In a uniform circular motion, the kinetic energy of the body is a constant.. The electric field E due to a charged solid sphere at any point inside the sphere is directly proportional to
Trang 1
INDIRECT THEORY NOTES
Dimensions and Units
1 A quantity can have unit without dimensions e.g angle and solid angle But a quantity cannot have dimensions without units
2 Some pairs which have same dimensions :
Torque and work (b) Angular momentum and Planck’s constant (c) Latent heat and gravitational potential (d) Momentum and impulse
One and two dimensional motion:
3 If two ends of a train accelerating uniformly crosses an observer with a velocity v1 and v2, then the middle point of the train will cross the observer with a velocity v =
2
v
v12+ 22
4 If a body travels two equal distances with different speeds v1 and v2, its average speed is v(average)
=
2
1
2
1
v
v
v
v
+
5 If a body travels three equal distances with speeds v1, v2 and v3, its average speed is v(average) =
1 3 3 2
2
1
3 2 1
v v v
v
v
v
v v
v
+ +
6 If a body falls freely, the distance covered by it in each subsequent second starting from first second will be in the ratio 1:3:5:7 etc That is S1:S2:S3: Sn = 1:3:5:7 (2n-1) This means that the distance travelled during the nth second Sn is proportional to 2n-1
7 If a body is thrown vertically up with an initial velocity u, it takes u/g second to reach maximum height and u/g second to return, if air resistance is negligible
8 The total time body remains in air when it is thrown vertically up with a velocity u is equal to
g
u
9 The maximum height reached by a body projected vertically up with an initial velocity u is
g
u2
10 If air resistance acting on a body is considered, the time taken by the body to reach maximum height
is less than the time to fall back the same height
11 Two vectors of equal magnitude F inclined at an angle θ, has a vector sum of magnitude 2Fcos⎢⎣⎡θ2⎥⎦⎤ Two vectors of equal magnitude F inclined at an angle θ, has a vector difference of magnitude is 2Fsin ⎢⎣⎡θ2⎥⎦⎤
12 If two equal vectors have a vector sum of magnitude equal to either of them, the angle between them
is 1200
13 If two equal vectors have a vector difference of magnitude equal to either of them, the angle between them is 600
14 If magnitude of sum of two vectors is equal to magnitude of their difference, the angle between them
is 90o
15 If T is the time of flight, h maximum height, R horizontal range of a projectile, α its angle of
projection, then the relations among these quantities
8
gT
h
2
= (1)
gT2= 2R tanα (2)
Rtanα = 4h (3)
The equation (1) is true for a body projected vertically up and also projected at an angle
Trang 216 For a given initial velocity, to get the same horizontal range, there are two angles of projection α and 90-α
17 The equation to the parabola traced by a body projected horizontally from the top of a tower of height
y, with a velocity u is y=gx2/2u2, where x is the horizontal distance covered by it from the foot of the tower
18 In a uniform circular motion, velocity and acceleration are constants only in magnitude Their directions change
19 In a uniform circular motion, the kinetic energy of the body is a constant
20 In a uniform circular motion, the work done by centripetal force in moving from any point in the
circle to any other point will be zero The centripetal force F is along the radius and the displacement s
(velocity) is along the tangent This makes W =F.s = Fscos900 = 0
21 The minimum velocity for a body at the lowest point to complete a vertical circle of radius r is 5rg The minimum velocity at highest point then is rg
22 The difference in tension at the highest and the lowest point in a vertical circle is 6mg, i.e 6 times the weight of the body
Force and motion
23 Newton’s second law is general The first and third laws can be deduced from it
24 Not all forces produce acceleration Only unbalanced forces do
25 The weight of a body W = mg, but is measured by the resistance, reaction or tension If the
resistance, reaction or tension becomes zero, the body feels weightlessness When a body falls freely,
only force of gravity acts on it There is no resistance, reaction or tension Hence, the body feels weightlessness
26 The acceleration of a lift a =
mass
weight apparent weight
, where the weights is in N If ‘a’ is positive lift is moving down, and if it is negative the lift is moving up
