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Chapter 1 - Nature of Physical World and Measurement - 11th Science Guide Samacheer Kalvi Solutions - Pdf Download [2024-2025]

Chapter 1 - Nature of Physical World and Measurement - 11th Science Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Question 5.
The measurement value of length of a simple pendulum is $20 \mathrm{~cm}$ known with $2 \mathrm{~mm}$ accuracy.

The time for 50 oscillations was measured to be $40 \mathrm{~s}$ within $1 \mathrm{~s}$ resolution. Calculate the percentage of accuracy in the determination of acceleration due to gravity ' $\mathrm{g}$ ' from the above measurement.
Answer:
Given,
Length of simple pendulum (1) $=20 \mathrm{~cm}$
absolute error in length $(\Delta \mathrm{l})=2 \mathrm{~mm}=0.2 \mathrm{~cm}$
Time taken for 50 oscillation $(t)=40 \mathrm{~s}$
error in time $\Delta \mathrm{T}=1 \mathrm{~s}$
Solution: Time period for one oscillation $(T)$
$\begin{array}{rlrl} & =\frac{t}{n}=\frac{40}{50} s \\ \therefore & \Delta \mathrm{T} & =\frac{\Delta t}{n} \\ \text { So, } & \frac{\Delta \mathrm{T}}{\mathrm{T}} & =\frac{\Delta t / n}{t / n}=\frac{\Delta t}{t}\end{array}$
$\Delta l, \Delta \mathrm{T}$ are least count errors
$
\frac{\Delta g}{g}=\frac{\Delta l}{l}+2 \frac{\Delta \mathrm{T}}{\mathrm{T}}=\frac{0.2}{20}+2\left(\frac{1}{40}\right)=\frac{1.2}{20}=0.06
$
Hence, the percentage error in $\mathrm{g}$ is $\left(\frac{\Delta g}{g}\right) \times 100=\left(\frac{\Delta l}{l}+2 \frac{\Delta \mathrm{T}}{\mathrm{T}}\right) \times 100 \%=0.06 \times 100 \%=6 \%$

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Question 5.
The measurement value of length of a simple pendulum is $20 \mathrm{~cm}$ known with $2 \mathrm{~mm}$ accuracy.

The time for 50 oscillations was measured to be $40 \mathrm{~s}$ within $1 \mathrm{~s}$ resolution. Calculate the percentage of accuracy in the determination of acceleration due to gravity ' $\mathrm{g}$ ' from the above measurement.
Answer:
Given,
Length of simple pendulum (1) $=20 \mathrm{~cm}$
absolute error in length $(\Delta \mathrm{l})=2 \mathrm{~mm}=0.2 \mathrm{~cm}$
Time taken for 50 oscillation $(t)=40 \mathrm{~s}$
error in time $\Delta \mathrm{T}=1 \mathrm{~s}$
Solution: Time period for one oscillation $(T)$
$\begin{array}{rlrl} & =\frac{t}{n}=\frac{40}{50} s \\ \therefore & \Delta \mathrm{T} & =\frac{\Delta t}{n} \\ \text { So, } & \frac{\Delta \mathrm{T}}{\mathrm{T}} & =\frac{\Delta t / n}{t / n}=\frac{\Delta t}{t}\end{array}$
$\Delta l, \Delta \mathrm{T}$ are least count errors
$
\frac{\Delta g}{g}=\frac{\Delta l}{l}+2 \frac{\Delta \mathrm{T}}{\mathrm{T}}=\frac{0.2}{20}+2\left(\frac{1}{40}\right)=\frac{1.2}{20}=0.06
$
Hence, the percentage error in $\mathrm{g}$ is $\left(\frac{\Delta g}{g}\right) \times 100=\left(\frac{\Delta l}{l}+2 \frac{\Delta \mathrm{T}}{\mathrm{T}}\right) \times 100 \%=0.06 \times 100 \%=6 \%$

Conceptual Questions - Chapter 1 - Nature of Physical World and Measurement - 11th Science Guide Samacheer Kalvi Solutions

Conceptual Questions
Question 1.

Why is it convenient to express the distance of stars in terms of light year (or) parsec rather than in $\mathrm{km}$ ?
Answer:
A parsec is 206, $265 \mathrm{AU}$ and is roughly the distance to the nearest stars. If we were to view a giant star with a diameter of $1 \mathrm{AU}$ at a distance of one parsec, it would appear to be just $1 / 3600^{\text {th }}$ of a degree in angular size. For comparison, the sun and moon are both half a degree in angular size when viewed from Earth.

Question 2.
Show that a screw gauge of pitch $1 \mathrm{~mm}$ and 100 divisions is more precise than a vernier caliper with 20 divisions on the sliding scale.
Answer:
Least count of screw gauge $=\frac{\text { Pitch }}{\text { No. of divisions }}=\frac{1 \mathrm{~mm}}{100}=0.01 \mathrm{~mm}$
Least count of vernier caliper $=\frac{1}{20} \mathrm{~mm}=0.05 \mathrm{~mm}$
As shown, the least count of screw gauge is lesser then vernier caliper, hence screw gauge is more precise.
Question 4.
Having all units in atomic standards is more useful. Explain.
Answer:
An atomic mass unit (symbolized AMU or amu) is defined as precisely $1 / 12$ the mass of an atom of carbon-12. The carbon-12 (C-12) atom has six protons and six neutrons in its nucleus. In imprecise terms, one AMU is the average of the proton rest mass and the neutron rest mass. This is approximately $1.67377 \times 10^{-27}$ kilogram (kg), or $1.67377 \times 10^{-24}$ gram (g). The mass of an atom in AMU is roughly equal to the sum of the number of protons and neutrons in the nucleus.
The AMU is used to express the relative masses of, and thereby differentiate between, various isotopes of elements. Thus, for example, uranium-235 (U-235) has an AMU of approximately 235 , while uranium-238 (U-238) is slightly more massive. The difference results from the fact that U-238, the most abundant naturally occurring isotope of uranium, has three more neutrons than U-235, an isotope that has been used in nuclear reactors and atomic bombs.

Question 5 .
Why dimensional methods are applicable only up to three quantities?
Answer:
Understanding dimensions is of utmost importance as it helps us in studying the nature of physical quantities mathematically. The basic concept of dimensions is that we can add or subtract only those quantities which have same dimensions. Also, two physical quantities are equal if they have same dimensions. these basic ideas help us in deriving the new relation between physical quantities, it is just like units.

Additional Questions - Chapter 1 - Nature of Physical World and Measurement - 11th Science Guide Samacheer Kalvi Solutions

Additional Questions Solved
MultipleChoose Questions
Question 1.

The unit of surface tension ......
(a) $\mathrm{MT}^{-2}$
(b) $\mathrm{Nm}^{-2}$
(c) $\mathrm{Nm}$
(d) $\mathrm{Nm}^{-1}$
Answer:
(d) $\mathrm{Nm}^{-1}$
Question 2.
One atomus equal to ......
(a) $100 \mathrm{~ms}$
(b) $\frac{1}{6.25} \mathrm{~ms}$
(c) $160 \mathrm{~ms}$
(d) $160 \mathrm{~ms}$
Answer:
(c) $160 \mathrm{~ms}$
Question 3.
One light year is ......
(a) $3.153 \times 10^7 \mathrm{~m}$
(b) $1.496 \times 10^7 \mathrm{~m}$
(c) $9.46 \times 10^{12} \mathrm{~km}$
(d) $3.26 \times 10^{15} \mathrm{~km}$
Answer:
(c) $9.46 \times 10^{12} \mathrm{~km}$

Question 4.
One Astronomical unit is
(a) $3.153 \times 10^7 \mathrm{~m}$
(b) $1.496 \times 10^7 \mathrm{~m}$
(c) $9.46 \times 10^{12} \mathrm{~m}$
(d) $3.26 \times 10^{15} \mathrm{~m}$
Answer:
(b) $1.496 \times 10^7 \mathrm{~m}$
Question 5 .
One parsec is .....
(a) $3.153 \times 10^7 \mathrm{~m}$
(b) $3.26 \times 10^{15} \mathrm{~m}$
(c) $30.84 \times 10^{15} \mathrm{~m}$
(d) $9.46 \times 10^{15} \mathrm{~m}$
Answer:
(c) $30.84 \times 10^{15} \mathrm{~m}$
Question 6.
One Fermi is .....

