Page 1
Important Formulae
1. Thermal Expansion
(i) ?l = l ? ? ?, ?s = s ? ? ? and
?V = V ? ? ?
(ii) ? = 2 ? and ? = 3 ?
for isotropic medium.
Effect of temperature on different physical quantities.
(i) On Density With increase in temperature volume of any substance increases while mass
remains constant, therefore density should decrease.
'
1
?
??
? ? ? ?
or ?’ = ?(1 - ? . ? ?) if ? . ? ? <<1
(ii) In Fluid Mechanics
Case 1 When a solid whose density is less than the density of liquid is floating, then a fraction of
it remains immersed. This fraction is
s
l
f
?
?
?
When temperature is increased, ?
s
and ?
l
both will decrease. Hence, fraction may increase,
decrease or remain same. At higher temperature,
l
s
1
f ' f
1
?? ? ? ??
?
??
? ? ??
??
If ?
l
> ?
s
, f’ > f or immersed fraction will increase.
Case 2 When a solid whose density is more than the density of liquid is immersed completely,
then upthrust will act on 100% volume of solid and apparent weight appears less than the actual
weight.
F
app
= W - F
Here, F = V
s
?
l
g
With increase in temperature V
s
will increase and ?
l
will decrease, while g will remain unchanged.
Therefore upthrust may increase, decrease or remain same. At some higher temperature,
s
l
1
F' F
1
?? ? ? ??
?
??
? ? ??
??
If ?
s
> ?
l
, upthrust will increase. Therefore, apparent weight will decrease.
(iii) Time period of pendulum
l
T2
g
?? or Tl ?
With increase in temperature, length of pendulum will increase. Therefore time period will
increase. A pendulum clock will become slow and it loses the time. At some higher temperature,
Page 2
Important Formulae
1. Thermal Expansion
(i) ?l = l ? ? ?, ?s = s ? ? ? and
?V = V ? ? ?
(ii) ? = 2 ? and ? = 3 ?
for isotropic medium.
Effect of temperature on different physical quantities.
(i) On Density With increase in temperature volume of any substance increases while mass
remains constant, therefore density should decrease.
'
1
?
??
? ? ? ?
or ?’ = ?(1 - ? . ? ?) if ? . ? ? <<1
(ii) In Fluid Mechanics
Case 1 When a solid whose density is less than the density of liquid is floating, then a fraction of
it remains immersed. This fraction is
s
l
f
?
?
?
When temperature is increased, ?
s
and ?
l
both will decrease. Hence, fraction may increase,
decrease or remain same. At higher temperature,
l
s
1
f ' f
1
?? ? ? ??
?
??
? ? ??
??
If ?
l
> ?
s
, f’ > f or immersed fraction will increase.
Case 2 When a solid whose density is more than the density of liquid is immersed completely,
then upthrust will act on 100% volume of solid and apparent weight appears less than the actual
weight.
F
app
= W - F
Here, F = V
s
?
l
g
With increase in temperature V
s
will increase and ?
l
will decrease, while g will remain unchanged.
Therefore upthrust may increase, decrease or remain same. At some higher temperature,
s
l
1
F' F
1
?? ? ? ??
?
??
? ? ??
??
If ?
s
> ?
l
, upthrust will increase. Therefore, apparent weight will decrease.
(iii) Time period of pendulum
l
T2
g
?? or Tl ?
With increase in temperature, length of pendulum will increase. Therefore time period will
increase. A pendulum clock will become slow and it loses the time. At some higher temperature,
T' = T(1 + ? ? ?)
1/2
or
1
T' T 1
2
??
? ? ? ? ?
??
??
if ? ? ? << 1
?T = (T' – T) =
1
2
T ? ? ?
Time lost / gained
?t =
T
t
T'
?
?
(iv) Thermal Stress If temperature of a rod fixed at both ends is increased, then thermal stresses
are developed in the rod.
At some higher temperature we may assume that the rod has been compressed by a length,
?l = l ? ? ? or strain
l
l
?
= ? ? ?
stress = Y × strain = Y ? ? ? (Y = Young's modulus of elasticity)
F = A × stress = YA ? ? ?
Rod applies this much force on wall to expand. In turn, wall also exerts equal and opposite pair of
encircled forces on rod. Due to this pair of forces only, we can say that rod is compressed.
2. Kinetic Theory of Gases Different equations used in kinetic theory of gases are listed below,
(i) pV = nRT =
m
RT
M
(m = mass of gas in gms)
(ii) Density
m
V
??
(general)
pM
RT
??
(for ideal gas)
(iii) Gas laws
(a) Boyle's law It is applied when T = constant, or process is isothermal. In this condition
pV = constant or p
1
V
1
= p
2
V
2
or
1
p
V
?
(b) Charles' law It is applied when p = constant or, process is isobaric.