27 A force is pseudo if (1) We cannot know where it is coming from, (2) It cannot be classified into one
of the four forces in nature and (3) The force disappears when we change the frame of reference (observer)
28 The frictional force acting on a body on an inclined plane is µsmgcosα , where α is the angle of the plane
29 Acceleration of a body when it slides down a rough inclined plane = g sinα - µs gcosα
30 Retardation produced by friction on a car in horizontal motion = µsg
31 The maximum safe speed of a cyclist in a curve of a radius r is µsrg
32 A car overturns in a curve of radius r, if its speed is greater than arg/ h, where 2a is the distance between wheels, h the height of centre of gravity from road
Work, Power, Energy
33 If E is kinetic energy and p momentum of a body, then p2 = 2mE
34 When kinetic energy of a body is made n times, its momentum increases to √n times
35 When momentum of a body is made n times, its kinetic energy increases to n2 times
36 If a light body and heavy body have equal momentum, then light body has greater kinetic energy
37 If a light body and a heavy body have equal kinetic energy, then heavy body has greater momentum
38 If a body moves with constant power, distance travelled by it (s) in a time (t) is related by the equation
s ∝ t3/2
39 If a body moves with constant power, its velocity (v) is related to distance travelled (x) by the formula v∝ x1/3
Trang 3Physics at a Glance 3
40 When water flows through a uniform tube with a constant velocity v, the power to maintain the flow
is proportional to cube of velocity
Rotational motion and Moment of inertia
41 A rigid body can rotate about a number of axes Its moment of inertia is minimum, when the axis passes through centre of mass
42 The angular displacement of a body (which rotates with uniform angular acceleration) in the 1st , 2nd ,
3rd second etc will be in the ratio 1:3:5:7 2n-1 This means θn ∝ 2n-1
43 The acceleration of a body rolling down an inclined plane of angle θ is
) r / k ( +
sin g
2 2
θ , where k is radius of gyration and r the radius of the body
44 The kinetic energy of rotation
2
1Iω2 = τθ, where τ is torque and θ the angular displacement
45 The power spent to rotate a body with a constant angular velocity ω is P = τω, where τ is torque
46 If L is angular momentum, E kinetic energy of rotation, I moment of inertia, then L2 = 2EI
47 For a rigid body of radius r and radius of gyration k, rolling forward, if E is the total energy, R kinetic energy of rotation, T kinetic energy of translation, the following equations hold
=
E
T
2
2
2
r
k
r
+ (1),
=
E
R
2
2
2
r
k
k
+ (2),
=
R
T
2
2
k
r
(3)
Gravitation
48 The gravitational field due to a solid sphere of radius R, at a point distant r is proportional to 2
r
1 for
r>R (i.e for point outside) It is proportional to r, for r<R (i.e for a point inside)
49 If earth contracts to
n
1 th of its present radius with its mass remaining constant, the duration of the
day will be 2
n
24
hour
50 If earth rotates with a period T =
g
R 2π = 84 minutes, g at equator will be effectively zero The corresponding angular velocity is 1.3x10-3 rad/s
Escape velocity
51 The escape velocity from the surface of earth is 11.2 km/s As the altitude increases, escape velocity decreases The escape velocity from the surface of moon is nearly 2.4 km/s
52 The escape velocity of a body is √2 times the first cosmic velocity
53 The escape velocity does not depend on direction of projection
54 The escape velocity at a distance x from centre of earth will be inversely proportional to√x
Satellites
55 The first cosmic velocity or velocity of a satellite very close to earth is nearly 8 km/s
56 The velocity of an earth satellite is always less than 8 km/s The velocity is inversely proportional to the square root of orbital radius [v=
r
GM ]
Trang 457 If the speed of an earth satellite is more than 8 km/s but less than 11.2 km/s, it moves in a hyperbolic path If its speed is more than 11.2 km/s, it escapes
58 The U is the potential energy of an earth satellite, K its kinetic energy, E its total energy, m mass of satellite, M mass of earth, r distance from the centre of earth, then,
r
GMm
−
K =
r
2
GMm,
r
2
GMm
59 When a planet is near sun, its velocity is more When it is far from sun, its velocity is less This follows from Kepler’s second law
Geo-stationary satellites
60 The altitude of geo-stationary satellite is nearly 36000 km 2) The orbital radius (assuming circular orbit) i.e distance from the centre of earth is nearly 42000 km 3) Its period is 24 h, which is the