(a) $10^{-9} \mathrm{~m}$
(b) $10^{-10} \mathrm{~m}$
(c) $10^{-12} \mathrm{~m}$
(d) $10^{-15} \mathrm{~m}$
Answer:
(d) $10^{-15} \mathrm{~m}$
Question 7.
One Angstrom is
(a) $10^{-9} \mathrm{~m}$
(b) $10^{-10} \mathrm{~m}$
(c) $10^{-12} \mathrm{~m}$
(d) $10^{-15} \mathrm{~m}$
Answer:
(b) $10^{-10} \mathrm{~m}$
Question 8 .
One solar mass is ....
(a) $2 \times 10^{30} \mathrm{~kg}$
(b) $2 \times 10^{30} \mathrm{~g}$
(c) $2 \times 10^{30} \mathrm{mg}$
(d) $2 \times 10^{30}$ tonne
Answer:
(d) $2 \times 10^{30}$ tonne
Question 9.
$\frac{1}{12}$ of the mass of carbon 12 atom is .....
(a) $1 \mathrm{TMC}$
(b) mass of neutron
(c) 1 amu
(d) mass of hydrogen
Answer:
(d) mass of hydrogen
Question 10.
The word physics is derived from the word
(a) scientist
(b) fusis
(c) fission
(d) fusion

Answer:
(b) fusis
Question 11.
The study of forces acting on bodies whether at rest or in motion is .....
(a) classical mechanics
(b) quantum mechanics
(c) thermodynamics
(d) condensed matter physics
Answer:
(a) classical mechanics
Question 12.
Mass of observable universe .....
(a) $10^{31} \mathrm{~kg}$
(b) $10^{41} \mathrm{~kg}$
(c) $10^{55} \mathrm{~kg}$
(d) $9.11 \times 10^{31} \mathrm{~kg}$
Answer:
(c) $10^{55} \mathrm{~kg}$
Question 13.
Mass of an electron
(a) $10^{-31} \mathrm{~kg}$
(b) $9.11 \times 10^{-31} \mathrm{~kg}$
(c) $1.6 \times 10^{-31} \mathrm{~kg}$
(d) $1.6 \times 10^{--}$
Answer:
(b) $9.11 \times 10^{-31} \mathrm{~kg}$
Question 14.
The study of production and propagation of sound waves .....
(a) Astrophysics
(b) Acoustics

(c) Relativity
(d) Atomic physics
Answer:
(b) Acoustics
Question 15.
The study of the discrete nature of phenomena at the atomic and subatomic levels.
(a) Quantum mechanics
(b) High energy physics
(c) Acoustics
(d) Classical mechanics
Answer:
(a) Quantum mechanics
Question 16.
The techniQuestion used to study the crystal structure of various rocks are
(a) diffraction
(b) interference
(c) total internal reflection
(d) refraction
Answer:
(a) diffraction
Question 17.
The astronomers used to observe distant points of the universe by .......
(a) Electron telescope
(b) Astronomical telescope
(c) Radio telescope
(d) Radar
Answer:
(c) Radio telescope

Question 18 .
The comparison of any physical quantity with its standard unit is known as
(a) fundamental quantities
(b) measurement
(c) dualism
(d) derived quantities
Answer:
(b) measurement
Question 19.
Fundamental quantities can also be known as ...... quantities.
(a) original
(b) physical
(c) negative
(d) base
Answer:
(d) base
Question 20.
Which one of the following is not a fundamental quantity?
(a) length
(b) luminous intensity
(c) temperature
(d) water current
Answer:
(d) water current
Question 21.
The system of unit not only based on length, mass, and time is
(a) FPS

(b) CGS
(c) MKS
(d) $\mathrm{SI}$
Answer:
(d) $\mathrm{SI}$
Question 22.
The coherent system of units .....
(a) CGS
(b) SI
(c) FPS
(d) MKS
Answer
(b) $\mathrm{SI}$
Question 23.
The triple point temperature of water is ......
(a) $-273.16 \mathrm{~K}$
(b) $0 \mathrm{~K}$
(c) $273.16 \mathrm{~K}$
(d) $100 \mathrm{~K}$
Answer:
(d) $100 \mathrm{~K}$
Question 24.
Which of the following is a unit of distance?
(a) Light year
(b) Leap year
(c) Dyne-sec
(d) Pauli
Answer:
(a) Light year
Question 25.
The unit of moment of force
(a) $\mathrm{Nm}^2$
(b) $\mathrm{Nm}$
(c) $\mathrm{N}$
(d) NJ rad
Answer:
(b) $\mathrm{Nm}$

Question 26.
1 radian is
(a) $2.91 \times 10^{-4} \mathrm{~m}$
(b) $57.27^{\circ}$
(c) $180^{\circ}$
(d) $\frac{\pi}{180}$
Answer:
(b) $57.27^{\circ}$

Question 27.
One degree of arc is
(a) $1^{\prime \prime}$
(b) $60^{\prime \prime}$
(c) $60^{\prime}$
(d) $60^{\circ}$
Answer:
(c) $60^{\prime}$
Question 28.
One degree of arc is equal to ........
(a) $1.457 \times 10^2 \mathrm{rad}$
(b) $1.457 \times 10^{-2} \mathrm{rad}$
(c) $1.745 \times 10^2 \mathrm{rad}$
(d) $1.745 \times 10^{-2} \mathrm{rad}$
Answer:
(b) $1.457 \times 10^{-2} \mathrm{rad}$
Question 29.
1 minute of arc is equal to ........
(a) $1.745 \times 10^{-2} \mathrm{rad}$
(b) $2.91 \times 10^{-4} \mathrm{rad}$
(c) $2.91 \times 10^4 \mathrm{rad}$
(d) $4.85 \times 10^{-6} \mathrm{rad}$
Answer:
(b) $2.91 \times 10^{-4} \mathrm{rad}$
Question 30 .
1 second of arc is equal to .........
(a) $\frac{1^{\circ}}{3600}$
(b) $4.85 \times 10^6 \mathrm{rad}$
(c) $\frac{1}{4.85} \times 10^{-6} \mathrm{rad}$
(d) $2.91 \div 10^{-4} \mathrm{rad}$

Answer:
(a) $\frac{1}{3600}$
Question 31.
1 second of arc is equal to ....
(a) $0.00027^{\circ}$
(b) $1.745 \times 10^{-2} \mathrm{rad}$
(c) $2.91 \times 10^{-4} \mathrm{rad}$
(d) $4.85 \times 10^{-6} \mathrm{rad}$
Answer:
(a) $0.00027^{\circ}$
Question 32.
Unit of impulse ....
(a) $\mathrm{NS}^2$
(b) NS
(c) $\mathrm{Nm}$
(d) $\mathrm{Kgms}^{-2}$
Answer:
(b) NS
Question 33.
The ratio of energy and temperature is known as ......
(a) Stefen's constant
(b) Boltzmann constant
(c) Plank's constant
(d) Kinetic constant
Answer:
(b) Boltzmann constant
Question 34.
The range of distance can be measured by using direct methods is .....
(a) $10^{-2}$ to $10^{-5} \mathrm{~m}$
(b) $10^{-2}$ to $10^2 \mathrm{~m}$
(c) $10^2$ to $1\left(\mathrm{~T} 5 \mathrm{~m}\right.$ (d) $10^{\prime \prime} 2$ to $105 \mathrm{~m}$
Answer:
(b) $10^{-2}$ to $10^2 \mathrm{~m}$

