ro heate 6hen it is prevente from expansion
is calle thermal stress ( )# 1: * * .=
7orce evelope ( )# 1:A * * .=
J. A 6ire is bent in the form of a ring 6ith a small gap of length "1 1x at t D. on
heating to "#t D if the gap increase to x#in length then the coefficient of linear
expansion of the 6ire material
( )# 1
1 # 1
x x
x t t
=
?. If t6o ros of ifferent materials have the ifference bet6een their lengths at
all temperatures then 1 1 # # 1 # # 11 1 or 1 &1 & = =
# 11 #
1 # 1 #
x xl an l
= =
: :
1". *he change in moment of inertia of a boy ue to rise in temperatureI #I t =
11. (etal tapesa- (etal tape sho6s correct reaing only at a temperature at 6hich it isconstructe.
b- A metal tape is grauate at "1t D an is use at"
#t D.
i- # 1If t t istance bet6een the ivisions ecreases then
a- Sbserve reaing ) is less than actual value
b- Dorrection to be applie ( )# 1D ) t t =
c- Dorrect reaing ) D.= + ii- It t# Q t1 istance bet6een the ivisions ecreases then
a- Sbserve reaing ) is greater than actual valueb- D ) t =
c- Dorrect reaing ) D.= 1#. Eariation of ensity 6ith temperature
( )( ) ( )" t t " 1 or 1 t= + =
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1%. A mercury barometer has a metal scale 6hich is calibrate at ""D the pressureas rea by it at t"D is ;t. *hen the pressure at "
"D 6ill be given by
" R; ;t 1 @ -t= R is coefficient of real expansion of mercury
is coefficient of linear expansion of metal
1. EPAN+(N (F ,+-'+6
1. R a a R a% relation bet6een an . = + = +
#. 7rom specific gravity bottle experiment
( )
( )( ) ( )
( )
a
# % 1 #a a
% 1 # 1 # # 1
mass of liqui expelle
mass of the liqui remaine change in temperature
6 6 m mor
6 6 * * m t t
=
= =
m1is nearly equal to m#an hence it can be 6ritten ( )1 #
R# # 1
m m.
m t t
=
%. If 1an #are the ensities at temperature t1an t#
( )
( )
( )
1 # R # 1
" t R
"t
1 #R
1 # # 1
1 * *
1
1 t
t t
= +
= +
=
=
'. *he unoccupie volume in a container remains same if the volume expansion ofthe container an that of liqui are same. *he conition for this is 1 1 s sE E . =
. *he temperature at 6hich the ensity of liqui is x! less than at
( )"
R
x" D is .1"" x
2. If a liqui column of height h1at temperature t1balances another liqui columnof height h# at temperature t#.
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( )# 1
a1 # 1
h h.
h t t
=
1". EPAN+(N (F BAE
M 0 +: olme and pressre coefficient of a as.
1. Eolume coefficient ( )"
""
E E E E 1 tE t = = +
# 1
1 # # 1
E EEt E t
=
@at constant pressure-.
#. Pressure coefficient ( )"
" t"
P PP P 1
P t
= = +
# 1
1 # # 1
P P
Pt P t
==
@ at constant valume-.
%. ( )"1>% ".""%22& c. for any gas = = =
M 0 ++: 7o5le?s la4:
'. A* constant temperature ( ) 1 1 # #1
P . or PE D PE PEE
= =
. *6o vessels of volume E1 an E# fille 6ith a gas at pressure P1 an P# are
connecte the common pressure( )1 1 # #
1 #
PE PEP .
E E+
=+
2. If the pressure of the gas is increase by x! then the ! ecrease in volume at
constant temperature is
1""x
.1"" x+>. If 11an 1#are the length of the air column 6ith open en up6ars an
o6n6ars respectively then ( ) ( ) # 11 ## 1
l l; h 1 ; h 1 . ; h
l l
++ = =
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; atm. Pressure.h length of mercury threa.
J. Fhen an air bubble at the epth h in a la8e rises to the top its volumeincreases. Assuming the temperature to be constant
If the volume becomes n timesh = ; @n 4 1- @; + h- E1= ;E#
An if raius becomes n times( ) ( )% % %1 #h ; n 1 ; h r ;r= + =
Fhere ; = 1 atm 1" m epth of 6ater = >2 cm of ;g.
If temperature is constant then( ) 1 #
1 #
; h E ;E* *
+=
M 0 +++: /harle?s la4:
?. At constant volume P * @for a given man of a gas-
( ) 1 #
1 #
P P Por constant .
* * *
= =
1". At constant pressure E * @for a given man of a gas-
( ) 1 #
1 #
E E Eor constant .
* * *= =
M 0 +: +deal as e;ation:
11. If one gram of gas is consiere.PE = r* @r = gas constant-.If one mole of gas is consiere
1 1 # #
1 #
PE PE RPE R* . r
* * (
= = = for n moles PE = nR*.
1#. ifference forms of ieal gas equation.a- PE = nR*b- PE = @m&(-R*c- P = @R*-&(m 4 mass of the gas 4 ensity of the gas ( 4 molecular 6eight.7or a given gas.
1 #
1 #
P Pm m
= @if E an * are constant-.
1 #
1 1 # #
P Pm* m * = @if E is constant-.
1 #
1 1 # #
P P
+* + * =
If * is constant1 #
1 #
P P.
=
If P is constant 1 1 # #* * .=
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1%. ! change in * = ! change P + ! change in E.1'. Fhen n1moles of a non reacting gas in the state @P1E1*1- is mixe 6ith n#moles
of gas in state @P#E#*#- if the resultant mixture is in the state @P E *-.
