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Units, Dimensions and
Error Analysis
m
1
...
1 Units and Dimensions
Chapter Snapshot
yC
Fundamental and Derived Quantities
q
ch
q
Units and Dimensions
Significant Figures
Error Analysis
ing
q
oa
To measure a physical quantity we need some
standard unit of that quantity
...
Thus, suppose I say that length of this
wire is 5 m
...
ww
w
...
However, not all the quantities are independent
...
Thus, we can define a set of
fundamental quantities and all other quantities may be
expressed in terms of the fundamental quantities
...
Unit
of all other quantities can be expressed in terms of the
units of these seven quantities by multiplication or
division
...
For example, if we take length
and time as the fundamental quantities, then speed is a
derived quantity and if we take speed and time as
fundamental quantities then length is a derived quantity
...
The
units defined for the fundamental quantities are called
fundamental units and those obtained for derived
quantities are called the derived units
...
It is
abbreviated as SI from the French name Le Systems
International d’ Unites
...
Table below gives the seven
fundamental quantities and their SI units
...
1
S
...
1
...
3
...
5
...
7
...
Their units are radian (rad)
and steradian (st) respectively
...
The unit of force is dyne and
that of work or energy is erg
...
The
unit of force in this system is poundal
...
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2
Objective Physics Volume 1
(ix) Steradian The steradian is the solid angle which
having its vertex at the centre of the sphere, cut-off an
area of the surface of sphere equal to that of a square
with sides of length equal to the radius of the sphere
...
2
Power of 10
Prefix
Symbol
6
3
-2
-3
-6
-9
- 12
mega
kilo
centi
mili
micro
nano
pico
M
k
c
m
m
n
p
Dimensions
Dimensions of a physical quantity are the powers to
which the fundamental quantities must be raised to
represent the given physical quantity
...
The dimensions of all other fundamental
quantities are zero
...
Generally mass is denoted
by M, length by L, time by T and electric current by A
...
The
physical quantity that is expressed in terms of the base
quantities is enclosed in square brackets
...
(i) can be written as
[density] = [ML–3 ]
Such an expression for a physical quantity in terms of
the fundamental quantities is called the dimensional
formula
...
, have no
units and dimensions
...
co
m
(i) Metre 1 m = 1,650,763
...
(ii) Second 1 s = 9, 192, 631, 770 time periods of a
particular radiation from cesium-133 atom
...
(iv) Ampere It is the current which when flows through
two infinitely long straight conductors of negligible
cross-section placed at a distance of 1 m in vacuum
produces a force of 2 ´ 10 -7 N/m between them
...
16 part of the thermodynamic
temperature of triple point of water
...
) as there are atoms in
12 g of carbon-12
...
013 ´ 105 N/m2
...
3 given below gives the dimensional
formulae and SI units of some physical quantities
frequently used in physics
...
No
...
2
...
pic
kM
(viii) Radian It is the plane angle between two radii of a
circle which cut-off on the circumference, an arc
equal in length to the radius
...
3
Physical Quantity
Velocity = displacement/time
Acceleration = velocity/time
[sin q] = [cos q] = [tan q] = [log x ]
= [ ex ] = [M0L0T0 ]
SI Units
m/s
m/s
Dimensional Formula
0
-1
[M LT ]
[M0LT -2 ]
2
3
...
Work = force ´ displacement
kg-m 2 /s 2 = N-m = joule or J
[ML2 T -2 ]
5
...
Torque = force ´perpendicular distance
N-m
[ML2 T -2 ]
7
...
Momentum = mass ´ velocity
kg-m/s
[MLT -1 ]
9
...
Angle = arc/radius
radian or rad
[M0 L0 T 0 ]
11
...
DL DV
or
L
V
Stress = force/area
13
...
PICKMYCOACHING
...
PICKMYCOACHING
...
No
...
Modulus of elasticity = stress/strain
N/m 2
[ML-1T -2 ]
15
...
Angular velocity = angle/time
rad/s
[M0 L0 T -1 ]
17
...
Surface tension = force/length
N/m
[ML0 T -2 ]
19
...
Coefficient of viscosity
N-s/m 2
[ML-1 T -1 ]
22
...
Specific heat (s)
J/kg-K
[M0 L2 T -2 q-1 ]
24
...
Gas constant (R )
J/mol-K
26
...
28
...
30
...
Electric potential
volt (V)
32
...
Permittivity of free space
[M–1L–3 T4 A2 ]
34
...
36
...
Coefficient of self-induction
m
20
...
co
[ML2 T -2 q-1 ]
[L q]
[MLT -3 q-4 ]
ing
watt/m 2 -K 4
[AT]
[MLT -3 A -1 ]
oa
ch
[ML2 T -3 A -1 ]
yC
[LTA]
[ML2 T –3 A –2 ]
[MT –2 A –1 ]
tesla (T) or weber/m 2 (Wb/m 2 )
henry (H)
kM
[ML2 T –2 A –2 ]
q
q
q
ww
w
...
5 ´ 1011 m
Light year
1 ly = distance travelled by light in vacuum in 1 year
= 9
...
07 ´ 1016 m = 3
...
1 Find the dimensional formulae of
(a) coefficient of viscosity h
(b) charge q
(c) potential V
(d) capacitance C, and
(e) resistance R
...
3 Pa
1 barn = 10 -28 m2
1 horse power = 746 W
1 pound = 453
...
4536 kg
where A denotes the area, v the velocity, l is the length, I the
electric current, t the time and U the energy
...
PICKMYCOACHING
...
PICKMYCOACHING
...
This is called the
‘Principle of Homogeneity’
...
On this basis we can judge whether a given
equation is correct or not
...
U = VIt
U
V =
It
\
or
[U] [ML2 T -2 ]
=
= [ML2 T -3 A -1 ]
[ I][t]
[ A ][ T]
[V ] =
(d)
q = CV
q
C=
V
[ q]
[ AT]
= [M -2 L-2 T 4 A 2 ]
=
[C ] =
[ V ] [ML2 T -3 A -1 ]
\
or
(e)
Example 1
...
V = IR
V
R=
I
or
[ R] =
2
-3
Dimensionally,
-1
[ V ] [ML T A ]
=
= [ML2 T -3 A -2 ]
[ I]
[ A]
[T] =
l
g
[L]
= [T]
[LT -2 ]
m
\
T = 2p
Solution
As in the above equation, the dimensions of both sides
are same
...
...
e
...
If the
fundamental units in one system are M1, L1 and T1
and in the other system are M2 , L2 and T2
respectively
...
For
1
example, in the physical expression s = ut + at2 , the
2
1
dimensions of s, ut and at2 all are same
...
kM
Here, n1 and n2 are the numerical values in two
system of units respectively
...
(i), we can
convert the numerical value of a physical quantity
from one system of units into the other system
...
2 The value of gravitational constant is
2
2
in SI units
...
pic
N-m /kg
G = 6
...
–11
Example 1
...