Page 3
Important Formulae
1. Thermal Expansion
(i) ?l = l ? ? ?, ?s = s ? ? ? and
?V = V ? ? ?
(ii) ? = 2 ? and ? = 3 ?
for isotropic medium.
Effect of temperature on different physical quantities.
(i) On Density With increase in temperature volume of any substance increases while mass
remains constant, therefore density should decrease.
'
1
?
??
? ? ? ?
or ?’ = ?(1 - ? . ? ?) if ? . ? ? <<1
(ii) In Fluid Mechanics
Case 1 When a solid whose density is less than the density of liquid is floating, then a fraction of
it remains immersed. This fraction is
s
l
f
?
?
?
When temperature is increased, ?
s
and ?
l
both will decrease. Hence, fraction may increase,
decrease or remain same. At higher temperature,
l
s
1
f ' f
1
?? ? ? ??
?
??
? ? ??
??
If ?
l
> ?
s
, f’ > f or immersed fraction will increase.
Case 2 When a solid whose density is more than the density of liquid is immersed completely,
then upthrust will act on 100% volume of solid and apparent weight appears less than the actual
weight.
F
app
= W - F
Here, F = V
s
?
l
g
With increase in temperature V
s
will increase and ?
l
will decrease, while g will remain unchanged.
Therefore upthrust may increase, decrease or remain same. At some higher temperature,
s
l
1
F' F
1
?? ? ? ??
?
??
? ? ??
??
If ?
s
> ?
l
, upthrust will increase. Therefore, apparent weight will decrease.
(iii) Time period of pendulum
l
T2
g
?? or Tl ?
With increase in temperature, length of pendulum will increase. Therefore time period will
increase. A pendulum clock will become slow and it loses the time. At some higher temperature,
T' = T(1 + ? ? ?)
1/2
or
1
T' T 1
2
??
? ? ? ? ?
??
??
if ? ? ? << 1
?T = (T' – T) =
1
2
T ? ? ?
Time lost / gained
?t =
T
t
T'
?
?
(iv) Thermal Stress If temperature of a rod fixed at both ends is increased, then thermal stresses
are developed in the rod.
At some higher temperature we may assume that the rod has been compressed by a length,
?l = l ? ? ? or strain
l
l
?
= ? ? ?
stress = Y × strain = Y ? ? ? (Y = Young's modulus of elasticity)
F = A × stress = YA ? ? ?
Rod applies this much force on wall to expand. In turn, wall also exerts equal and opposite pair of
encircled forces on rod. Due to this pair of forces only, we can say that rod is compressed.
2. Kinetic Theory of Gases Different equations used in kinetic theory of gases are listed below,
(i) pV = nRT =
m
RT
M
(m = mass of gas in gms)
(ii) Density
m
V
??
(general)
pM
RT
??
(for ideal gas)
(iii) Gas laws
(a) Boyle's law It is applied when T = constant, or process is isothermal. In this condition
pV = constant or p
1
V
1
= p
2
V
2
or
1
p
V
?
(b) Charles' law It is applied when p = constant or, process is isobaric.
In this condition,
V
T
= constant
or
12
12
VV
TT
? or V ? T
(c) Pressure law of Gay Lussac's law It is applied when V = constant or process is isochoric. In
this condition,
p
T
= constant
or
12
12
pp
TT
? or p ? T
(iv) Four speeds,
ART
v
M
?
AkT Ap
m
??
?
Here, m = mass of one gas molecule.
A = 3 for rms speed of gas molecules
8
2.5 ??
?
for average speed of gas molecules
= 2 for most probable speed of gas molecules
p
V
C
C
?? for speed of sound in a gas
(v)
2
rms
1 mn
pv
3V
?
(vi)
2
pE
3
?
Here, E = total translational kinetic energy per unit volume
(vii) f = degree of freedom
= 3 for monoatomic gas
= 5 for diatomic and linear polyatomic gas
= 6 for nonlinear polyatomic gas
Note (a) Vibrational degree of freedom is not taken into consideration.
(b) Translational degree of freedom for any type of gas is three,
(viii) Total internal energy of a gas is,
nf
U RT
2
?
Here, n = total number of gram moles.
Page 4
Important Formulae
1. Thermal Expansion
(i) ?l = l ? ? ?, ?s = s ? ? ? and
?V = V ? ? ?
(ii) ? = 2 ? and ? = 3 ?
for isotropic medium.
Effect of temperature on different physical quantities.
(i) On Density With increase in temperature volume of any substance increases while mass
remains constant, therefore density should decrease.
'
1
?
??
? ? ? ?
or ?’ = ?(1 - ? . ? ?) if ? . ? ? <<1
(ii) In Fluid Mechanics
Case 1 When a solid whose density is less than the density of liquid is floating, then a fraction of
it remains immersed. This fraction is
s
l
f
?