Question 35 .
Which of the following is in increased order?
(a) exa, tera, hecto
(b) tera, exa, hecto
(c) giga, tera, exa
(d) hecto, exa, giga
Answer:
(c) giga, tera, exa
Question 36.
$10^{-18}$ is called as .....
(a) nano
(b) pico
(c) femto
(d) atto
Answer:
(d) atto
Question 37.
A radio signal sent towards the distant planet, returns after " $\mathrm{t}$ " $\mathrm{s}$. If "c" is the speed of radio waves then the distance of the planet and from the earth is
(a) $c \frac{\boldsymbol{t}}{2}$
(b) $c t^2$
(c) $2 c t$
(d) $c^2 \frac{t^2}{2}$
Answer:
(a) $c \frac{t}{2}$

Question 38 .
Find odd one out ....
(a) Newton
(b) metre
(c) candela
(d) Kelvin
Answer:
(a) Newton
Question 39.
The shift in the position of an object when viewed with two eyes, keeping one eye closed at a time is known as ...
(a) basis
(b) fundamental
(c) parallax
(d) pendulum
Answer:
(c) parallax

Question 40.
Chandrasekar limit is ..... times the mass of the sun.
(a) 1.2
(b) 1.4
(c) 1.6
(d) 1.8
Answer:
(b) 1.4
Question 41.
The smallest physical unit of time is
(a) second
(b) minute
(c) microsecond
(d) shake
Answer:
(d) shake
Question 42 .
Size of atomic nucleus is .....
(a) $10^{-10} \mathrm{~m}$
(b) $10^{-12} \mathrm{~m}$
(c) $10^{-14} \mathrm{~m}$
(d) $10^{-18} \mathrm{~m}$
Answer:
(c) $10^{-14} \mathrm{~m}$
Question 43.
Time interval between two successive heart beat is in the order of
(a) $10^{\circ} \mathrm{s}$
(b) $10 \mathrm{~s}$
(c) $10^2 \mathrm{~s}$
(d) $10^{-3} \mathrm{~s}$
Answer:
(a) $10^{\circ} \mathrm{s}$
Question 44.
Half life time of a free neutron is in the order of
(a) $10^{\circ}$
(b) $10^1 \mathrm{~s}$
(c) $10^2 \mathrm{~s}$

(d) $10^3 \mathrm{~s}$
Answer:
(d) $10^3 \mathrm{~s}$
Question 45.
The uncertainty contained in any measurement is
(a) rounding off
(b) error
(c) parallax
(d) gross
Answer:
(b) error
Question 46.
Zero error of an instrument is a ......
(a) Systematic error
(b) Random error
(c) Gross error
(d) Both (a) and (b)
Answer:
(a) Systematic error
Question 47.
Error in the measurement of radius of a sphere is $2 \%$. Then error in the measurement of surface
area is ....
(a) $1 \%$
(b) $2 \%$
(c) $3 \%$
(d) $4 \%$
Answer:
(d) $4 \%$

Question 48 .
Imperfections in experimental procedure gives ..... error.
(a) random
(b) gross
(c) systematic
(d) personal
Answer:
(c) Systematic
Question 49.
Random error can also be called as ....
(a) personal error
(b) chance error
(c) gross error
(d) system error
Answer:
(b) chance error

Question 50 .
To get the best possible true value of the quantity has to be taken.
(a) rms value
(b) net value
(c) arithmetic mean
(d) mode
Answer:
(c) arithmetic mean
Question 51.
The error caused due to the shear carelessness of an observer is called as ...... error.
(a) Systematise
(b) Gross
(c) Random
(d) Personal
Answer:
(b) Gross
Question 52 .
The uncertainty in a measurement is called as ....
(a) error
(b) systematic
(c) random error
(d) gross error
Answer:
(a) error
Question 53.
The difference between the true value and the measured value of a quantity is known as .....
(a) Absolute error
(b) Relative error
(c) Percentage error
(d) Systemmatic error
Answer:
(a) Absolute error
Question 54.
If $a_1, a_2, a_3 \ldots a_n$ are the measured value of a physical quantity "a" and $a_m$ is the true value then absolute error .....
(a) $a_m=\Delta a_n+a_n$
(b) $\Delta a_n=a_m+a_n$
(c) $\Delta a_n=a_m-a_m$
(d) $\Delta a_n=a_m$

Answer:
(d) $\Delta a_n=a_m-a_n$
Question 55.
If ' $\mathrm{a}_{\mathrm{m}}$ ' and ' $\Delta \mathrm{a}_{\mathrm{m}}$ ' are true value and mean absolute error respectively, then the magnitude of the quantity may lie between
(a) $a_m+a_n$ to $a_m-a_n$
(b) $a_m-\Delta a_m$ to $a_m+\Delta a_m$
(c) $2 a_m$ to $\Delta a_m$
(d) 0 to $2 a_m$
Answer:
(b) $a_m-\Delta a_m$ to $a_m+\Delta a_m$
Question 56.
The ratio of the mean absolute error to the mean value is called as ......
(a) absolute error
(b) random error
(c) relative error
(d) percentage error
Answer:
(c) Relative error
Question 57.
Relative error can also be called as .......
(a) fractional error
(b) absolute error
(c) percentage error
(d) systematic error
Answer:
(a) fractional error

Question 58.
A measured value to be close to targeted value, percentage error must be close to
(a) 0
(b) 10
(c) 100
(d) $\propto$
Answer:
(a) 0
Question 59.
The maximum possible error in the sum of two quantities is equal to
(a) $\mathrm{Z}=\mathrm{A}+\mathrm{B}$
(b) $\Delta \mathrm{Z}=\Delta \mathrm{A}+\Delta \mathrm{B}$
(c) $\Delta \mathrm{Z}=\Delta \mathrm{A} / \Delta \mathrm{B}$
(d) $\Delta \mathrm{Z}=\Delta \mathrm{A}-\Delta \mathrm{B}$
Answer:
(b) $\Delta \mathrm{Z}=\Delta \mathrm{A}+\Delta \mathrm{B}$

Question 60.
The maximum possible error in the difference of two quantities is ......
(a) $\mathrm{Z}=\mathrm{A}+\mathrm{B}$
(b) $\Delta \mathrm{Z}=\Delta \mathrm{A}-\Delta \mathrm{B}$
(c) $\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=\frac{\Delta \mathrm{A}}{\mathrm{A}}+\frac{\Delta \mathrm{B}}{\mathrm{B}}$
(d) $\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=$
Answer:
(c) $\frac{\Delta \mathbf{Z}}{\mathrm{Z}}=\frac{\Delta \mathrm{A}}{\mathrm{A}}+\frac{\Delta \mathrm{B}}{\mathrm{B}}$
Question 61.
The maximum fractional error in the division of two quantities is ....
(a) $\mathrm{Z}=\mathrm{A}+\mathrm{B}$
(b) $\Delta \mathrm{Z}=\Delta \mathrm{A}-\Delta \mathrm{B}$
(c) $\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=\frac{\Delta \mathrm{A}}{\mathrm{A}}+\frac{\Delta \mathrm{B}}{\mathrm{B}}$
(d) $\frac{\Delta}{2}$
Answer:
(c) $\frac{\Delta \mathbf{Z}}{\mathbf{Z}}=\frac{\Delta \mathbf{A}}{\mathbf{A}}+\frac{\Delta \mathbf{B}}{\mathbf{B}}$
Question 62.
The fractional error in the $\mathrm{n}^{\text {th }}$ power of a quantity is .....
(a) $\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=n \frac{\Delta \mathrm{A}}{\mathrm{A}}$
(b) $\frac{Z}{\Delta Z}=n \frac{\mathrm{A}}{\Delta \mathrm{A}}$
(c) $\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=\frac{1}{n} \frac{\Delta \mathrm{A}}{\mathrm{A}}$
(d) $\frac{\mathrm{Z}}{\Delta \mathrm{Z}}=\frac{1}{n}$.
Answer:
(a) $\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=n \frac{\Delta \mathrm{A}}{\mathbf{A}}$
Question 63.
A physical quantity is given as $\mathrm{y}=\frac{a b^3}{c^2}$. If $\Delta \mathrm{a}, \Delta \mathrm{b}, \Delta \mathrm{c}$ are absolute errors, the possible fractional error in $\mathrm{y}$ is .....
(a) $\frac{\Delta y}{y}=\frac{\Delta a \Delta b}{2 \Delta c}$
(b) $\frac{\Delta y}{y}=\frac{\Delta a}{a}+3 \frac{\Delta b}{b}+2 \frac{\Delta c}{c}$