Solution From principle of homogeneity
[ a] = [ v]
or
Here,
–1
3
é L1 ù é T1 ù
êL ú ê T ú
ë 2û ë 2û
M1 = 1 kg, M2 = 1 g = 10
Further,
kg
[ d ] = [t] = [ T]
[ c]
= [ v] or [ c] = [ v][ d + t]
[ d + t]
or
Substituting in the above equation, we get
3
é 1 m ù é1 s ù
ê10–2 m ú ê1 s ú
ë
û ë û
–2
or
n2 = 6
...
67 ´ 10-8 dyne cm2 /g2
...
67 ´ 10-11 ê –3
ú
ë10 kg û
[ b] =
or
–2
n1 = 6
...
Using Eq
...
e
...
Let us take an example
...
5 The frequency (f) of a stretched string depends
upon the tension F (dimensions of force), length l of the string
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Chapter 1 · Units, Dimensions and Error Analysis
and the mass per unit length m of string
...
Solution Suppose, that the frequency f depends on the
tension raised to the power a, length raised to the power b and
mass per unit length raised to the power c
...
Thus,
or
[ f] = [F]a [ l]b [m ]c
Rules for Counting Significant Figures
[M0 L0 T –1 ] = [MLT –2 ]a [L]b [ML–1 ]c
or
[M0 L0 T –1 ] = [M a + c La + b - c T -2 a]
For dimensional balance, the dimensions on both sides
should be same
...
and b = - 1
Substituting these values in Eq
...
0042 has two significant digits
...
For example, 4
...
002300 have four significant
figures each
...
Thus, 426 m
= 42600 cm = 426000 mm has three significant figures
...
For
example, x = 2567 has four significant figures
...
For example, 6
...
ing
1
1
, c=2
2
For counting significant figures, we use the following
rules
...
For example, when we measure the length of a
straight line using a metre scale and it lies between 7
...
5 cm, we may estimate it as l = 7
...
This
expression has three significant figures out of these 7 and
4 are precisely known but the last digit 3 is only
approximately known
...
pic
The method of dimensions has the following
limitations
(i) By this method the value of dimensionless constant
cannot be calculated
...
(iii) If a physical quantity depends on more than three
factors, then relation among them cannot be
established because we can have only three
equations by equalising the powers of M, L and T
...
2 Significant Figures
Significant figures in the measured value of a physical
quantity tell the number of digits in which we have
confidence
...
‘‘All accurately known digits in a measurement plus
the first uncertain digit together form significant figures
...
30 m = 23
...
00230 km
When we are writing 2300 mm, then from Rule -4, we
would conclude erroneously that the number has two
significant figures, while in fact it has three significant
figures and a mere change of units cannot change the
number of significant figures
...
Rule 6 The power of 10 is irrelevant to the
determination of significant figures
...
30 m = 2
...
30 ´ 103 mm
= 2
...
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6
Objective Physics Volume 1
Rule 7 A choice of change of different units does not
change the number of significant digits or figures in a
measurement
...
03 cm has three significant
figures
...
0703 m or 70
...
All these measurements
have the same number of significant figures (digits 7, 0
and 3) namely three
...
Rule
5
1
5
2
0
...
co
2
...
6 ´ 10 10
(ii) Multiplication or Division Suppose in the
measured values to be multiplied or divided, the
least number of significant digits be n, then in the
product or quotient, the number of significant
digits should also be n
...
2 ´ 36
...
064 » 44
The least number of significant digits in the measured
values are two
...
Therefore, the answer is 44
...
8431373 » 110
10
...
e
...
Hence, the result when rounded off to
two significant digits becomes 110
...
8431373 » 108
10
...
7
Example 12
...
2 = 2
...
4
Rounding off a Digit
Note In this case answer becomes 108
...
However, if it is more than 5, then the cut-off digit is
increased by 1
...
24 is rounded off to 6
...
328 is rounded off to 5
...
ing
Number of significant figures
Here, the least number of significant digits after
the decimal is one
...
4
(when rounded off to smallest number of decimal
places)
...
2 + 3
...
789 = 11
...
4
kM
Rule 2 If the insignificant digit to be dropped is 5 then
the rule is
(i) it the preceding digit is even, the insignificant
digit is simply dropped
...
pic
(ii) if the preceding digit is odd, the preceding digit is
raised by 1
...
265 is rounded off to x = 6
...
275 is rounded off to
x = 6
...
Algebraic Operations with Significant
Figures
In addition , subtraction, multiplication or division the
final result should not have more significant figures than
the original data from which it was obtained
...
The chain will obviously break
at the weakest link
...
(i) Addition and Subtraction Suppose in the
measured values to be added or subtracted, the
least number of significant digits after the decimal
is n
...
1
...
Measured
value is always somewhat different from the true value
...
Errors can be classified in two ways
...
Systematic
errors and random errors fall in this group
...
Absolute error, mean absolute error and relative (or
fractional) error lie on this group
...
(i) Systematic errors These are the errors whose
causes are known to us
...
Following are few causes of these
errors
...
These errors can be reduced by
using more accurate instruments and applying
zero correction, when required
...
For example, in measuring time
period of simple pendulum error may creap
because no consideration is taken of air
resistance
...
(c) Change in temperature, pressure, humidity
etc
...
PICKMYCOACHING
...
PICKMYCOACHING
...
620 + 2
...
630 + 2
...
626
am =
5
= 2
...
626 cm
(rounding off to three decimal places)
(b) Taking am as the true value, the absolute errors in
different observations are,
Da1 = 2
...
620 = + 0
...
626 – 2
...
001 cm
Da3 = 2
...
630 = – 0
...
626 – 2
...
002 cm
...
626 – 2
...
000 cm
(c) Mean absolute error,
| Da1| + | Da2| + | Da3| + | Da4| + | Da5|
Damean =
5
0
...
001 + 0
...
002 + 0
...
0026
= 0
...
Usually the mean value am is
taken as the true value
...
(e) percentage error, and
(f) express the result in terms of percentage error
...
Relevant corrections can be made to
minimise their effects
...
Hence, it is not possible to remove
them completely
...
For example, the reading of a
sensitive beam balance may change by the
vibrations caused in the building due to persons
moving in the laboratory or vehicles running
nearby
...
The mean value would be very close
to the most accurate reading
...
Thus,
|Da| + |Da| + K + |Dan|
1
2
Damean =
n
ww
w
...
(v) Relative or fractional error The ratio of mean
absolute error to the mean value of the quantity
measured is called relative or fractional error
...
e
...
6 The diameter of a wire as measured by a screw
gauge was found to be 2
...
625, 2
...
628 and 2
...
Calculate
(a) mean value of diameter,
(b) absolute error in each measurement,
(c) mean absolute error,
(d) fractional error,
=±
0
...
626
= ± 0
...
001 ´ 100
= ± 0
...
626 cm ± 0
...
Further, let Da is the absolute error in the
measurement of a, Db is the absolute error in the
measurement of b and Dx is the absolute error in
the measurement of x
...
Therefore, the
maximum absolute error in x is
Dx = ± ( Da + Db )
i
...
, the maximum absolute error in sum and
difference of two quantities is equal to sum of the
absolute errors in the individual quantities
...
PICKMYCOACHING
...
PICKMYCOACHING
...