?
?
When temperature is increased, ?
s
and ?
l
both will decrease. Hence, fraction may increase,
decrease or remain same. At higher temperature,
l
s
1
f ' f
1
?? ? ? ??
?
??
? ? ??
??
If ?
l
> ?
s
, f’ > f or immersed fraction will increase.
Case 2 When a solid whose density is more than the density of liquid is immersed completely,
then upthrust will act on 100% volume of solid and apparent weight appears less than the actual
weight.
F
app
= W - F
Here, F = V
s
?
l
g
With increase in temperature V
s
will increase and ?
l
will decrease, while g will remain unchanged.
Therefore upthrust may increase, decrease or remain same. At some higher temperature,
s
l
1
F' F
1
?? ? ? ??
?
??
? ? ??
??
If ?
s
> ?
l
, upthrust will increase. Therefore, apparent weight will decrease.
(iii) Time period of pendulum
l
T2
g
?? or Tl ?
With increase in temperature, length of pendulum will increase. Therefore time period will
increase. A pendulum clock will become slow and it loses the time. At some higher temperature,
T' = T(1 + ? ? ?)
1/2
or
1
T' T 1
2
??
? ? ? ? ?
??
??
if ? ? ? << 1
?T = (T' – T) =
1
2
T ? ? ?
Time lost / gained
?t =
T
t
T'
?
?
(iv) Thermal Stress If temperature of a rod fixed at both ends is increased, then thermal stresses
are developed in the rod.
At some higher temperature we may assume that the rod has been compressed by a length,
?l = l ? ? ? or strain
l
l
?
= ? ? ?
stress = Y × strain = Y ? ? ? (Y = Young's modulus of elasticity)
F = A × stress = YA ? ? ?
Rod applies this much force on wall to expand. In turn, wall also exerts equal and opposite pair of
encircled forces on rod. Due to this pair of forces only, we can say that rod is compressed.
2. Kinetic Theory of Gases Different equations used in kinetic theory of gases are listed below,
(i) pV = nRT =
m
RT
M
(m = mass of gas in gms)
(ii) Density
m
V
??
(general)
pM
RT
??
(for ideal gas)
(iii) Gas laws
(a) Boyle's law It is applied when T = constant, or process is isothermal. In this condition
pV = constant or p
1
V
1
= p
2
V
2
or
1
p
V
?
(b) Charles' law It is applied when p = constant or, process is isobaric.
In this condition,
V
T
= constant
or
12
12
VV
TT
? or V ? T
(c) Pressure law of Gay Lussac's law It is applied when V = constant or process is isochoric. In
this condition,
p
T
= constant
or
12
12
pp
TT
? or p ? T
(iv) Four speeds,
ART
v
M
?
AkT Ap
m
??
?
Here, m = mass of one gas molecule.
A = 3 for rms speed of gas molecules
8
2.5 ??
?
for average speed of gas molecules
= 2 for most probable speed of gas molecules
p
V
C
C
?? for speed of sound in a gas
(v)
2
rms
1 mn
pv
3V
?
(vi)
2
pE
3
?
Here, E = total translational kinetic energy per unit volume
(vii) f = degree of freedom
= 3 for monoatomic gas
= 5 for diatomic and linear polyatomic gas
= 6 for nonlinear polyatomic gas
Note (a) Vibrational degree of freedom is not taken into consideration.
(b) Translational degree of freedom for any type of gas is three,
(viii) Total internal energy of a gas is,
nf
U RT
2
?
Here, n = total number of gram moles.
(ix)
V
dU
C
dT
? (where U = internal energy of one mole of a gas =
f
2
RT)
V
f
CR
2
?
R
1
?
??
(x) C
p
= C
V
+ R =
f
1 R R
21
?? ? ??
??
?? ??
??
??
??
(xi)
p
V
C
2
1
Cf
? ? ? ?
(xii) Internal energy of 1 mole in one degree of freedom of any gas is
1
RT
2
.
(xiii) Translational kinetic energy of one mole of any type of gas is
3
RT
2
.
(xiv) Rotational kinetic energy of 1 mole of monoatomic gas is zero of dia or linear polyatomic
gas is
2
RT
2
or RT, of non-linear polyatomic gas is
3
RT
2
.
(xv) Mixture of non-reactive gases
(a) n = n
1
+ n
2
(b) p = p
1
+ p
2
(c) U = U
1
+ U
2
(d) ?U = ?U
1
+ U
2
(e)
12
1 V 2 V
V
12
n C n C
C
nn
?
?
?
(f)
12
1 p 2 p
pV
12
n C n C
C C R
nn
?
? ? ?
?
(g)
p
12
V 1 2
C
nn n
or
C 1 1 1
? ? ? ?
? ? ? ? ? ?
(h)
1 1 2 2
12
n M n M
M
nn
?
?
?
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