(c) $\frac{\Delta y}{y}=\frac{\Delta a}{a}+\left(\frac{\Delta b}{a}\right)^3+\left(\frac{\Delta c}{c}\right)^2$
(d) $\frac{\Delta y}{y}=\frac{\Delta a}{a}+\frac{\Delta b}{a}+\frac{\Delta c}{c}$
Answer:
(b) $\frac{\Delta y}{y}=\frac{\Delta a}{a}+3 \frac{\Delta b}{b}+2 \frac{\Delta c}{c}$
Question 64.
Number of significant digits in $3256 \ldots$
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4
Question 65.
Number of significant digits in 32005 ......
(a) 1
(b) 2
(c) 5
(d) 2
Answer:
(c) 5

Question 66 .
Number of significant digits in $2030 \ldots$
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4
Question 67.
Number of significant digits in $2030 \mathrm{~N}$.....
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4
Question 68.
Number of significant digits in $0.0342 \ldots \ldots$
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(c) 3

Question 69 .
Number of significant digit in $20.00 \ldots \ldots$
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4
Question 70.
Number of significant digit in 0.030400
(a) 6
(b) 5
(c) 4
(d) 3
Answer:
(b) 5
Question 71.
The force acting on a body is measured as $4.25 \mathrm{~N}$. Round it off with two significant figure ..
(a) 4.3
(b) 4.2
(c) both
(a) or (b)
(d) 4.25
Answer:
(b) 4.2
Question 72.
The quantities a, b, c are measured as $3.21,4.253,7.2346$. The $\operatorname{sum}(\mathrm{a}+\mathrm{b}+\mathrm{c})$ with proper significant digits is .....
(a) 14.6976
(b) 14.697
(c) 14.69
(d) 14.6
Answer:
(c) 14.69
Question 73.
The dimensions of gravitational constant $\mathrm{G}$ are ...
(a) $\mathrm{ML}^2 \mathrm{~T}^{-2}$
(b) $\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}$
(c) $\mathrm{M}^3 \mathrm{~L}^{-2} \mathrm{~T}^{-1}$
(d) $\mathrm{ML}^2 \mathrm{~T}^{-2}$

Answer:
(b) $\mathbf{M}^{-1} \mathbf{L}^3 \mathbf{T}^{-2}$
Question 74.
The ratio of one nanometer to one micron is
(a) $10^{-3}$
(b) $10^3$
(c) $10^{-9}$
(d) $10^{-6}$
Answer:
(b) $10^3$
Question 75.
Which of the following pairs does not have same dimension?
(a) Moment of inertia and moment of force
(b) Work and torque
(c) Impulse and momentum
(d) Angular momentum and Plank's constant
Answer:
(a) Moment of inertia and moment of force
Question 76.
Two quantities A and B have different dimensions. Which of the following is physically meaningful?
(a) $\mathrm{A}+\mathrm{B}$
(b) $A-B$
(c) $\mathrm{A} / \mathrm{B}$
(d) None
Answer:
(c) $\mathrm{A} / \mathrm{B}$
Question 77.
The dimensional formula for moment of inertia ......
(a) $\mathrm{M} \mathrm{L}^0 \mathrm{~T}^{-2}$
(b) $\mathrm{M} \mathrm{L}^{-1} \mathrm{~T}^2$
(c) $\mathrm{ML}^2 \mathrm{~T}^0$
(d) $\mathrm{ML}^2 \mathrm{~T}^0$
Answer:
(d) $\mathbf{M L}^2 \mathbf{T}^0$

Question 78.
Which of the following is having same dimensional formula?
(a) Work and power
(b) Radius of gyration and displacement
(c) Impulse and force
(d) Frequencies and wavelength
Answer:
(b) Radius of gyration and displacement
Question 79.
Which of the following quantities is expressed as force per unit area?
(a) Pressure
(b) Stress
(c) Both (a) and (b)
(d) None
Answer:
(c) Both (a) and (b)
Question 80.
In equation of motion $\mathrm{S}=u t+\frac{1}{2} k t^2$
(a) $\left[\mathrm{L} \mathrm{T}^{-1}\right]$
(b) $\left[\mathrm{L} \mathrm{T}^{-2}\right]$
(c) $[\mathrm{T}]$
(d) $\left[\mathrm{L}^{-1} \mathrm{~T}\right]$
Answer:
(b) $\left[\mathbf{L} \mathbf{T}^{-2}\right]$
Question 81.
The dimensional formula for heat capacity ......
(a) $\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$
(b) $\cdot\left[\mathrm{ML}^2 \mathrm{~K}^{-1}\right]$
(c) $\left[\mathrm{ML}^2 \mathrm{~T}^2 \mathrm{~K}^{-1}\right]$
(d) $\left[\mathrm{ML}^2 \mathrm{~T}^2\right.$

Answer:
(d) $\left[\mathbf{M L}^2 \mathbf{T}^2 \mathbf{K}^{-1}\right]$
Question 82.
The product of Avogadro constant and elementary charge is known as constant.
(a) Planck's
(b) Avagadro
(c) Boltzmann
(d) Faraday
Answer:
(d) Faraday
Question 83 .
The force $F$ is given by $F=a t+b t^2$ where $t$ is time. The dimensions of ' $a$ ' and ' $b$ ' respectively are
(a) $\left[\mathrm{M} \mathrm{L}^{-3}\right]$ and $\left[\mathrm{MLT}^{-4}\right]$
(b) $\left[\mathrm{MLT}^{-4}\right]$ and $\left[\mathrm{MLT}^{-3}\right]$
(c) $\left[\mathrm{MLT}^{-1}\right]$ and $\left[\mathrm{MLT}^{-2}\right]$
(d) $\left[\mathrm{MLT}^{-2}\right]$ and $\left[\mathrm{MLT}^{-0}\right]$
Answer:
(d) $\left[\mathrm{MLT}^{-2}\right]$ and $\left[\mathrm{MLT}^{-0}\right]$

Question 84 .
Dimensions of impulse are
(a) $\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]$
(b) $\left[\mathrm{MLT}^{-2}\right]$
(c) $\left[\mathrm{MLT}^{-1}\right]$
(d) $\left[\mathrm{ML}^2 \mathrm{~T}^0\right]$
Answer:
(c) $\left[\mathrm{MLT}^{-1}\right]$
Question 85 .
If speed of light (c), acceleration due to gravity (g) and pressure (P) are taken as fundamental units, the possible relation to gravitational constant $(G)$ is ....
(a) $c^0 g \mathrm{p}^{-3}$
(b) $c^2 g^3 \mathrm{p}^{-2}$
(c) $c^0 g^2 \mathrm{p}^{-1}$
(d) $c^2 g^2 p^{-2}$
Answer:
(c) $c^0 g^2 p^{-1}$
Question 86.
Equivalent of one joule is ......
(a) $\mathrm{Nm}^2$
(b) $\mathrm{kg} \mathrm{m}^2 \mathrm{~s}^{-2}$
(c) $\mathrm{kg} \mathrm{m} \mathrm{s}^{-1}$
(d) $\mathrm{N} \mathrm{kg} \mathrm{m}^2$
Answer:
(b) $\mathrm{kg} \mathrm{m}^2 \mathrm{~s}^{-2}$
Question 87.
Pick out the dimensionless quantity .....
(a) force
(b) specific gravity
(c) planck's constant
(d) velocity
Answer:
(b) specific gravity
Question 88.
Odd one out
(a) strain
(b) refractive index
(c) numbers
(d) stress