same as that of earth 4) Its angular velocity is the same as that of earth’s spin 5) But its speed is not
the same as that of earth Its speed is nearly 3 km/s 6) Its orbital plane is equatorial 7) It is launched from west to east
Molecular forces
Differences between inter-molecular and inter-atomic forces:
61 The inter-molecular forces are weaker by about
100
1
compared to inter-atomic forces
62 The inter-molecular forces depend on the orientation of the molecules and their separation, while inter- atomic forces depend only on the separation
Similarities between intermolecular and inter-atomic forces
63 Both are electro-magnetic forces
64 Both are attractive and repulsive The force is attractive when they are far and repulsive when they are near
65 The distance-force graph of the molecules and that of the atoms will have same shape However, the equilibrium distance for molecules is more than that for atoms
Properties of matter
66 The breaking stress of a material depends only on the material (i.e modulus of elasticity) and not on the dimensions of the material
67 If S is the stress and Y is Young’s modulus, the energy density of the wire E is equal to
Y 2
S2
68 If α is the longitudinal strain and E is the energy density of a stretched wire, Y Young’s modulus of the wire, then E is equal to
2
1 Yα2
69 The couple to twist a wire of length L, radius r and rigidity (shear) modulus n through one radian is
L
2
nr4
π
70 The theoretical values of Poisson’s ratio σ is between –1 and + 0.5
71 The angle of contact of a liquid with a semi-soluble impurity is less than that of solvent
72 The surface tension of a liquid is zero at boiling point The surface tension is zero at critical temperature
73 If a capillary tube were made with silver, there would be neither capillary rise nor capillary depression (angle of contact 900)
Trang 5Physics at a Glance 5
74 If two bubbles of radii r1 and r2 coalesce under constant temperature (isothermal conditions), the radius of resulting bubble will be r12+r22
75 A hollow sphere with a small hole of radius r in it can be immersed in a liquid of density d and surface tension S to maximum depth h, so that water will not enter into it Here h is given by
rdg
S 2
76 If a drop of water of radius R is broken into n identical drops, the work done in the process is 4πR2S(n1/3-1), where S is surface tension
77 The ratio of surface energy of the big drop to that of one smaller drop in the above case is n2/3
78 Two capillary tubes each of radius r are joined in parallel The rate of flow is Q If they are replaced
by single capillary tube of radius R for the same rate of flow, then R = 21/4r
79 If radius of a drop is doubled its terminal velocity increases to 4 times
80 If n liquid drops each having a terminal velocity v coalesce to form a bigger drop, the terminal velocity of the resulting drop will be n2/3v
Hydrodynamics and Hydrostatics
81 When a body is floats in the liquid, the fraction of immersed volume = density of body / density of liquid
82 A body floats in a liquid with a fraction of its volume immersed If the system falls freely under gravity, the immersed fraction remains the same The buoyant force on the body becomes zero
83 The standard atmospheric pressure is about 10 m of water
84 The viscosity and surface tension of a liquid decrease on heating The viscosity of a gas increases on heating
85 When n drops each of radius r coalesce, generally the radius of the resulting drop will be n1/3r
86 The horizontal distance travelled by a liquid coming out a narrow orifice in a vessel is x = 2 h1h2 , where h1 is the depth of the orifice from the top surface of the liquid and h2 the height of orifice from the bottom of the vessel
87 When ice floating in a beaker of water melts, the level of water remains the same
88 If ice contains denser materials (e.g iron) water level falls when it melts If ice contains less dense material (e.g wood) the level remains the same
89 The swinging of a cricket ball in air is due to Bernoulli’s theorem and difference in air speeds at the two sides
Thermal Physics
90 Temperature at which Fahrenheit and Celsius thermometers show the same reading is –400
91 Temperature at which Fahrenheit and Kelvin scale give the same reading is nearly 574
92 In a compensated pendulum, the change in the length of the two rods should be same This gives L1α1