Dx
Da Db
Hence,
±
=±
+
x
a
b
Dx
æ Da Db ö æ Da Db ö
Possible values of
are ç
+
÷, ç
÷,
x
b ø è a
b ø
è a
æ Da Db ö
æ Da Db ö
+
ç÷ and ç ÷
...
7 The volumes of two bodies are measured to be
or
V1 = (10
...
02) cm3 and V2 = (6
...
01) cm3
...
V1 = (10
...
02) cm3
Solution
V2 = (6
...
01) cm3
and
DV = ± (DV1 + DV2 )
= ± (0
...
01) cm3 = ± 0
...
2 + 6
...
6 cm3
V1 – V2 = (10
...
4) cm3 = 3
...
6 ± 0
...
8 ± 0
...
co
m
Or, the maximum value of fractional error in
division of two quantities is equal to the sum of
fractional errors in the individual quantities
...
pic
Dx
Db ö
æ Da
= ±ç
+
÷
x
a
b ø
è
kM
Hence, maximum possible value of
yC
Therefore, maximum fractional error in product of
two (or more) quantities is equal to sum of
fractional errors in the individual quantities
...
8 The mass and density of a solid sphere are
measured to be (12
...
1) kg and (4
...
2) kg/m3
...
Solution Here, m ± Dm = (12
...
1) kg
and
r ± Dr = (4
...
2) kg/m3
m 12
...
6
= 2
...
7 m3
(rounding off to one decimal place)
Da ö
÷
a ø
Db ö
÷
b ø
Dx ö æ
Da ö æ
Db ö
æ
ç1 ±
÷ = ç1 ±
÷ ç1 ±
÷
x ø è
a øè
b ø
è
(iv) Error in quantity raised to some power
an
Let
x= m
b
Then, ln ( x ) = n ln ( a) - m ln ( b)
Differentiating both sides, we get
db
da
dx
=n
-m
b
a
x
In terms of fractional error we may write,
Dx
Da
Db
±
=± n
+m
b
a
x
Therefore, maximum value of
Dx
Db ö
æ Da
= ± çn
+m
÷
b ø
x
è a
ch
oa
(ii) Errors in a product Let x = ab
Then, ( x ± Dx ) = ( a ± Da)( b ± Db)
Dx ö
Da ö æ
Db ö
æ
æ
or x ç1 ±
÷ = ab ç1 ±
÷ ç1 ±
÷
x ø
a øè
b ø
è
è
Dx
Db Da Da Db
or
(as x = ab)
1±
=1 ±
±
±
×
x
b
a
a b
Dx
Da Db Da Db
or
±
=±
±
±
×
x
a
b
a b
Da Db
is a small quantity, so can be
Here,
×
a b
neglected
...
a
b ø
b ø
è
è a
1±
Now,
-1
DV
æ Dm Dr ö
=±ç
+
÷
r ø
V
è m
or
æ Dm Dr ö
DV = ± ç
+
÷ ´V
r ø
è m
aö
æ
ç as x = ÷
bø
è
Db
<<1, so expanding binomially, we get
b
Db ö
Dx ö æ
Da ö æ
æ
ç1 ±
÷ = ç1 ±
÷ ç1 +
÷
x ø è
a øè
b ø
è
0
...
1
=± ç
+
÷ ´ 2
...
4 4
...
14
\
V ± DV = (2
...
14) m3
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Chapter 1 · Units, Dimensions and Error Analysis
9
1
æ1
ö
= ± ç ´ 1 + ´ 2÷ = ± 1
...
9 Calculate percentage error in determination of
time period of a pendulum
T = 2p
l
g
Least Count
The minimum measurement that can be measured
accurately by an instrument is called the least count
...
The least count of a watch having seconds hand is
1 s
...
Solution
DT
´ 100 = ±
T
ö
æ 1 Dl
1 Dg
ç ´
÷
ç 2 l ´ 100 + 2 ´ g ´ 100÷
ø
è
Key-Terms for Concepts
Least count of vernier callipers
ìValue of 1 part of ü ìValue of one part onü
=í
ý –í
ý
î main scale ( s ) þ î vernier scale ( v ) þ
Value of 1 part on main scale ( s )
Number of parts on vernier scale ( n )
q
Least count =
q
Least count of screw guage
m
q
Least count of vernier callipers = 1MSD – 1VSD
where, MSD = Main Scale Division
VSD = Vernier Scale Division
Pitch ( p )
Number of parts on circular scale ( n )
ing
...
Solution Let
oa
Example 1
...
LHS = s = displacement = [M0 LT0 ]
ww
w
...
Writing the dimensions of RHS and LHS in Eq
...
(i), we get
0
= [M LT ][T] =[M LT ]
F = kmv2 r -1 = k
As LHS = RHS, formula is dimensionally correct
...
Write the dimensions of a and b in the relation,
b - x2
P=
at
where P is power, x the distance and t the time
...
The centripetal force F acting on a particle
moving uniformly in a circle may depend upon mass (m),
velocity (v) and radius (r) of the circle
...
…(i)
F = k (m)x (v)y (r )z
mv2
r
mv2
r
(where k = 1)
Example 4
...
00 m
(c) 0
...
(b) 62
...
(c) 0
...
(d) 1200 N has four significant figures
...
Round off to four significant figures :
(a) 45
...
0082
Solution (a) 45
...
008
Example 6
...
75 ´103 cm to 4
...
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Objective Physics Volume 1
Solution
a = 6
...
52 ´ 102 cm
=
0
...
01+ 0
...
03 + 0
...
02
= 0
...
45 ´ 10 cm
(d) Fractional error =
(upto 2 places of decimal)
= ± 0
...
75 ´ 10 + 0
...
02
=
nm
1
...
0132 ´ 100) = ± 1
...
20 ´ 10 cm
Example 7
...
7 cm and radius
Example 9
...
1 ± 0
...
Calculate its surface area with error limits
...
1)2
è7ø
l = 21
...
46 mm = 0
...
046)2 (21
...
1443 cm3 = 0
...
e
...
46 mm )
...
The refractive index (n) of glass is found to have
the values 1
...
50, 1
...
54 and 1
...
Calculate
(a) the mean value of refractive index,
= 26
...
4 cm2
S = (55
...
4) cm2
the following observations
...
1 ± 0
...
1 ± 0
...
oa
(d) fractional error, and
Solution
yC
(e) percentage error
...
49 + 1
...
52 + 1
...
48
nm =
5
(b) Taking nm as the true value, the asbolute errors in
different observations are,
ww
w
...
5 ´ 55
...
1
Example 10
...
505 = 1
...
co
Further,
= 55
...
4 cm2
DS
Dr
=2
S
r
æ Dr ö
DS = 2 ç ÷ (S)
è r ø
m
0
...
Calculate the volume of the wire to correct significant
figures
...
51 - 1
...
01
Dn5 = 1
...
48 = + 0
...
00998 + 0
...
00997]
Dn3 = 1
...
52 = - 0
...
1) (20
...
3 cm
(50
...
1)
é 0
...
2
0
...
2 ù
=±ê
+
+
50
...
1 50
...
1ú
û
ë
Dn1 = 1
...
49 = + 0
...
51 - 1
...
03
1 1 1
= +
f v u
uv
f=
u+v
= ± (0
...