Answer:
(d) stress
Question 89.
A wire has a mass $0.3 \pm 0.003 \mathrm{~g}$, radius $0.5 \pm 0.005 \mathrm{~mm}$ and length $6+0.06 \mathrm{~cm}$. The maximum percentage error in the measurement of its density is .......
(a) $1 \%$
(b) $2 \%$
(c) $3 \%$
(d) $4 \%$
Answer:
(d) $4 \%$
Question 90.
The dimensions of planck constant equals to that of .....
(a) energy
(b) momentum
(c) angular momentum
(d) power
Answer:
(c) angular momentum
Question 91
Given that $y=\mathrm{A} \sin \left(\frac{2 \pi}{\lambda}(c t-x)\right)$. Where $y$ and $x$ are measured in metres. Wh following statements is true?
(a) The unit of $\lambda$ is same as that of $x$ and $A$
(b) The unit of $\lambda$ is same as that of $x$ but not of $A$
(c) The unit of $\mathrm{c}$ is same as that of $2 \pi / \lambda$
(d) The unit of (ct-x)is same as that $2 \pi / \lambda$
Answer:
(a) The unit of $\lambda$ is same as that of $x$ and $A$
Question 92.
The number of significant figures in 0.06900 is .......
(a) 2
(b) 2
(c) 4
(d) 5
Answer:
(c) 4

Question 93.
The numbers 3.665 and 3.635 on rounding off to 3 significant figures will give
(a) 3.66 and 3.63
(b) 3.66 and 3.64
(c) 3.67 and 3.63
(d) 3.67 and 3.64
Answer:
(b) 3.66 and 3.64
Question 94.
Which of the following measurements is most precise?
(a) $4.00 \mathrm{~mm}$
(b) $4.00 \mathrm{~cm}$
(c) $4.00 \mathrm{~m}$
(d) $4.00 \mathrm{~km}$
Answer:
(a) $4.00 \mathrm{~mm}$
Question 95 .
The mean radius of a wire is $2 \mathrm{~mm}$. Which of the following measurements is most accurate?
(a) $1.9 \mathrm{~mm}$
(b) $2.25 \mathrm{~mm}$
(c) $2.3 \mathrm{~mm}$
(d) $1.83 \mathrm{~mm}$
Answer:
(a) $1.9 \mathrm{~mm}$
Question 96.
If error in measurement of radius of sphere is $1 \%$. What will be the error in measurement of volume?
(a) $1 \%$
(b) $\frac{1}{3} \%$

(c) $3 \%$
(d) $10 \%$
Answer:
(c) $3 \%$
Question 97.
Dimensions $\left[\mathrm{M} \mathrm{L}^{-1} \mathrm{~T}^{-1}\right]$ are related to
(a) torque
(b) work
(c) energy
(d) Coefficient of viscosity
Answer:
(d) Coefficient of viscosity
Question 98.
Heat produced by a current is obtained a relation $\mathrm{H}=\mathrm{I}^2 \mathrm{RT}$. If the errors in measuring these quantities current, resistance, time are $1 \%, 2 \%, 1 \%$ respectively then total error in calculating the energy produced is
(a) $2 \%$
(b) $4 \%$
(c) $5 \%$
(d) $6 \%$
Answer:
(c) $5 \%$

Question 99.
Length cannot be measured by ....
(a) fermi
(b) angstrom
(c) parsec
(d) debye
Answer:
(d) debye
Question 100 .
The pressure on a square plate is measured by measuring the force on the plate and the length of the sides of the plate by using the formula $\mathrm{p}=\backslash\left(\left[/\right.\right.$ latelfrac $\left.\{\backslash \operatorname{mathrm}\{\mathrm{F}\}\}\left\{1^{\wedge}\{2\}\right\} \mathrm{x}\right]$. If the maximum errors in the measurement of force and length are $4 \%$ and $2 \%$ respectively, then the maximum error in the measurement of pressure is .......
(a) $1 \%$
(b) $2 \%$
(c) $8 \%$
(d) $10 \%$
Aswer:
(c) $8 \%$
Question 101.
Which of the following cannot be verified by using dimensional analysis?
$1 \mathrm{mv}^2$
(a) $s=u t+\frac{1}{2} a t$
(b) $y=a \sin w t$
(c) $\mathrm{F}=\frac{m v^2}{r}$
(d) $\mathrm{F}=m a$
Answer:
(b) $y=a \sin w t$
Question 102 .
Percentage errors in the measurement of mass and speed are $3 \%$ and $2 \%$ respectively. The error in the calculation of kinetic energy is .......

(a) $2 \%$
(b) $3 \%$
(c) $5 \%$
(d) $7 \%$
Answer:
(d) $7 \%$
Question 103.
More number of readings will reduce
(a) random error
(b) systematic error
(c) both (a) and (b)
(d) neither (a) nor (b)
Answer:
(a) random error
Question 104 .
If the percentage error in the measurement of mass and momentum of a body are $3 \%$ and $2 \%$ respectively, then maximum possible error in kinetic energy is
(a) $2 \%$
(b) $3 \%$
(c) $5 \%$
(d) $7 \%$
Answer:
(d) $7 \%$
Question 105 .
In a vernier caliper, $\mathrm{n}$ divisions of vernier scale coincides with $(\mathrm{n}-1)$ divisions of main scale. The least count of the instrument is ........
(a) $\frac{1}{n} \mathrm{MSD}$
(b) $\frac{n}{n+1} \mathrm{MSD}$
(c) $\frac{n+1}{n} \mathrm{MSD}$
(d) $\frac{n+1}{n-1} \mathrm{MS}$
Answer:
(a) $\frac{1}{n} \mathrm{MSD}$

Question 106.
The period of a simple pendulum is recorded as $2.56 \mathrm{~s}, 2.42 \mathrm{~s}, 2.71 \mathrm{~s}$ and $2.80 \mathrm{~s}$ respectively. The average absolute error is
(a) $0.1 \mathrm{~s}$
(b) $0.2 \mathrm{~s}$
(c) $1.0 \mathrm{~s}$
(d) $0.11 \mathrm{~s}$
Answer:
(d) $0.11 \mathrm{~s}$
Question 107.
In a system of units, if force (F), acceleration (A) and time (T) are taken as fundamental units then the dimensional formula of energy is
(a) $\left[\mathrm{FA}^2 \mathrm{~T}\right]$
(b) $\left[\mathrm{FAT}^2\right]$
(c) $\left[\mathrm{F}^2 \mathrm{AT}\right]$
(d) $[\mathrm{FAT}]$
Answer:
(b) $\left[\mathrm{FAT}^2\right]$
Question 108.
The random error in the arithmetic mean of 50 observations is ' $\mathrm{a}$ ', then the random error in the arithmetic mean of 200 observations a would be
(a) $4 a$
(b) $16 a^2$
(c) $\frac{a}{4}$
(d) $\frac{a}{2}$
Answer:
(c) $\frac{a}{4}$

Question 109.
Which of the following is not dimensionless?
(a) Relative permittivity
(b) Refractive index
(c) Relative density
(d) Relative velocity
Answer:
(d) Relative velocity
Question 110.
If $\mathrm{V}$-velocity, $\mathrm{K}$ - kinetic energy and $\mathrm{T}$ - time are chosen as the fundamental units, then what is the dimensional formula for surface tension?
(a) $\left[\mathrm{K} \mathrm{V}^{-2} \mathrm{~T}^{-2}\right]$
(b) $\left[\mathrm{K}^2 \mathrm{~V} \mathrm{~T}^{-2}\right]$
(c) $\left[\mathrm{K} \mathrm{V}^2 \mathrm{~T}^2\right]$
(d) $\left[\mathrm{K} \mathrm{V}^{-2} \mathrm{~T}^{\text {}}\right.$
Answer:
(a) $\left[\mathrm{K} \mathrm{V}^{-2} \mathrm{~T}^{-2}\right]$
Short Answer Questions (1 Mark)
Question 1.