= L2α2, where L1, L2 are their lengths and α1, α2, their linear expansivity respectively
93 Gas constant in CGS system is R= 2 cal/mol 0C
94 Latent heat of fusion of ice in CGS system = 80 cal/g
95 The latent heat of steam in the CGS system is 540 cal/g
96 If equal mass of ice at 0oC and steam at 100oC are mixed, the resulting temperature will be 100oC
97 If equal mass of ice at 0oC is mixed with water at or below 80oC, the resulting temperature will be
0oC
98 If x g of ice at 00C is mixed with x g of water above 800C but below 1000C, the resulting temperature will always be less than 100C If θ is the temperature of water, (between 800C and 1000C) then final temperature of the mixture will be
x 80
− θ
in 0 C
99 If x g of ice at 00C melts when y g of dry steam at 1000C is added to it, then x = 8y
Trang 6Thermodynamics
100 The ratio of increase in internal energy to total energy supplied to a gas is equal to
γ
1 (γ ratio of specific heats)
101 The ratio of external work to the total energy supplied to a gas =
γ
−
γ 1
102 The ratio of external work to increase in internal energy is γ-1
103 (a) TABLE OF SPECIFIC HEAT OF GASES
in J/mol K
CP
in J/mol K γ = V
p C C
Monatomic
2
3
R
2
5
R
3
5
=1.67
Diatomic
2
5
R
2
7
R
5
7
=1.4
3
4
=1.33 (b) TABLE OF SPECIFIC HEAT OF MIXTURES
THIS TABLE GIVES SPECIFIC HEAT WHEN TWO GASES OF DIFFERENT MASSES ARE MIXED
If N1 mol of gas of specific heat at constant volume C1 and specific heat at constant pressure D1 is mixed with N2 mol of gas of specific heat at constant volume C2 and specific heat at constant pressure D2 For this mixture,
CV in J/mol K CP in J/ mol K
V
P
C
C
= γ
2 1
2 2 1
1
N
N
C N
C
N
+
+
2 1 2 2 1 1
N + N D N + D N
2 2 1 1
2 2 1 1
C N + C N
D N + D N
104 Using the above table, for a mixture of 1 mol of monatomic gas and 1 mol of diatomic gas, i) Value
of CP will be 3R
ii) Value of CV will be 2R
iii) Value of will be 1.5 γ
105 Specific heat is 0 during an adiabatic change since no heat is supplied Specific heat will be infinite during an isothermal change since there is no temperature rise
106 Triple point of water is 273.16 K at a pressure of 4.58 mm of Hg or 610 Pa or 0.006 atmosphere At a pressure less than this, ice can turn into steam at a temperature lower than 00C
107 If earth had no atmosphere, it would have been terribly cool
108 The mean free path of gas molecules decreases with increase in pressure and (or) temperature
109 Cooking vessels should be made with material of low specific heat and high thermal conductivity
110 The time to increase thickness of ice in a pond from x1 to x2 is proportional to x2-x1
Wave motion (General)
111 When a wave reflects from a medium, quantities, which remain constant, are its frequency, wavelength and speed The quantities, which change, are its amplitude, phase and intensity
112 When a wave refracts from one medium to another medium, the quantity that remains constant is
frequency Those, which change, are wavelength and speed
Trang 7Physics at a Glance 7
113 There are two velocities associated with a wave
1) Wave velocity given by v = fλ or v = ω/k
2) The particle velocity is the velocity of simple harmonic motion of the source that produces the wave
114 The wave velocity is constant for a given medium
115 The particle-velocity changes Its maximum value is ωA or 2π fA, where A is amplitude and ω angular frequency, f the frequency
116 A wave (front) from a point source is spherical near the source It is plane far away from the source
117 The wave (front) from a line source is cylindrical
118 The standard form of stationary wave equation is y = 2A sin ωt cos kx (i.e product of sin and cos terms)
119 If the wave equation has f(vt+x) or f(ωt+kx), the wave is propagating along the negative x-direction If the wave equation has f(vt-x) or f(ωt-kx), it is propagating along the positive x x-direction
120 When two waves of amplitude a1 and a2 and phase difference φ superpose, the amplitude of resulting wave is a = a12 +a22 +2a a1 2cosφ
121 If a1=a2,the amplitude of resulting wave is 2a cos⎢⎣⎡φ2⎥⎦⎤ The intensity of resulting wave = 4a
2cos2
⎥⎦
⎤
⎢⎣
⎡φ
2 = Iocos
2
⎥⎦
⎤
⎢⎣
⎡φ
2 where Io = 4a
2 is the maximum intensity
122 A path difference of x is equal to a phase difference of
λ
πx
2 radian A time difference of t is equal to
a phase difference of
T t 2π
(T period)
Acoustics
123 When a source moves with a speed more than the speed of sound, the waves produced are shock waves They are conical