0299 ´ 14
...
428 = 0
...
3 ± 0
...
PICKMYCOACHING
...
PICKMYCOACHING
...
Fill in the blanks
(a) The volume of a cube of side 1 cm is equal to … m 3
...
0 cm and height 10
...
(c) A vehicle moving with a speed of 18 kmh-1
covers…… m in 1s
...
3
...
0 ms -2 = 3 ´ 10 -3 km ´ ç
h -1 ÷
60 ´ 60
è
ø
= 3 ´ 10 -3 ´ ( 3600)2 kmh-2
= 3
...
9 ´ 10 4 kmh-2
(d) G =6
...
67 ´ 10 -11 (105 dyne) (102 cm)2 (103 g) -2
= 6
...
67 ´ 10 -8 (g cm s -2 ) cm2 g -2
= 6
...
A calorie is a unit of heat or energy and it equals
about 4
...
Suppose we
employ a system of units in which the unit of mass
equals a kg, the unit of length b m , and the unit of
time is g s
...
2 a -1 b-2 g 2 in terms of new units
...
5099 ´ 10 4 mm2
oa
yC
ww
w
...
3
Density of water = 1 g cm-3
density of lead
Relative density of lead =
density of water
\ Density of lead
= relative density of lead ´ density of water
= 11
...
3 g cm-3
In SI system density of water = 103 kg m-3
\ Density of lead = 11
...
13 ´ 10 4 kg m-3
2
...
(a) 1 kg m 2 s-2 =…… g cm 2 s-2
(b) 1 m = …… ly (light year)
(c) 3
...
67 ´ 1011Nm 2 kg -2 = …… cm 3 s-2 g -1
Solution
(a) 1 kg m2s -2 = 1 (103 g) (102 cm)2 s -2
= 103 ´ 10 4 g cm2s -2 = 107 g cm2 s -2
(b) 1 light year (ly) = 9
...
057 ´- 16 ly
\
9
...
co
V = (10 -2 m)3 = 10 -6 m3
= 1
...
2 J =4
...
2 ê
ú ê ú
ú ê
ë a kg û ë b m û ë g s û
n 2 = 4
...
2 a -1 b -2 g 2 in new system
...
A new unit of length is chosen such that the speed of
light in vacuum is unity
...
Time taken by light of sun of reach the earth,
t = 8 min 20s
...
Which of the following is the most precise device for
measuring length?
(a) A vernier callipers with 20 divisions on the sliding
scale
...
(c) An optical instrument that can measure length to
within a wavelength of light
...
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Objective Physics Volume 1
9
...
The percentage errors
a, b, c and d as follows P =
cd
of measurement in a, b, c and d are 1%, 3%, 4% and 2%
respectively
...
763, to what value
should you round off the result?
(a) Least count of vernier callipers
= 1 MSD - 1 VSD
19
1
= 1 MSD MSD =
MSD
20
20
1
=
mm
20
1
=
cm
200
= 0
...
001 cm
(c) Wavelength of light, l » 10 -5 cm = 0
...
co
ch
ing
6
...
007 m 2
(b) 2
...
2370 g cm
(d) 6
...
032 Nm -2
(f) 0
...
A book with many printing errors contains four
different formulae for the displacement y of a particle
under going a certain periodic motion :
2pt
(a) y = a sin
T
(b) y = a sin vt
a
(c) y = sin (t / a)
T
2 pt
2 pt ö
æ a öæ
(d) y = ç
+ cos
÷ ç sin
÷
T
T ø
è 2 øè
oa
yC
7
...
234 m, 1
...
01 cm
respectively
...
kM
Solution Here length, l = 4
...
(ii)
ú
Db
Dd
´ 100 = 3 %,
´ 100 = 2% ú
b
d
û
\From Eqs
...
e
...
763 = 3
...
\ Least count of optical instrument = 0
...
Solution The number of significant figures is as given below
...
pic
Breadth, b = 1
...
0201 m = 2
...
234 ´ 1005 + 1005 ´ 0
...
0201 ´ 4
...
...
7209468 m2
As the least number of significant figures in thickness is 3,
Area = 8
...
234 ´ 1005 ´ 0
...
0855 m3
...
The mass of a box measured by a grocer’s balance is
2
...
Two gold pieces of masses 20
...
17 g
are added to the box
...
300 + 0
...
02017) kg
= 2
...
Total mass = 2
...
17 - 20
...
02 g
Since there are two significant figures, so the difference
in masses to the correct significant figures is 0
...
(where a = maximum displacement of the particle,
v = speed of the particle, T = time period of motion)
...
Solution The argument of a trigonometrical function i
...
, angle is
dimensionless
...
11
...
1 Å
= 10 -10 m
...
5 Å
...
PICKMYCOACHING
...
PICKMYCOACHING
...
5Å = 0
...
14 ´ (0
...
236 ´ 10 -31 m3
According to Avogadro’s hypothesis, one mole of hydrogen
contains
N = 6
...
023 ´ 1023 ´ 5
...
154 ´ 10 -7 m3
@ 3 ´ 10 -7 m3
ww
w
...
4 ´ 10 -3 m3
=
Atomic volume 3
...
1 ´10 4 = 7 ´ 10 4
The large value of the ratio shows that the inter molecular
separation in a gas in much larger than size of a mulecule
...
The nearest star to our solar system is 4
...
How much is this distance in terms of parsecs?
Solution
Distance = 4
...
29 ´ 9
...
46 ´1015 m)
=
m
Solution Average radius of sodium atom,
r = 2
...
5 ´ 10 -10 m
4
\ Volume of sodium atom = pr 3
3
4
= ´ 3
...
5 ´ 10 -10 )3
3
= 65
...
023 ´ 1023 ´ 5
...
154 ´ 10 -7 m3
15
...
5 Å (Use the known
values of Avogadro's number, and the atomic mass of
sodium)
...
Are the two densities of
the same order of magnitude? If so, why?
ch
Molar volume of one mole of hydrogen
= 22
...
4 ´ 10 -3 m3
r = radius of one molecule of hydrogen
d
= =0
...
5 ´ 10 -10 m
Volume of one molecule of hydrogen
4
4
= pr 3 = p (0
...
236 ´ 10 -31 m3
1 mole has 6
...
1 part in -11 to
10
6
...
34 ´ 1012
...
One mole of an ideal gas at standard temperature and
pressure occupies 22
...
What is the
ratio of molar volume to the atomic volume of a mole
of hydrogen? (Take the size of hydrogen molecule to
be about 1 Å)
...
155 ´ 10 9s
Difference in time = 0
...
2
=
3
...
34 ´ 10 -12
= 10 ´ 10 -12 » 10 -11
4
...
46 ´ 1015
parsec
3
...
08 ´ 1016 m)
= 1
...
32parsec
14
...
02s
...
One mole contains 6
...
82 ´ 10 -26 kg
M =
6
...
82 ´ 10 -26
r=
=
kgm-3
V 65
...
64 ´ 103 kgm-3
Density of sodium in crystalline phase = 970 kgm-3
= 0
...
64 ´ 103
=
Density of sodium of crystalline phase 0
...