A new unit of length is chosen such that the speed of light in vaccum is unity. What is the
distance between the sun and the earth in terms of the new unit if light takes $8 \mathrm{~min}$ and $20 \mathrm{~s}$ to cover this distance.
Answer:
Speed of light in vacuum, $\mathrm{c}=1$ new unit of length $\mathrm{s}^{-1}$
$\mathrm{t}=8 \mathrm{~min} .20 \mathrm{sec},=500 \mathrm{~s}$
$\mathrm{x}=\mathrm{ct}=1$ new unit of length $\mathrm{s}^{-1} \times 500 \mathrm{~s}$
$x=500$ new unit of length

Question 2.
If $x=a+b t+c t^2$, where $x$ is in metre and $t$ in seconds, what is the unit of $c$ ?
Answer:
The unit of left hand side is metre so the units of $\mathrm{ct}^2$ should also be metre.
Since $t^2$ has unit of $\mathrm{s}^2$, so the unit of $\mathrm{c}$ is $\mathrm{m} / \mathrm{s}^2$.
Question 3.
What is the difference between $\mathrm{mN}, \mathrm{Nm}$ and $\mathrm{nm}$ ?
Answer:
$\mathrm{mN}$ means milli newton, $1 \mathrm{mN}=10^{-3} \mathrm{~N}, \mathrm{Nm}$ means Newton meter, nm means nano meter.
Question 4.
The radius of atom is of the order of $1 \mathrm{~A}^{\circ} \&$ radius of nucleus is of the order of fermi. How many magnitudes higher is the volume of the atom as compared to the volume of nucleus?
Answer:
$
\frac{V_{\text {atom }}}{V_{\text {nucleus }}}=\frac{4}{3} \pi \frac{\left(10^{-10} \mathrm{~m}\right)^3}{\left(10^{-15} \mathrm{~m}\right)^3}=10^{15}
$
Question 5.
How many kg make 1 unified atomic mass unit?
Answer:
$
1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg}
$
Question 6.
Name some physical quantities that have same dimension.
Answer:
Work, energy and torque.
Question 7.
Name the physical quantities that have dimensional formula $\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]$.
Answer:
Stress, pressure, modulus of elasticity.
Question 8.
Give two examples of dimensionless variables.

Answer:
Strain, refractive index.
Question 9.
State the number of significant figures in
(i) $0.007 \mathrm{~m}^2$
(ii) $2.64 \times 1024 \mathrm{~kg}$
(iii) $0.2370 \mathrm{~g} \mathrm{~cm}^{-3}$
(iv) $0.2300 \mathrm{~m}$
(v) 86400
(vi) $86400 \mathrm{~m}$
Answer:
(i) 1 ,
(ii) 3 ,
(iii) 4 ,
(iv) 4 ,
(v) 3 ,
(vi) 5 since it comes from a measurement the last two zeros become significant.
Question 10.
Given relative error in the measurement of length is 0.02 , what is the percentage error?
Answer:

$2 \%$.
Question 11.
A physical quantity $P$ is related to four observables $a, b, c$ and $d$ as follows :
$
\mathrm{P}=\frac{a^3 b^2}{d \sqrt{c}}
$
The percentage errors of measurement in a, b, c and $\mathrm{d}$ are $1 \%, 3 \%, 4 \%$ and $2 \%$ respectively. What is the percentage error in the quantity $\mathrm{P}$ ?
Answer:
Relative error in $\mathrm{P}$ is given by
$
\begin{aligned}
\frac{\Delta \mathrm{P}}{\mathrm{P}}=3 \frac{\Delta a}{a}+2 \frac{\Delta b}{b}+\frac{1}{2} \frac{\Delta c}{c}+\frac{\Delta d}{d} \\
\frac{\Delta \mathrm{P}}{\mathrm{P}} \times 100=3\left(\frac{\Delta a}{a} \times 100\right)+2\left(\frac{\Delta b}{b} \times 100\right)+\frac{1}{2}\left(\frac{\Delta c}{c} \times 100\right)+\frac{\Delta d}{d} \times 100
\end{aligned}
$

$
=(3 \times 1 \%)+(2 \times 3 \%)+\left(\frac{1}{2} \times 4 \%\right)+(1 \times 2 \%)=13 \%
$
Question 12 .
A boy recalls the relation for relativistic mass $(\mathrm{m})$ in terms of rest mass $\left(\mathrm{m}_0\right)$ velocity of
particle $\mathrm{V}$, but forgets to put the constant c (velocity of light). He writes
$
m=\frac{m_0}{\left(1-v^2\right)^{1 / 2}}
$
correct the equation by putting the missing ' $\mathrm{c}$ '.
Answer:
Since quantities of similar nature can only be added or subtracted, $v^2$ cannot be subtracted from 1 but $\mathrm{v}^2 / \mathrm{c}^2$ can be subtracted from 1 .
$
m=\frac{m_0}{\sqrt{1-v^2 / c^2}}
$

Question 13.
Name the technique used in locating.
(a) an under water obstacle
(b) position of an aeroplane in space.
Answer:
(a) SONAR $\rightarrow$ Sound Navigation and Ranging.
(b) RADAR $\rightarrow$ Radio Detection and Ranging.
Question 14.
Deduce dimensional formulae of-
(i) Boltzmann's constant
(ii) mechanical equivalent of heat.
Answer:
(i) Boltzmann Constant:
$
k=\frac{\text { Heat }}{\text { Temperature }} \Rightarrow[k]=\frac{\left[\mathrm{ML}^2 \mathrm{~T}^2\right]}{[\mathrm{K}]}=\left[\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]
$
(ii) $[\mathrm{J}]=\left[\frac{\text { Work }}{\text { Heat }}\right]=\frac{\left[\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2}\right]}{\left[\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2}\right]}=\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^0\right]$
Question 15 .
Give examples of dimensional constants and dimensionless constants.
Answer:
Dimensional Constants : Gravitational constant, Plank's constant. Dimensionless Constants : it, e.
ShortAnswer Questions (2 Marks)

Question 16 .
The vernier scale of a travelling microscope has 50 divisions which coincide with 49 main scale divisions. If each main scale division is $0.5 \mathrm{~mm}$. Calculate the minimum inaccuracy in the measurement of distance.
Answer:
Minimum inaccuracy $=$ Vernier constant
$
\begin{aligned}
& =1 \mathrm{MSD}-1 \mathrm{VS} \cdot \mathrm{D} \\
& =1 \mathrm{MSD}-\frac{49}{50} \mathrm{MSD} \\
& =\frac{1}{50}(0.5 \mathrm{~mm})=0.01 \mathrm{~mm}
\end{aligned}
$
Question 17.
If the unit of force is $100 \mathrm{~N}$, unit of length is $10 \mathrm{~m}$ and unit of time is $100 \mathrm{~s}$. What is the unit of Mass in this system of units?
Answer:
$
\begin{aligned}
{[\mathrm{F}] } & =\left[\mathrm{MLT}^{-2}\right] \\
{[\mathrm{M}] } & =\frac{[\mathrm{F}]}{[\mathrm{L}]\left[\mathrm{T}^{-2}\right]}=\frac{[100 \mathrm{~N}]}{[10 \mathrm{~m}][100 \mathrm{~s}]^{-2}}=10^5 \mathrm{~kg} .
\end{aligned}
$
Question 18.
State the principle of homogeneity. Test the dimensional homogeneity of equations
(i) $s=u t+\frac{1}{2} a t^2$
(ii) $\mathrm{S}_n=u+\frac{a}{2}(2 n-1)$
Answer:
$
[s]=\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^0\right]
$
Dimension of R.H.S. $=[u t]+\left[a t^2\right]$
$
=\left[\mathrm{LT}^{-1} \cdot \mathrm{T}\right]+\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^{-2} \cdot \mathrm{T}\right]=\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^0\right]
$
as Dimensions of L.H.S. = Dimensions of R.H.S.
$\therefore$ The equation to dimensionally homogeneous.