124 The frequency of a tuning fork decreases on heating
125 If a wire is divided into lengths in the ratio a:b:c, the ratio of frequencies of these lengths will be bc:ca:ab, for the same tension
126 If a vibrating body of frequency f moves towards a wall with a speed Us, small compared to the speed
of sound, an observer hears beats This is due to Doppler effect Here beats are produced due to sound reaching directly from the source and the sound reflected The number of beats per second =
V
fU
where V is the velocity of sound
127 If both listener and source are at rest, but the medium moves, there will be no Doppler effect
128 Doppler effect formula in light:
c
v
d = λ
λ
, where dλ is change in wavelength of a spectral line of original wave length λ and v, the speed of the source and c is the speed of light
Simple Harmonic Motion: (some periods)
129 When a spring of force constant ‘k’ is cut into n equal parts, force constant of each part will be nk
130 The period of an earth satellite very close to earth is T =
g
R 2π = 84 minutes approximately:
131 The period of longest pendulum suspended in the vicinity of earth is equal to T =
g
R 2π = 84 minutes approximately:
Trang 8132 The period of rotation of earth so that ‘g’ at the equator will be apparently zero and a person feels weightlessness, is T =
g
R 2π = 84 minutes approximately
133 The period of oscillation of a body dropped in a diametrical tunnel dug across earth is T
=
g
R
2π = 84 minutes approximately:
134 Period of a test-tube floating in a liquid = T =
Adg
m 2π , m mass of the test tube, A its cross sectional area, d density of the liquid
135 The period of a liquid oscillating in a U-tube = T =
g
L 2π where L is total length of liquid column
136 A uniform cubic plank of side L is floating in a liquid of density n times its density ( n > 1) If it is immersed slightly and released, it will execute simple harmonic motion of period T = T =
ng
L
137 The loss or gain of a seconds pendulum ( or clock) in a dT = ⎥
⎦
⎤
⎢
⎣
⎡
− L
dL g
dg
increase in length, dg is increase in value of acceleration due to gravity g Give + sign to dg or dL if
it is increase, - sign if it is decrease If the answer carries positive sign, it is gain If it carries negative sign it is loss
Electrostatics/Preliminaries
138 Charge is a scalar It is added algebraically (with positive and negative signs)
139 Coulomb’s law is obeyed for all distances except that in nuclear range Nuclear forces do not obey inverse square (Coulomb’s) law
140 The coulomb force between two point charges depends only on the charges, their separation and the medium It is independent of other charges present
141 The dielectric constant of a conductor is infinite
142 ε0 has two units C2N-1m-2 and Fm-1 Its value is 8.85x10-12 and for rough calculations it can be taken as 9 x10-12
143 The charge of electron in SI is equal to -1.6x10-19 C In CGS electrostatic units (esu) it is equal to -4.8x10-10 In CGS electromagnetic units (emu) it is equal to -1.6x10-20
144 The specific charge of electron is nearly 1.7x1011 C/kg
Electrostatic field
145 The electrostatic field has two units NC-1 and Vm-1
146 The number of lines of force coming out of a unit positive charge is
0
1
ε = 1.11 x 10
11
147 If a charge q coulomb is placed at the centre of cube of side a, the number of lines of force coming out through one side of a cube is q/6ε0 This follows from Gauss’s theorem
148 If a cube is placed in uniform electric field the net flux through it will be zero This also follows from Gauss’s theorem
149 The electric field (E) due to a line of charge is proportional to
r 1
150 The electric field (E) due to a point charge is proportional to 2
r 1
Trang 9Physics at a Glance 9
151 The electric field (E) due to a dipole is proportional to 3
r
1
152 The electric field (E) due to a uniformly charged flat sheet is constant at all points This means it does not depend on distance It is proportional to r0
153 The electric field (E) due to a charged solid sphere at any point inside the sphere is directly proportional to r, where r is the distance from its centre
154 The electric field is uniform in a region; if (a) the number of lines of force crossing unit area normally, is same at all points and (b) the lines of force are parallel The first condition (a) makes the magnitude of the field to be the same, while the second condition (b) makes the direction of the field