66
Both densities are of the same order i
...
of the order of 103
...
16
...
In a submarine equipped with a SONAR the
time delay between generation of a probe wave and
the reception of its echo after reflection from an
enemy submarine is found to be 77
...
What is the
distance of the enemy submarine? (Speed of sound in
water = 1450 m s-1 )
...
PICKMYCOACHING
...
PICKMYCOACHING
...
5 s
2
Speed of sound, v = 1450 ms -1
puzzling features which have not yet been
satisfactorily explained
...
0 billion years to
reach us?
Solution Time taken, t = 3 ´ 10 9 years
= 3 ´ 10 9 ´ 365 ´ 24 ´ 60 ´ 60 s
Distance of enemy submarine,
\
S = vt = 1450 ´ 38
...
825 km
Velocity of light, c = 3 ´ 10 8 ms -1
\Distance of quasar from earth = ct
= 3 ´ 10 8 ´ 3 ´ 10 9 ´ 365 ´ 24 ´ 3600 m
17
...
These objects (known as quasars) have many
= 2
...
8 ´ 1022 km
m
Objective Problems (Level 1)
ww
w
...
Parallactic second is equal to
(a) 9
...
07 ´ 1016m
(c) 1
...
A new unit of length is chosen such that the speed of light in
vacuum is unity
...
A student measures the thickness of a human hair by looking
at it through a microscope of magnification 100
...
5 mm
...
0035 mm
(b) 0
...
01 m
(d) 0
...
Which of the following is the most precise device for
measuring length?
(a) a vernier calipers with 20 divisions on the sliding scale
(b) an optical instrument that can measure length to within
a wavelength of light
(c) a screw gauge of pitch 1 mm and 100 divisions on the
circular scale
(d) All the above are equally precise
Units and Dimensions
8
...
In the SI system, the unit of temperature is
(a) degree centigrade
(b) kelvin
(c) degree celsius
(d) degree fahrenheit
10
...
Which of the following does not posses the same dimensions
as that of pressure?
(a) Stress
(b) Bulk modulus
(c) Thrust
(d) Energy density
12
...
Which one of the following is not the dimensionless
quantity?
(a) Planck’s constant
(b) Dielectric constant
(c) Solid angle
(d) Strain
oa
1
...
“Parsec” is the unit of
(a) time
(b) distance
(c) frequency
(d) angular acceleration
3
...
46 ´ 1010 km
(b) 9
...
46 ´ 10 km
(d) 9
...
co
Measurement of Length and Time
14
...
Which of the following is not equal to watt?
(a) joule/second
(b) ampere ´ volt
(c) (ampere)2 ´ ohm
(d) ampere/volt
16
...
Wb/m2 is equal to
(a) dyne
(b) tesla
(c) watt
(d) henry
18
...
Which of the following has the dimensions of pressure?
(a) [ML-2 T -2 ]
(b) [M -1L-1]
-2
(c) [MLT ]
(d) [ML-1T -2 ]
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Chapter 1 · Units, Dimensions and Error Analysis
ch
ing
...
Assuming that the mass m of the largest stone that can be
moved by a flowing river depends upon the velocity v of the
water, its density r and the acceleration due to gravity g
...
If p represents radiation pressure, c represent speed of light
and Q represents radiation energy striking a unit area per
second, then non-zero integers x , y and z such that p x Q y c z
is dimensionless are
(a) x = 1, y = 1, z = - 1
(b) x = 1, y = - 1, z = 1
(c) x = - 1, y = 1, z = 1
(d) x = 1, y = 1, z = 1
38
...
Which of
the following is the correct change in the other units?
(a) Unit of time is doubled
(b) Unit of mass is doubled
(c) Unit of momentum is doubled
(d) Unit of energy is doubled
39
...
The dimensional representation of specific resistance in
terms of charge Q is
(b) [ML2 T -2Q2 ]
(a) [ML3 T -1Q -2 ]
-2 -1
(c) [MLT Q ]
(d) [ML2 T -2Q -1]
41
...
If C and R denote capacity and resistance, the dimensions of
CR are
(a) [M 0L0 T]
(b) [ML0 T]
(c) [M 0L0 T2 ]
(d) not expressible in terms of M, L and T
43
...
The dimensions of h are
(a) [ML-3 ]
(b) [MLT -2 ]
-1
(c) [MT ]
(d) [ML-1T -1]
æx
ö
44
...
The
dimension of k will be
(a) [T -2 ]
(b) [T -1]
(c) [T]
(d) [LT]
45
...
The
dimensions of a and b are
(a) [MLT -4] and [MLT]
(b) [MLT -1] and [MLT 0 ]
-3
-4
(c) [MLT ] and [MLT ]
(d) [MLT -3 ] and [MLT 0 ]
46
...
pic
kM
yC
oa
20
...
Dimensions of impulse are
(a) [ML-2 T -3 ]
(b) [ML-2 ]
-1
(c) [MLT ]
(d) [MLT -2 ]
22
...
Dimensions of surface tension are
(a) [M2L2 T -2 ]
(b) [M2LT -2 ]
-2
(c) [MT ]
(d) [MLT -2 ]
24
...
Which of the following is the dimensions of the coefficient of
friction?
(b) [M 0L0 T 0 ]
(a) [M2L2 T]
(c) [ML2 T -2 ]
(d) [M2L2 T -2 ]
26
...
[ML T ] stand for dimensions of
(a) work
(b) torque
(c) linear momentum
(d) coefficient of viscosity
28
...
The dimensions of the ratio of angular to linear momentum
us
(b) [MLT -1]
(a) [M 0LT 0 ]
2 -1
(d) [M -1L-1T -1]
(c) [ML T ]
30
...
[ML2 T -3 A -1] is the dimensional formula for
(a) capacitance
(b) resistance
(c) resistivity
(d) potential difference
32
...
Given that y = a cos ç - qx ÷ , where t represents time in
èp
ø
following statements is true?
(a) The unit of x is same as that of q
(b) The unit of x is same as that of p
(c) The unit of t is same as that of q
(d) The unit of t is same as that of p
34
...
Which of the following is dimensionally correct?
(a) Pressure = energy per unit area
(b) Pressure = energy per unit volume
(c) Pressure = force per unit volume
(d) Pressure = momentum per unit volume per unit time
15
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Objective Physics Volume 1
52
...
56
...
ww
w
...
kM
yC
54
...
...
ing
49
...
62
...
001 cm and listed the
measurements
...
320 cm
(b) 5
...
32 cm
(d) 5
...
The length, breadth and thickness of rectangular sheet of
metal are 4
...
005 m and 2
...
The
volume of the sheet to correct significant figures is
(a) 0
...
086 m3
(c) 0
...
08 m3
64
...
425 cm, 7
...
0 cm
...
635 cm
(b) 30
...
63 cm
(d) 30
...
Subtract 0
...
26 J and express the result with
correct number of significant figures
(a) 7
...
06
(c) 7
(d) None of these
66
...
88 by 0
...
8068
(b) 64
...
81
(d) 65
...
When 97
...
54, the correct result is
(a) 38
...
394
(c) 65
...
4
68
...
16 mm
...
08
(b) 0
...