(ii) $\mathrm{S}_{\mathrm{n}}=$ Distance travelled in $\mathrm{n}^{\text {th }}$ sec that is $\left(\mathrm{S}_{\mathrm{n}}-\mathrm{S}_{\mathrm{n}-1}\right)$
$
\begin{aligned}
\therefore \quad \mathrm{S}_n & =u \times 1+\frac{a}{2}[2 n-1] \\
{\left[\mathrm{LT}^{-1}\right] } & =\left[\mathrm{LT}^{-1}\right]+\left[\mathrm{LT}^{-2}\right][\mathrm{T}] \\
{\left[\mathrm{LT}^{-1}\right] } & =\left[\mathrm{LT}^{-1}\right] \\
\text { L.H.S. } & =\text { R.H.S. }
\end{aligned}
$
Hence this is dimensionally correct.
Question 19.
In Vander Wall's gas equation $\left(\mathrm{P}+\frac{a}{\mathrm{~V}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}$. Determine the dimensions of
Answer:
Since dimensionally similar quantities can only be added
$
\begin{aligned}
& {[\mathrm{P}]=\left[\frac{a}{\mathrm{~V}^2}\right] \Rightarrow[a]=\left[\mathrm{PV}^2\right]=\left[\mathrm{M}^1 \mathrm{~L}^5 / \mathrm{T}^{-2}\right]} \\
& {[b]=[\mathrm{V}]=\left[\mathrm{L}^3\right] .}
\end{aligned}
$
Question 20.
Magnitude of force experienced by an object moving with speed $v$ is given by $F=\mathrm{kv}^2$. Find dimensions of $\mathrm{k}$.
Answer:
$
[\mathrm{K}]=\frac{[\mathrm{F}]}{\left[v^2\right]}=\frac{\left[\mathrm{M}^1 \mathrm{~L}^1 \mathrm{~T}^{-2}\right]}{\left[\mathrm{LT}^{-1}\right]^2}=\frac{\left[\mathrm{M}^1 \mathrm{~L}^1 \mathrm{~T}^{-2}\right]}{\left[\mathrm{M}^0 \mathrm{~L}^2 \mathrm{~T}^{-2}\right]}=\left[\mathrm{M}^1 \mathrm{~L}^{-1}\right]
$
Question 21.
A book with printing error contains four different formulae for displacement. Choose the correct formula/formulae
(a) $y=a \sin \frac{2 \pi}{\mathrm{T}} t$
(b) $y=a \sin v t$
(c) $y=\frac{a}{\mathrm{~T}} \sin \left(\frac{t}{a}\right)$
(d) $y=\frac{a}{T}\left(\sin \frac{2 \pi}{\mathrm{T}} t+\cos \frac{2 \pi}{\mathrm{T}} t\right)$

Answer:
The arguments of sine and cosine function must be dimensionless so (a) is the probable
correct formulae. Since
(a) $y=a \sin \left(\frac{2 \pi}{\mathrm{T}} t\right), \therefore\left[\frac{2 \pi t}{\mathrm{~T}}\right]=\left[\mathrm{T}^0\right]$ is dimensionless.
(b) $y=a \sin v t, \quad \because[v t]=[\mathrm{L}]$ is dimensional so this equation is incorrect.
(c) $y=\frac{a}{t} \sin \left(\frac{t}{a}\right), \because\left[\frac{t}{a}\right]$ is dimensional so this is incorrect.
(d) $y=\frac{a}{t}\left(\sin \frac{2 \pi}{\mathrm{T}} t+\cos \frac{2 \pi t}{\mathrm{~T}}\right):$ Though $\frac{2 \pi t}{\mathrm{~T}}$ dimensionless $\frac{a}{\mathrm{~T}}$ does not have dim displacement so this is also incorrect.

Numerical Questions-2 - Chapter 1 - Nature of Physical World and Measurement - 11th Science Guide Samacheer Kalvi Solutions

Numerical Questions
Question 1

Determine the number of light years in one metre.
Answer:
$
\begin{aligned}
& 1 \mathrm{ly}=9.46 \times 10^{15} \mathrm{~m} \\
& 1 \mathrm{~m}=\frac{1}{9.46 \times 10^{15}}=1.057 \times 10^{-16} \mathrm{ly}
\end{aligned}
$

Question 2
The mass of a box measured by a grocer's balance is $2.3 \mathrm{~kg}$. Two gold pieces $20.15 \mathrm{~g}$ and $20.17 \mathrm{~g}$ are added to the box.
(i) What is the total mass of the box?
(ii) The difference in masses of the pieces to correct significant figures.
Answer:
(i) Mass of box $=2.3 \mathrm{~kg}$
Mass of gold pieces $=20.15+20.17=40.32 \mathrm{~g}=0.04032 \mathrm{~kg}$.
Total mass $=2.3+0.04032=2.34032 \mathrm{~kg}$
In correct significant figure mass $=2.3 \mathrm{~kg}$ (as least decimal)
(ii) Difference in mass of gold pieces $=0.02 \mathrm{~g}$
In correct significant figure ( 2 significant fig. minimum decimal) will be $0.02 \mathrm{~g}$.
Question 3 .
$5.74 \mathrm{~g}$ of a substance occupies $1.2 \mathrm{~cm}^3$. Express its density to correct significant figures. Answer:
Density $=\frac{\text { Mass }}{\text { Volume }}=\frac{5.74}{1.2}=4.783 \mathrm{~g} / \mathrm{cm}^3$
Here least significant figure is 2, so density $=4.8 \mathrm{~g} / \mathrm{cm}^3$.

Question 4
If displacement of a body $\mathrm{s}=(200 \pm 5) \mathrm{m}$ and time taken by it $\mathrm{t}=(20+0.2) \mathrm{s}$, then find the percentage error in the calculation of velocity.

Answer:
Percentage error in measurement of displacement $=\frac{5}{200} \times 100$
Percentage error in measurement of time $=\frac{0.2}{20} \times 100$
$\therefore$ Maximum permissible error $=2.5+1=3.5 \%$
Question 5
If the error in measurement of mass of a body be $3 \%$ and in the measurement of velocity be $2 \%$. What will be maximum possible error in calculation of kinetic energy.
Answer:
$
\begin{aligned}
& \text { K.E. }=\frac{1}{2} m v^2 \\
& \therefore \frac{\Delta k}{k}=\frac{\Delta m}{m}+\frac{2 \Delta v}{v} \Rightarrow \frac{\Delta k}{k} \times 100=\frac{\Delta m}{m} \times 100+2\left(\frac{\Delta v}{v}\right) \times 100 \\
& \therefore \text { Percentage error in K.E. }=3 \%+2 \times 2 \%=7 \%
\end{aligned}
$
Question 6
The length of a rod as measured in an experiment was found to be $2.48 \mathrm{~m}, 2.46 \mathrm{~m}, 2.49 \mathrm{~m}$, $2.50 \mathrm{~m}$ and $2.48 \mathrm{~m}$. Find the average length, absolute error and percentage error. Express the result with error limit.

Answer:
$
\begin{aligned}
& \text { Average length }=\frac{2.48+2.46+2.49+2.50+2.48}{5}=\frac{12.41}{5}=2.48 \mathrm{~m} \\
& \text { Mean absolute error }=\frac{0.00+0.02+0.01+0.02+0.00}{5}=\frac{0.05}{5}=0.01 \mathrm{~m} \\
& \text { Percentage error }=\frac{0.01}{2.48} \times 100 \%=0.04 \times 100 \%=0.40 \% \\
& \text { Correct length }=(2.48 \pm 0.01) \mathrm{m} \\
& \text { Correct length }=(2.48 \mathrm{~m} \pm 0.40 \%)
\end{aligned}
$
Question 28.
A physical quantity is measured as $\mathrm{Q}=(2.1 \pm 0.5)$ units. Calculate the percentage error in (1) $\mathrm{Q}^2(2) 2 Q$.
Answer:
$
\begin{gathered}
\mathrm{P}=\mathrm{Q}^2 \\
\frac{\Delta p}{p}=\frac{2 \Delta \mathrm{Q}}{\mathrm{Q}}=2\left(\frac{0.5}{2.1}\right)=\frac{1.0}{2.1}=0.476 \\
\frac{\Delta p}{p} \times 100 \%=47.6 \%=48 \% \\
\mathrm{R}=2 \mathrm{Q} \\
\frac{\Delta \mathrm{R}}{\mathrm{R}}=\frac{\Delta \mathrm{Q}}{\mathrm{Q}} \Rightarrow \frac{\Delta \mathrm{R}}{\mathrm{R}} \frac{0.5}{2.1}=0.238 \\
\frac{\Delta \mathrm{R}}{\mathrm{R}} \times 100 \%=24 \%
\end{gathered}
$