1 1 # #
1 #
PE PE PE.
* * *+ =
1. If t6o ifferent gases of same mass occupy same volume at same pressure then
(1&*1= (#&*#@(1 (# are molecular 6eights-
1D. T)ERM(6*NAM+/
M 0 +: Hole?s la4:
1. *he 6or8 one on a system @6- is irectly proportional to heat prouce in thesystem.
6 H or 6 WH$ W '.# B&cal W 1 in ,I system = = =
#. A boy of mass falls from height h an if it is potential energy converts intoheat then
( )gh
mgh ms t t ,. shoul+ be in ,.I unitss
= =
If x ! of energy is converte into heat then
x mgh ms t.1""
=
%. An ice bloc8 of mass ( falls from a height h an if its energy converts into heatthen
(gh(gh m) m
)) ,I Cnits.
m mass of the ice melte+
=
*he ice completely melts if h = %%.2 8m.
'. Fhen a boy of mass m moving 6ith velocity v is stoppe an all its 0 convertsinto heat energy then
( )#
#1 m ms t t , ,.I units .# #s
= =
. If a boy of mass m is ragge on a rough surface through a istance x then
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gxmg ms t t
,
= =
If it is an ice bloc8 then1mgx m ). =
(ass of the ice melte 1mgx
m .)
=
M 0 ++: First la4 of thermod5namics:2. H = u + 6.
H + ve if heat is supplie to the system. ve if heat is absorbe from the system.
u + ve if temperature increases. ve if temperature ecreases.( )# 16 P v v= = + ve if 6or8 is one by the gas.
ve 6or8 is one on the gas.
M 0 +++: vpD D and relation 8et4een them:
>. A gas at constant
,pecific heat at constant pressure p1 +
cm +t
= @at constant pressure-.
(olar ,pecific heat at constant pressure p1
Dn t
=
,pecific heat at constant pressure1 +
cm +t
=
(olar ,pecific heat at constant pressure1
D .n t
=
p p v vD (c $ D (c .= =J. p v p vD D R c c R&(. = =
?. *he follo6ing table sho6 the values of p vD 9D .
T5pe of the as vD pD
(onoatomic %R#
R
#
1.2>%
=
iatomic R#
>R
#>
1.'
=
Polyatomic@*rihigher-
2R#
JR#
' 1.%%%
=
M 0 +: /onstant olme process: =+sochoric@
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b-1 #
1 #
P P* *
=
c- 6 = ".- H = C = n Dv*.
M 0 : /onstant pressre process: =+so8aric@
1". a-1 #
1 #
E E* *
=
b- u = nDv *.
c- 6 = ( )# 1P E E
- H = nDp *.11. *he fraction of heart energy supplie 6hich is store as internal energy
u 1.
H=
1#. *he fraction of heat energy supplie 6hich is use in oing external 6or8
6 11 .H
=
M 0 +: +sothermal process:
1%. a- 1 1 # #PE PE .=
b- u = * = ".
# 1
1 #
#
1
#
1
v Pv P
e e
vv
evv
1"
c- 6 nR* log nR* log
PE log
#.%"% nR* log .
= = =
=
=
- ( ) ( )Dp Dv R$ Dp&Dv r$ Dv R& 1 $ Dp R 1 = = = =
e- ,lope of isothermal curve = P&E.f- Isothermal bul8 moules = .P.
M 0 ++: Adia8atic process:
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( )
1 1 # #
1 1 11 1 # #
1 1 11 1 # #
p v
1 #1 1 # #
1'. a- PE cons tan t PE P E
b-*E Dons tant *E * E
c-* P Donstant * P * P
+-+H " D D "
nR * * PE PEe-+6 +C .1 1
f-A+iabatic bul8 mo+ulus P.
,lope of a+iabatic curve P&E.
= =
= =
= =
= = =
= = =
=
=
1. 7or a mixture of t6o gases1 1 # # 1 1 # #
1 # 1 #
n Dp n Dp n D n DDp D
n n n n
+ + = =
+ +Dp&Dv =
12. ;eat capacity or thermal capacity or 6ater equivalent @ms- = H& t
19. T)ERMA, RA6+AT+(N
1. AbsorptivitytHHa Hra Reflectivity r *ransimitivity t
H H H= = =
a + r + t = 1
#. 0nergy emitte by a bloc8 boy from unit surface area in unit time is given by'0 * .= 0 is also calle emissive po6er.
7or the boies '0 e * 6here 0 emissivity.=
%. *otal energy emitte by a boy in a time t is given by'0 e A* t.=
7or a sphere ( )# '0 e ' R * t.=
i.e. # '0 R * .
'. *he net heat lost or gaine by a boy at temperature * in unit time
( )'s'0 e A * * .=
. 3e6tonGs la6 of cooling ( )< s+H
* * .+t
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s* *emperature of the surrounings.2. If the temperature of a boy ecrease from *1to *#in a time t then accoring
to 3e6tonGs la6 of cooling.
1 # 1 #s
* * * * * .
t #
+ = ". If m is the 6avelength corresponing to maximum monochromatic emissive
po6er of a boy at temperature * the ( )m m1 or * b*
= @b is a constant-.
b = #.?% x 1"%m8.J. 7or a spherical boy at temperature *
c- Rate of heat loss ' #H H
e A* R .t t
= @If * is same-.
- Rate of cooling is given by'
'ms* * e A* * 1e A*t t ms t R
= =
@If * is same-