0804
(d) 0
...
What is the number of significant figure in
(3
...
80) ´ 105 ?
(a) 5
(b) 4
(c) 3
(d) 2
70
...
0 ´ 10 -6 ) ( 5
...
0 ´ 10 -14
-13
(c) 2
...
The dimensions of a, b and c
are respectively
(a) [LT -2 ], [L] and [T]
(b) [L2 ], [T] and [LT2 ]
2
(c) [LT ], [LT] and [L]
(d) [L], [LT] and [T2 ]
1
What is the units of k =
?
4 pe0
(a) C2 N-1m-2
(b) Nm2C -2
2 2
(c) Nm C
(d) Unitless
Pressure gradient has the same dimensions as that of
(a) velocity gradient
(b) potential gradient
(c) energy gradient
(d) None of these
The unit of permittivity of free space, e0 is
(a) coulomb/newton-metre
(b) newton-metre2 /coulomb2
(c) coulomb2 /newton-metre2
(d) coulomb2 /(newton-metre)2
Dimensions of electrical resistance are
(a) [ML2 T -3 A -1]
(b) [ML2 T -3 A -2 ]
(c) [ML3 T -3 A -2 ]
(d) [ML-1L3 T3 A3 ]
The magnetic moment has dimensions of
(a) [LA]
(b) [L2 A]
-1
(c) [LT A]
(d) [L2 T -1A]
The dimensional representation of specific resistance in
terms of charge Q is
(a) [ML3 T -1Q -2 ]
(b) [ML2 T -2Q2 ]
-2 -1
(c) [MLT Q ]
(d) [ML2 T -2Q -1]
47
...
The significant figures of the number 6
...
What is the number of significant figures in 0
...
The number of significant figures in 11
...
In which of the following numerical values, all zeros are
significant?
(a) 0
...
2
(c) 2020
(d) None of these
71
...
3 kg
...
15 g and 20
...
The total mass of
the box to correct significant figures is
(a) 2
...
34 kg (c) 2
...
31 kg
72
...
2 kg from 34 kg
...
8 kg
(b) 33
...
0 kg
73
...
45 cm
...
763 ´ 10 cm
(d) None of these
Error Analysis
74
...
05 ± 0
...
What is the length of
the two rods?
(a) (22
...
05) cm
(b) (22
...
1) cm
(c) (22
...
05) cm
(d) (22
...
2) cm
75
...
2 ± 0
...
The percentage error in
the volume of the ball is approximately
(a) 11%
(b) 4%
(c) 7%
(d) 9%
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Chapter 1 · Units, Dimensions and Error Analysis
84
...
Imagine a system of unit in which the unit of mass is 10 kg,
length is 1 km and time is 1 min
...
6
(c) 36 ´ 105
(d) 36 ´ 10 -5
86
...
In measuring electric energy, 1kWh is equal to
(a) 3
...
6 ´ 10 6 J
6
(c) 7
...
Out of the following four dimensional quantities, which one
qualifies to be called a dimensional constant?
(a) Acceleration due to gravity
(b) Surface tension of water
(c) Weight of a standard kilogram mass
(d) The velocity of light in vacuum
89
...
With usual notation, the following equation, said to give the
distance
covered
in
the
second
...
e
...
pic
kM
yC
oa
78
...
If error in measuring diameter of a circle is 4%, the error in
measuring radius of the circle would be
(a) 2%
(b) 8%
(c) 4%
(d) 1%
80
...
0 ± 0
...
0 ± 0
...
What is the percentage error in the
equivalent resistance when they are connected in parallel?
(a) 2%
(b) 5%
(c) 7%
(d) 10%
81
...
If the error in measuring the above are
1%, 2% and 1% respectively
...
A force F is applied on a square plate of side L
...
What is the permissible error in pressure?
(a) 8%
(b) 6%
(c) 4%
(d) 2%
Miscellaneous Problems
m
If percentage errors in A, B, C , D are 2%, 1%, 3% and 4%
respectively
...
A body travels uniformly a distance of (13
...
2) m in a time
(4
...
3) s
...
45 ± 0
...
45 ± 0
...
45 ± 0
...
45 ± 0
...
co
A3 B3
C D
ing
Q=
83
...
If the relative percentage error in the
measurement of l is 1%, then the relative percentage error
in measurement of V is
(a) 18%
(b) 6%
(c) 3%
(d) 1%
ch
76
...
Then, X has dimensional
formula same as that of
(a) resistance
(b) charge
(c) voltage
(d) current
1
...
The length of a strip measured with a metre rod is 10
...
Its width measured with a vernier calipers is 1
...
The
least count of the metre rod is 0
...
01 cm
...
The length of cylinder is measured with a metre rod having
least count 0
...
Its diameter is measured with vernier
calipers having least count 0
...
Given that length is 5
...
0 cm
...
5%
(b) 2
...
5%
(d) 4%
4
...
PICKMYCOACHING
...
PICKMYCOACHING
...
Dimensions of ‘ohm’ are same as
h
h2
h
h2
(a)
(b)
(c) 2
(d) 2
e
e
e
e
(where h is Planck’s constant and e is charge)
dx
éx -aù
6
...
13
...
m
where a = constant
...
Suppose the
earth’s radius decreases by 2%
...
Assertion and Reason
kM
yC
oa
10
...
...
11
...
pic
Directions (Q
...
1-17) These questions consists of two
statements each printed as Assertion and Reason
...
(a) If both Assertion and Reason are correct and Reason in
the correct explanation of Assertion
...
(c) If Assertion is true but Reason is false
...
(e) If both Assertion and Reason are false
...
Assertion Pressure has the dimensions of energy density
...
2
...
Reason This is because trigonometrical ratios have no
dimensions
...
In this system 1 N
force will be equal to
(a) 1 new unit
(b) 129
...
6 new unit
(d) 60 new unit
If force F , length L and time T are taken as fundamental
units, the dimensional formula for mass will be
(a) [FL-1T2 ]
(b) [FLT - 2 ]
-1 -1
(c) [FL T ]
(d) [FL5 T2 ]
é æ 2p
öù
Given that y = A sin ê ç
( ct - x ) ÷ ú , where y and x are
øû
ëè l
measured in metre
...
Here, p is number of segments in the string
2l ê m ú
ë û
and l is the length
...
(b)
ch
(a) 4x
3
...
1
DE Dm 2Dv
Reason KE or E = mv 2 ,
=
+
2
E
m
v
4
...
3%
...
6%
...
A
r
5
...
Reason Both light year and wavelength represent time
...
Assertion Number of significant figures in 0
...
500 are three
...
7
...
7 m and l = 0
...
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Chapter 1 · Units, Dimensions and Error Analysis
Magnetic dipole moment ´ moment induction
Moment of inertia
10
...
14
...
Dimensional formula [M 0L0 T]
Reason The given dimension is that of frequency
...
Density
1
Reason Acceleration has the dimensions of
...
Assertion
æ ± Da ö
ç
÷-m
è a ø
æ ± Db ö
ç
÷
è b ø
ing
12
...
...
Match the Columns
4
...
Column II
Column I
kM
Column I
Pressure
Energy density
Angle
Energy
ww
w
...
