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NCERT Exemplar Class 11 Physics Solutions Chapter 2 Units and Measurement

NCERT Exemplar Class 11 Physics Solutions Chapter 2 Units and Measurement

Edited By Safeer PP | Updated on Aug 08, 2022 06:13 PM IST

NCERT Exemplar Class 11 Physics solutions Chapter 2 offers reliable and comprehensive answers to all the questions and queries related to measurement and the units used to measure quantities. NCERT Exemplar Class 11 Physics chapter 2 solutions have been curated by experts and throw light on the internationally accepted measurement standards. The chapter deals with different types of units, the procedure to go about measuring an abstract or a physical quantity, and the plausible errors in a measuring instrument.

NCERT Exemplar Class 11 Physics solutions chapter 2 PDF Download could be availed by the students for a better understanding of concepts and help revise the chapter for a flawless academic performance in final exams as well as competitive exams.

NCERT Exemplar Class 11 Physics Solutions Chapter 2: MCQ I


The number of significant figures in 0.06900 is:
a) 5
b) 4
c) 2
d) 3


The answer is the option (b) 4
Explanation: The two zeros before 6 are insignificant in the number 0.06900. Thus, there are 4 significant numbers.


The sun of the numbers 436.32, 227.2,and 0.301 in appropriate significant figures is:


The answer is the option (c) 663.8
Explanation: After addition, the sum is rounded off to the minimum number of decimal places of the numbers that are added. Hence, option (c) is correct


The mass and volume of a body are 4.237\; g and 2.5\; cm^{3} , respectively. The density of the material of the body in correct significant figures is:
a) 1.6048 \; g/cm^{3}
b) 1.69 \; g/cm^{3}
c) 1.7 \; g/cm^{3}
d) 1.695 \; g/cm^{3}


The answer to a multiplication or division is rounded off to the same number of significant figures as possessed by the least precise term used in the calculation. The final result should retain as many significant figures as are there in the original number with the least significant figures. In the given question, density should be reported to two significant figures .
\begin{aligned} &\text { Density }=\frac{4.237 \mathrm{g}}{2.5 \mathrm{cm}^{3}}=1.6948\\ \end{aligned}
After rounding off the number, we get density =1.7


The mass and volume of a body are 4.237\; g and 2.5\; cm^{3} , respectively. The density of the material of the body in correct significant figures is:
a) 1.6048 \; g/cm^{3}
b) 1.69 \; g/cm^{3}
c) 1.7 \; g/cm^{3}
d) 1.695 \; g/cm^{3}


The answer is the option (c) 1.7 \; g\; cm^{-3}
Explanation: When we are performing multiplication or division, we should keep in mind to retain as many significant figures in the original no. with the least significant figures.
We know that Density = mass/ volume
=1.7\; g\; cm^{-3}
So, we get the above no. after rounding up 2 significant figures.


The numbers 2.745 and 2.735 on rounding off to 3 significant figures will give:
a) 2.75 and 2.74
b) 2.74 and 2.73
c) 2.75 and 2.73
d) 2.74 and 2.74


The answer is the option (d) 2.74 \; and\; 2.74
Explanation:(i) 2.745 \rightarrow 2.74
Here the fourth digit is five and is preceded by, viz., an even no. Hence, it becomes 2.74.
(ii) 2.745 \rightarrow 2.74
Here the fourth digit is 5, but the preceding digit is 3, viz., an odd no. Hence, it is increased by 1, and we get 2.74.


The length and breadth of a rectangular sheet are 16.2 \; cm and 10.1 \; cm respectively. The area of the sheet in appropriate significant figures and error is:
a) 164\pm 3\; cm^{2}
b) 163.62\pm 2.6\; cm^{2}
c) 163.6\pm 2.6\; cm^{2}
d) 163.62\pm 3\; cm^{2}


The answer is the option (i) 164\pm 3\; cm^{2}
Explanation: Given :
I=16.2\; cm,\; and \; \Delta I=0.1
b=10.1\; cm,\; \Delta b=0.1
I=16.1\pm 0.1\; and\; b=10.1\pm 0.1
Now, area (a) =I\times b
=16.2\times 10.1
=163.62\; cm^{2}
=164\; cm^{2}
\Delta \frac{A}{A}=\frac{\Delta I}{I}+\frac{\Delta b}{b}
\frac{\Delta A}{164}=(\frac{10.1\times0.1+16.2\times0.1}{16.2\times10.1})
Thus, \Delta A=2.63\; cm^{2}
We get \Delta A= 3\; cm^{2} by rounding off \Delta I\; and\; \Delta b
Thus, \Delta A=(163\pm 3)cm^{2}