Question 29.
When the planet Jupiter is at a distance of 824.7 million $\mathrm{km}$ from the earth, its angular diameter is measured to be $35.72^{\prime \prime}$ of arc. Calculate diameter of Jupiter.
Answer:
$
\begin{aligned}
& \theta=35.72^{\prime \prime} \\
& l^{\prime \prime}=4.85 \times 10^{-6} \mathrm{radian} \Rightarrow 35.72^{\prime \prime}=35.72 \times 4.85 \times 10^{-6} \mathrm{rad} \\
& d=\mathrm{D} \theta=824.7 \times 35.72 \times 4.85 \times 10^{-6}=1.4287 \times 10^5 \mathrm{~km}
\end{aligned}
$
Question 30.
A laser light beamed at the moon takes $2.56 \mathrm{~s}$ and to return after reflection at the moon's surface. What will be the radius of lunar orbit?
Answer:
$
\mathrm{t}=2.54 \mathrm{~s}
$
$\therefore t=$ time taken by laser beam to go to the moon $=\frac{t}{2}$
Distance between earth and moon $=d=c \times \frac{t}{2}=3 \times 10^8 \times \frac{2.56}{2}=3.84 \times 10^8 \mathrm{~m}$
Question 31.
Convert

$\text { (i) } 3 \mathrm{~m}^{-2} \mathrm{~s}^{-2} \mathrm{~km} \mathrm{~h}^{-2}$
(ii) $\mathrm{G}=6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^2 \mathrm{~kg}^{-2}$ to $\mathrm{cm}^3 \mathrm{~g}^{-1} \mathrm{~s}^{-2}$
Answer:
(i)
$
3 \mathrm{~ms}^{-2}=\left(\frac{3}{1000} \mathrm{~km}\right)\left(\frac{1}{60 \times 60} \mathrm{hr}\right)^{-2}
$

$
\begin{aligned}
& =\frac{3 \times(60 \times 60)^2}{1000}=3.8880 \times 10^4 \mathrm{~km} \mathrm{~h}^{-2}=3.9 \times 10^4 \mathrm{~km} \mathrm{~h}^{-2} \\
& \text { (ii) } \mathrm{G}=6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^2 \mathrm{~kg}^{-2} \\
& =6.67 \times 10^{-11}\left(\mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-2}\right)\left(\mathrm{m}^2 \mathrm{~kg}^{-2}\right) \\
& =6.67 \times 10^{-11} \mathrm{~kg}^{-1} \mathrm{~m}^3 \mathrm{~s}^{-2} \\
& =6.67 \times 10^{-11}(1000 \mathrm{~g})^{-1}(100 \mathrm{~cm})^3\left(\mathrm{~s}^{-2}\right) \\
& =6.67 \times 10^{-11} \times \frac{1}{1000} \times 100 \times 100 \times 100 \mathrm{~g}^{-1} \mathrm{~cm}^3 \mathrm{~s}^{-2} \\
& =6.67 \times 10^{-8} \mathrm{~g}^{-1} \mathrm{~cm}^3 \mathrm{~s}^{-2} \text {. } \\
&
\end{aligned}
$
Question 32 .
A calorie is a unit of heat or energy and it equals $4.2 \mathrm{~J}$ where $1 \mathrm{~J}=1 \mathrm{~kg} \mathrm{~m}^2 \mathrm{~s}^{-2}$. Suppose we employ a system of units in which unit of mass is $\alpha \mathrm{kg}$, unit of length is $\beta \mathrm{m}$, unit of time $\gamma \mathrm{s}$. What will be magnitude of calorie in terms of this new system.
Answer:
$
\begin{aligned}
& n_2=n_1\left[\frac{\mathrm{M}_1}{\mathrm{M}_2}\right]^a\left[\frac{\mathrm{L}_1}{\mathrm{~L}_2}\right]^b\left[\frac{\mathrm{T}_1}{\mathrm{~T}_2}\right]^c=4.2\left(\frac{\mathrm{kg}}{\alpha \mathrm{kg}}\right)^1\left(\frac{m}{\beta m}\right)^2\left(\frac{s}{\gamma s}\right)^{-2} \\
& n_2=4.2 \alpha^{-1} \beta^{-2} \gamma^2
\end{aligned}
$

Question 33.
The escape velocity $\mathrm{v}$ of a body depends on-
(i) the acceleration due to gravity ' $\mathrm{g}$ ' of the planet,
(ii) the radius $\mathrm{R}$ of the planet. Establish dimensionally the relation for the escape velocity.
Solution:
$v \propto g^a \mathrm{R}^b \Rightarrow v=k g^a \mathrm{R}^b, \mathrm{~K} \rightarrow$ dimensionless proportionality constant
$\begin{aligned} {[v] } & =[g]^a[\mathrm{R}]^b \\ \Rightarrow \quad\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^{-1}\right] & =\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^{-2}\right]^a\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^0\right]^b\end{aligned}$
equating powers
$
\begin{aligned}
1 & =a+b \\
-1 & =-2 a \Rightarrow a=\frac{1}{2} \\
b & =1-a=1-\frac{1}{2}=\frac{1}{2} \\
\therefore \quad v & =k \sqrt{g R}
\end{aligned}
$
Question 34.
The frequency of vibration of a string depends of on,
(i) tension in the string
(ii) mass per unit length of string,
(iii) vibrating length of the string. Establish dimensionally the relation for frequency.
Answer:

$
\begin{aligned}
& n \propto \mathrm{I}^a \mathrm{~T}^b m^c, \quad[\mathrm{I}]=\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^0\right] \\
& {[\mathrm{T}] }=\left[\mathrm{M}^1 \mathrm{~L}^1 \mathrm{~T}^{-2}\right](\text { force) } \\
& {[\mathrm{M}] }=\left[\mathrm{M}^1 \mathrm{~L}^{-1} \mathrm{~T}^0\right] \\
& {\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^{-1}\right] }=\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^0\right]^a\left[\mathrm{M}^1 \mathrm{~L}^1 \mathrm{~T}^{-2}\right]^b\left[\mathrm{M}^0 \mathrm{~L}^{-1} \mathrm{~T}^0\right]^c \\
& b+c=0 \\
& a+b-c=0 \\
&-2 b=-1 \Rightarrow \mathrm{b}=\frac{1}{2} \\
& c=-\frac{1}{2} a=1 \\
& n \propto \frac{1}{l} \sqrt{\frac{\mathrm{T}}{m}}
\end{aligned}
$
Question 35.
One mole of an ideal gas at STP occupies $22.4 \mathrm{~L}$. What is the ratio of molar volume to atomic volume of a mole of hydrogen? Why is the ratio so large? Take radius of hydrogen molecule to be $1^{\circ} \mathrm{A}$.
Answer:
$
\mathrm{A}^0=10^{-10} \mathrm{~m}
$
Atomic volume of 1 mole of hydrogen
$=$ Avagadro's number $\times$ volume of hydrogen molecule
$
\begin{aligned}
& =6.023 \times 10^{23} \times \frac{4}{3} \times \pi \times\left(10^{-10} \mathrm{~m}\right)^3 \\
& =25.2 \times 10^{-7} \mathrm{~m}^3
\end{aligned}
$
Molar volume $=22.4 \mathrm{~L}=22.4 \times 10^{-3} \mathrm{~m}^3$
$\frac{\text { Molar volume }}{\text { Atomic volume }}=0.89 \times 10^4 \approx 10^4$
This ratio is large because actual size of gas molecule is negligible in comparison to the inter molecular separation.

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