Suppose force ( F ), area ( A ) and time (T ) are the fundamental
units, then the match the following columns
...
Match the following two columns
...
Match the following columns
...
Match the following columns
...
e
...
Reason The above relation is valid when Da << a and
Db << b
...
Reason Both systematic and random errors are based on the
cause of error
...
Reason Absolute error is the difference between the real
value and the measured value of a physical quantity
...
Reason Least count is the maximum measurement that can
be measured accurately by an instrument
...
Reason Sphere has minimum surface area
...
Reason The least count of screw gauge is directly
proportional to the number of divisions on circular scale
...
8
...
Reason Smaller the unit of measurement smaller is its
numerical value
...
Assertion L / R and CR both have same dimensions
...
19
(C) Velocity
Column II
1/ 2 -1
(p)
[A
(q)
Column II
(A)
GM e M s
(p)
[M2L2 T -3 ]
(B)
3RT
M
(q)
[ML3 T -2 ]
(C)
F2
(r)
[L2 T -2 ]
(s)
None
2 2
q B
(D)
GM e
Re
[FA 1/ 2 ]
(r)
Column I
[FA 1/ 2 T 2 ]
T ]
Entrance Corner
1
...
SI unit of power is
[J& K CET 2011]
[J& K CET 2011]
(a) Joule
(b) Erg
(c) Newton
(d) Watt
3
...
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...
PICKMYCOACHING
...
The dimensional formula of a is [JCECE 2010]
(b) [M -1L5 T -2 ]
(a) [ML5 T -2 ]
-1 -2
(c) [ML T ]
(d) [ML-5 T -2 ]
[CBSE AIPMT 2011]
-1
(a) [L T]
(b) [LT ]
(c) [L-1/2 T1/2 ]
(d) [L1/2 T -1/2 ]
5
...
9
...
19
...
21
...
What
are the dimensions of a, b and c respectively?
13
...
15
...
pic
12
...
18
...
17
...
(a) coefficient of viscosity
(b) impulse
(c) momentum
(d) spring constant
(e) frequency
The dimension of impulse is
[J&K CET 2011]
(b) [ML2 T -1]
(a) [MLT -1]
(c) [ML-1T -1]
(d) [MT -1]
If C be the capacitance and V be the electric potential, then
the dimensional formula of CV 2 is
[KCET 2011]
(b) [MLT -2 A -1]
(a) [ML2 T -2 A 0 ]
(d) [ML-3 TA]
(c) [M 0LT -2 A 0 ]
What is the dimension of surface tension? [WB JEE 2011]
(a) [MLT 0 ]
(b) [MLT -1]
0 -2
(c) [ML T ]
(d) [ML0 T -2 ]
[Guj
...
The correct relation between its
fundamental frequency f , the length L and the diameter D
is
[KCET 2010]
1
1
(b) f µ
(a) f µ
LD
L D
1
1
(c) f µ 2
(d) f µ
D
LD 2
The dimensions of resistance are same as those of
...
[KCET 2010]
h2
h2
(b)
(a) 2
e
e
h
h
(c) 2
(d)
e
e
16
...
The
[AFMC 2010]
dimensional formula of b will be
(a) [M 0L2 T 0 ]
(b) [ML2 T]
(c) [ML0 T -1]
(d) [M 0L2 T -1]
The dimension of electromotive force in terms of current A
is
[BVP 2010]
(b) [ML2 T -2 A -2 ]
(a) [ML-2 A -2 ]
(c) [ML2 T -2 A -2 ]
(d) [ML2 T -3 A -1]
1
is
The dimensional formula of
[Guj
...
co
4
...
If E , M , L and G denote energy, mass, angular momentum
and gravitation constant respectively, then the quantity
( E 2 L2 / M 5G 2 ) has the dimensions of
[AMU 2010]
(a) angle
(b) length
(c) mass
(d) None of these
23
...
If the radius of the capillary tube is increased
by 10%, then the rate of flow of liquid will change nearly by
[WB JEE 2010]
(a) + 10%
(b) + 46%
(c) - 10%
(d) - 40%
24
...
If increase in linear momentum of a body is 50%, then
change in its kinetic energy is
[Manipal 2010]
(a) 25%
(b) 125%
(c) 150%
(d) 50%
26
...
The pressure must be increased by
[BVP 2010]
(a) 4%
(b) 4
...
86%
(c) 8%
27
...
[AMU 2010]
(a) Every measurement by any measuring instrument has
some errors
(b)
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Chapter 1 · Units, Dimensions and Error Analysis
Torque
Stress
Modulus of elasticity
Power
Surface tension
Newton metre
Newton metre -2
Newton metre -2
Newton metre/second-1
Newton metre -2
29
...
An object is moving through the liquid
...
Then
dimensional formula of constant of proportionality is
[UP CPMT, Punjab PMET 2009]
31
...
2%
(c) 12
...
1%
32
...
The error in the estimation of
kinetic energy obtained by measuring mass and speed will
be
[AIIMS 2009]
(a) 12%
(b) 10%
(c) 2%
(d) 8%
33
...
(d)
11
...
(d)
12
...
pic
Objective Problems (Level 2)
4
...
24
...
44
...
64
...
84
...
(b)
13
...
15
...
35
...
55
...
75
...
(c)
(d)
(b)
(b)
(c)
(b)
(a)
(a)
(d)
6
...
26
...
46
...
66
...
86
...
13
...
33
...
53
...
73
...
oa
(b)
(d)
(c)
(d)
(a)
(b)
(a)
(c)
(a)
yC
2
...
22
...
42
...
62
...
82
...
11
...
31
...
51
...
71
...
(b) [MLT -1]
(d) [ML0 T -1]
m
(a)
(b)
(c)
(d)
(e)
(a) [ML-1T -1]
(c) [M 0LT -1]
...
Which one of the following quantities has not been
expressed in proper units?
[Kerala CEE 2009]
21
7
...
27
...
47
...
67
...
87
...
18
...
38
...
58
...
78
...
(b)
(a)
(c)
(c)
(b)
(b)
(b)
(c)
(d)
9
...
29
...
49
...
69
...
89
...
20
...
40
...
60
...
80
...
(c)
(b)
(d)
(a)
(a)
(c)
(a)
(c)
(c)
4
...
(a)
5
...
(c)
6
...
(a)
8
...
(c)
10
...
(c)
14
...
(a)
15
...
(c)
16
...
(b)
17
...
(c)
9
...
(d)
8
...
(d)
28
...
(b)
19
...
(d)
10
...
(a)
30
...
(a)
11
...
(a)
12
...
(a)
13
...
(A ® q, B ® p, C ® r, D ® s)
2
...
(A ® q, B ® r, C ® p)
4
...
(A ® q, B ® r, C ® r, D ® s)
Entrance Corner
1
...
21
...
(c)
(a)
(a)
(d)
2
...
22
...
(d)
(c)
(d)
(d)
3
...
23
...
(c)
(a)
(b)
(d)
4
...
(a)
24
...
(d)
15
...
(b)
6
...
(a)
26
...
(a)
17
...