Which of the following pairs of physical quantities does not have same dimensional formula?
a) work and torque
b) angular momentum and Planck’s constant
c) tension and surface tension
d) impulse and linear momentum


The answer is the option (c) Tension & surface tension
Explanation : We know that,
Work = force \times displacement
& Torque = force \times distance
Hence, their dimensions are the same.
(b) Dimensions of-
Angular momentum =[ML^{2}T^{-1}]
& Planck’s constant =[ML^{2}T^{-1}]
Hence, their dimensions are the same.
(c) Dimensions of-
Tension =[MLT^{-2}]
Surface tension=[ML^{0}T^{-2}]
Their dimensions are not the same, hence opt (c).
(d) Dimensions of-
Impulse =[MLT^{-1}]
Momentum =[MLT^{-1}]
Thus, their dimensions are also the same.


Measure of two quantities along with the precision of respective measuring instrument is
A=2.5\; m/s\pm 0.5\; m/s
B= 0.10\; s\; \pm 0.01\;s
The value of AB will be
a) (0.25 \pm 0.08) \; m
b) (0.25 \pm 0.5) m
c) (0.25 \pm 0.05) m
d) (0.25 \pm 0.135) m


The answer is the option (a) (0.25\pm 0.08)m
Explanation : By using the given values of A & B,
X=AB=2.5\times 0.10=0.25\; m
\frac{\Delta x}{x}=\frac{\Delta A}{A}+\frac{\Delta B}{B}
Thus, \Delta x=0.075\cong 0.08
Thus, AB=(0.25\pm 0.08)m


You measure two quantities as A = 1.0 \; m \pm 0.2 \; m, B = 2.0 \; m \pm 0.2 \; m. We should report correct value for √ AB as:
a) 1.4 \; m \pm 0.4 \; m
b) 1.41 \; m \pm 0.15\; m
c) 1.4 \; m \pm 0.3 \; m
d) 1.4 \; m \pm 0.2\; m


The answer is the option (d) 1.4\pm 0.2\; m
There are two significant figures in 1.0 & 2.0
\sqrt{AB}=\sqrt{1.0\times 2.0}=\sqrt{2}
=1.414\; m
After roundig up 1.0 &2.0 we het two sigificat features, i.e.
Now, \frac{\Delta X}{X}=\frac{1}{2}[\frac{\Delta A}{A}+\frac{\Delta B}{B}]
\frac{\Delta X}{1.4}=0.1\left [ \frac{2.0+1.0}{1.0\times 2.0} \right ]
\Delta X=0.1(\frac{3}{2})((1.4)
=0.21\ m after rounding up
Thus, option (d) is the right answer.


Which of the following measurements is most precise?
a)5.00 mm
b) 5.00 cm
c) 5.00 m
d) 5.00 km


The answer is the option (a)5.00 mm
Explanation: The least unit is mm, and all the measurements are up to two decimal places. Hence, 5.00 mm is the most precise.


The mean length of an object is 5\; cm. Which of the following measurements is most accurate?
a) 4.9\; cm
b) 4.805\; cm
c) 5.25\; cm
d) 5.4\; cm


The answer is the option (a) 4.9 cm
Explanation : \left | \Delta a_{1} \right |=\left | 5-4.9 \right |=0.1cm,\left |\Delta a_{2} \right |=\left | 5-4.805 \right |=0.195\; cm
\left | \Delta a_{3} \right |=\left | 5-5.25 \right |=0.25cm,\left |\Delta a_{4} \right |=\left | 5-5.4\right |=0.4\; cm
Thus,\left | \Delta a_{1} \right | it is the mmost acurate.


Young’s modulus of steel is 1.9 \times 10^{11} N/m^{2}. When expressed in CGS units of dynes/cm2, it will be equal to:
a) 1.9 \times 10^{10}
b) 1.9 \times 10^{11}
c) 1.9 \times 10^{12}
d) 1.9 \times 10^{13}


The answer is the option (c) 1.9\times 10^{12}
Explanation : Young's modulus (Y) =1.9\times 10^{11}N/m
Converting into dyne/cm2-
Y=\frac{1.9\times 10^{11}\times 10^{5}dynes}{10^{4}cm^{2}}
=1.9\times 10^{11+5-4}
=1.9\times 10^{12}dyne/cm^{2}


If momentum (P), area (A), and time (T) are taken to be fundamental quantities, then energy has the dimensional formula
b) (P^{2}A^{1}T^{1})
c) (P^{1}A^{-1/2}T^{1})
d) (P^{1}A^{1/2}T^{-1})