to be the same at all points
155 To find the direction of electric field at a point, imagine a unit positive charge at the point Find the magnitude of force on it This gives the magnitude of field Find the direction of motion of that charge This gives the direction of electric field
Electrostatic potential
156 Electric potential (V) due to a point charge at a distance r is proportional to
r 1
157 Electric potential (V) due to a dipole at a distance r is proportional 2
r 1
158 In a uniform electric field a dipole experiences only a torque In a non uniform electric field a dipole experiences both a force and a torque
159 When a wire connects two charged conductors, charges flow from higher potential to lower potential
until the potentials are equal and not from higher charge to lower charge
Capacitance
160 When n identical capacitors each of capacitance C are connected in parallel, the effective capacitance
is CP = nC If n identical capacitors each of capacitance C are in series, the effective capacitance is
Cs =
n
C
161 If n identical capacitors are joined in parallel and in series the ratio of effective capacitance will be
S
p
C
C
= n2 and
p
S C
C
= 2
n 1
162 If three capacitors of capacitance C1, C2 and C3 are connected in series to a potential difference V, the potential across the capacitors in the respective order will be in the ratio C2 C3 : C3 C1 : C1 C2.
163 N capacitors are connected in parallel to a potential difference V If those capacitors are now joined
in series without disturbing their charges, the potential difference across the combination will be nV
164 When a conductor of capacity C is charged to a potential V, the work done by the battery is CV2 and energy of the conductor is equal to
2
1
CV2 This means half of the energy of the battery appears heat during charging
165 If a wire connects two conductors of capacitance C1 and C1, charged to potentials V1 and V2, charges flow from higher potential to lower potential In the process, some electrostatic energy is lost The loss of electrostatic energy is given by (V V)
C C
C C 2
2 1 2 1
2
+
166 If two condensers of capacitance C1 and C2, charged to potentials V1 and V2 are connected in parallel, the loss of electrostatic energy will be same as given in the previous equation
Trang 10General
167 If electric field at a point is zero, electric potential at the point need not be zero This means electric potential can exist without electric field For e.g in figure 1, the electric field at the centre point O is zero, but the electric potential is not zero
168 If electric potential at a point zero, electric field need not be zero This means electric field can exist
without electric potential For example in fig 2, the electric potential at the centre point O is zero, but the electric field at the point is not zero
Fig.2
Fig.1
169 The following table gives how various quantities change when a dielectric slab is introduced between
the plates of a parallel plate condenser: Quantities: Q:charge, C capacitance,V potential,U electrostatic energy, E electric field
How the quantity changes Quantity
When battery is connected When battery is disconnected
170 In a charged conductor, there is outward mechanical force acting This force is given by
0
2 2ε
σ
N/m2, where σ is the surface density of charge (charge per unit area)
171 The capacity of the earth is nearly 700 µF
172 The approximate electric field above which air becomes ionised is equal to 3x106 N/C
About charged drops
173 When n identical drops each charged to a potential V coalesce, the potential of the resulting drop will
be n2/3V
174 When n identical drops each of capacity C coalesce, the capacity of the resulting drop will be n1/3C
175 When n identical drops each carrying a charge q coalesce, charge of the resulting drop will be nq
176 When n identical drops each of surface density of charge σ coalesce, the surface density of charge of the resulting drop will be n1/3 σ
177 When n identical drops each of potential energy U coalesce, the potential energy of resulting drop =
n5/3U
178 A charged soap bubble always expands whether positively or negatively charged
Ohm’s law and applications:
179 The number of electrons crossing when 1 A of current flows through a conductor is 6.25 x 1018 per second
180 One way of writing Ohm’s law is j = σE, where j is current density, σ conductivity and E, electric field