(d)
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Objective Physics Volume 1
Solutions
Objective Problems (Level 1)
éF ù
éF ù
and [ b ] = 2
45
...
Leap year, year and shake are the units of time
...
1 light year = ( 3 ´ 105 ) ( 365) ( 24) ( 3600)
47
...
Solid angle, strain and dielectric constant are dimensionless
constant
...
Since ( mvr ) = n ×
2p
and
E = hn
So, unit of h = joule second = angular momentum
17
...
21
...
2
48
...
Velocity gradient is change in velocity per unit length
...
416 ´ 1012 km
8
...
50
...
26
...
év ù
51
...
Relative density =
54
...
m µ v r g
...
Since p x Q y c z is dimensionless
...
pic
[ML-1T -2 ] x [MT -3 ]y [LT -1] z = [M 0L0 T 0 ]
Only option (b) satisfies this expression
So
x = 1, y = - 1, z = 1
38
...
Since, R = , where r is specific resistance
...
i = i 0 {1 - e -t /( L / R )}
L
Where is time constant and its dimension is same as for
R
time
...
CR is time constant
...
wk is dimensionless
...
M = NIA
57
...
Momentum is doubled
...
From definition of time constant t = RC , where R is
resistance and C is capacitance
...
co
24
...
68
...
16 mm
Hence,
A = p R2
22
´ ( 0
...
080384
Since radius has two significant figure so answer also will
have two significant figures
...
080
73
...
=
75
...
2 cm
V =
4 3
pR
3
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Chapter 1 · Units, Dimensions and Error Analysis
DV
æ DR ö
=3 ç
÷
V
è R ø
23
1
L2
GM
: K = Iw2 =
2
2
2I
R
Further, L will remain constant
...
g =
æ DV ö
æ 0
...
2 ø
= 11%
78
...
H = i R t
\ % error in H = 2 (% error in i)
+ (% error in R) + (% error in t)
F
F
82
...
f =
2p LC
yC
1
T
=
2pf 2p
kM
Thus, LC has the dimensions of time
...
Volume of cylinder
ww
w
...
l
ct
Hence, is also dimensionless and unit of ct is same as that
l
of x
...
Also unit of y is same
as that of A, which is also that unit of x
...
m is mass per unit length
...
Here,
ch
= ( 4%) + 2 ( 2%)
LC =
1
mv 2
2
[E]
[ m] = 2 = [Ev -2 ]
[v ]
12
...
6 units
...
[FL-1T2 ] = [ MLT -2 ] [L-1] [ T2 ] = [ M]
11
...
co
K1 =
ing
\
æ DV ö
æ DD ö
æ DL ö
ç
÷ ´ 100 = 2 ç
÷ ´ 100 + ç ÷ ´ 100
è V ø
è D ø
è L ø
Match the Columns
3
...
01ö
æ 0
...
0 ø
è 0
...
5%
4
...
So error will reduce by 1/4 factor
...
Dimension of (ohm) R
h
= 2 ; (e = charge = current ´ time)
e
[ Et ]
=
[ it ]2
=
-2
-1
2
P
æE ö
= ( R ) as P = ç ÷
i2
èt ø
2
-2
-1
8
...
The SI unit of activity of a radioactive sample is Becqueral
...
The SI unit of power is watt
...
The SI unit of thermal conductivity is Wm-1 K -1
...
This expression for speed of light and the dimensions of
speed of light are [LT -1]
...
Surface tension has the same dimension as that of spring
constant
...
The dimension of impulse = [MLT -1]
...
We know,
Energy, E = CV 2
Dimensions of CV 2 = Dimensions of energy, E
= [ ML2 T -2 ]
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24
Objective Physics Volume 1
F
l
Dimensions of F
Dimension of L
[MLT2 ]
T =
[L]
= [ML0 T -2 ]
Dimensions of T =
9
...
Its SI unit is
A ´ m2 or N-m/T or JT -1
...
Unit of electrical conductivity is mho / m or siemem / m
...
Taking T = 2p
R3
GM
[ML-1T -2 ] [L6] = [ML5 T -2 ]
aZ
should be dimensionless
17
...
co
2p
m
Substituting the dimensions,
LHS, T = [ T]
RHS,
= [T]2 = [ T]
R3
GM
ing
LHS = RHS for T = 2p
Thus,
2 -1
é h ù [ML T ]
=
= [ML2 T -3 A -2 ]
2ú
êe û
[AT]2
ë
16
...
Electromotive force = potential difference
W [ML2 T -2 ]
V =
=
Þ
q
[AT]
Þ
12
...
The capacitance C of a conductor is defined as the ratio of
charge q given to raise the potential V of the conductor
...
e
...
We know, T =
coulomb2
joule
=
(ampera - sec)2
ampere2 - sec
=
newton - metre (kg - m sec -2 ) ´ metre
=
ampera2 - sec 4
kg - metre2
ww
w
...
1 T
14
...
Resistance, R = =
=
i qi
[A2 T]
Þ
Þ
f =
R = [ML2 T -3 A -2 ]
T
pr
= [ML2 T2 A -1]
19
...
Given p =
RT
e - aV / RT
V -b
aV
is dimensionless
...
21
...
PICKMYCOACHING
...
PICKMYCOACHING
...
The dimension of E = [ ML2 T -2 ]
1-
Þ
Dimensions of M = [ M]
Þ
p1 V2 24
=
=
p2 V1 25
Þ
Dimensions of L = [ ML2 T -1]
p2 - p1 25
1
=
-1=
p1
24
24
Dimensions of G = [ M -1L3 T -2 ]
\Dimensions of
é E2L2 ù [ML2 T -2 ]2 [ML2 T -1]2
= [ML2 T -2 ]
ê 5 2ú =
MG û
[M]5 [M -1L3 T -2 ]2
ë
100
= 4
...
When the relative error is expressed in percentage, we cal it
percentage error
...
Volume of liquid coming out of the tube per second
Þ
Þ
V =
ppr 4
8hl
V2 é r2 ù
=
V1 ê r1 ú
ë û
V2
1
V
24
=
Þ 2 =
V1 25
V1 25
28
...
1
1
29
...
co
30
...
When T is constant, pV = constant
...
Thus percentage increase in pressure
( 100 - 90) ´ 100
=
= 11
...
Kinetic energy K = mv 2
2
Fractional error in kinetic energy
DK Dm 2 Dv
=
+
K
m
v
Percentage error in kinetic energy is
Dm
2 Dv
=
´ 100 +
´ 100
m
v
Dm
2Dv
As we know,
´ 100 = 2% and
´ 100 = 3%
m
v
So, percentage error in kinetic energy
= 2 + 2 ´ 3 = 2 + 6 = 8%
DT 1 Dl
l
or
33
...
1) = 1
...
4641 - V
=
=
= 46%
Þ
V
V
V
p2
24
...
2 p
...
=
= 44%
( p 2 / 2m )
25
...
pic
Now we have p1 = p , p2 = p1 + 50% of p1 = 15p1
...
25 K
K 2 p2
p1
So change in KE = 2
...
26
...
02 ´ T = 0
...
01 s
2è l ø
2
For a day, DT = 24 ´ 60 ´ 60 ´ 0
...
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