The answer is the option d) (P^{1}A^{1/2}T^{-1})
Explanation: Let us consider [P^{a}A^{b}T^{c}] as the formula for energy for fundamental quantities P, A & T.
Thus, dimensional formula of-
Energy(E) = [P^{a}A^{b}T^{c}]
Momentum (P) = [MLT^{-1}]
Area (A) =[L^{2}]
Time(T) = [T^{1}]
Now, E = f.s
= [MLT^{-2}L].[ ML^{2}T^{-2}]
= [MLT^{-1}]^{a}[L^{2}]^{b}[T]^{c} . [M^{a}L^{2+2b}T^{-a+c}]
Now, let us compare the powers,
a=1\; \; \; \; \; \; \; \; a+2b=2\; \; \; \; \; \; \; \; \; -a+c=-2
\; \; \; \; \; \; \; \; 1+2b=2\; \; \; \; \; \; \; \; \; -1+c=-2
\; \; \; \; \; \; \; \; 2b=2-1\; \; \; \; \; \; \; \; \; c=-2+1
\; \; \; \; \; \; \; \; b=\frac{1}{2}\; \; \; \; \; \; \; \; \; c=-1
Thus, [P^{1}A^{1/2}T^{-1}] is the dimensional formula of energy.

NCERT Exemplar Class 11 Physics Solutions Chapter 2: MCQII


On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct:
a) y=a\; \sin \; \frac{2\pi t}{T}
b) y=a\; \sin \; vt
c) y=\frac{a}{T}\sin (\frac{t}{a})
d) y=a\sqrt{2}[\sin (\frac{2\pi t}{t})-\cos (\frac{2\pi t}{T})]


The answer is the option (b) y=a\; \sin \; vt and (c) y=\frac{a}{T}\sin (\frac{t}{a})
Explanation: (b) Here, v.t is an angle, whose dimensions are – [LT^{-1}] [T] = [L]
Thus, it is not true that sin vt is dimensionless.
(c) Here the dimension of amplitude a/T on the R.H.S. is equal to \frac{[L]}{[T]} = [LT^{-1}], viz., not equal to the dimensions of y and the angle \frac{t}{a} = [LT^{-1}], which means that they are not dimensionless.


If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity?
a) \frac{(P - Q)}{R}
b) PQ - R
c)\frac{ PQ}{R}
d)\frac{ (PR-Q^{2})}{R}
e) \frac{(R + Q)}{P}


The answer is the option a) \frac{(P - Q)}{R} and e) \frac{(R + Q)}{P}
(i) The different physical quantities (P-Q) & (R+Q) in option a & e and never be added or subtracted. Thus, they will be meaningless.
(ii) In opt (d), dimensions of PR & Q^{2} may be equal.
(iii) in opt (b), the dimension of PQ may be equal to the dimension of R.
(iv) There is no addition or subtraction, which gives the possibility of opt (c).
Hence, opt (a) & (e).


Photon is quantum of radiation with energy E = hv where v is frequency and h is Planck’s constant. The dimensions of h are the same as that of:
a) linear impulse
b) angular impulse
c) linear momentum
d) angular momentum


The answer is the option (b) Angular Impulse and (d) Angular momentum
Explanation: It is given that E = hv
Hence, h (Planck’s constant) =\frac{E}{v}
Now, Linear Impulse = F.t
& Angular Impulse = = \tau.dt= \frac{dL}{dt}.dt
= dL= mvr
= [M][LT^{-1}][L]
= [ML^{2}T^{-1}]
Now, Linear momentum = mv
= [MLT^{-1}]
& Angular momentum (L) = mvr

= [ML^{2}T^{-1}]
Thus, the dimensional formulae of Planck’s constant, Angular Impulse and Angular Momentum are the same.


If Planck’s constant (h) and speed of light in vacuum (c) are taken as two fundamental quantities, which of the following can in addition be taken to express length, mass, and time in terms of the three chosen fundamental quantities?
a) mass of electron (me)
b) universal gravitational constant (G)
c) charge of electron (e)
d) mass of proton (mp)


The answer is the option (a) Mass of electron (me) and (b) Universal gravitational constant (G) and (d) Mass of proton (mp)
Explanation: Dimensions of –
h (Planck’s constant)
= [ML^{2}T^{-1}]
c (Speed of light in vacuum)
= \frac{s}{t}
= [LT^{-1}]
Thus, dimension of hc
= [ML^{2}T^{-1}][LT^{-1}]
= [ML^{3}T^{-2}]
G = \frac{Fr^{2}}{M_{1}M_{2}}
= \frac{[ML^{3}T^{-2}]}{[M][M]}
= [M^{-1}L^{3}T^{-2}]
\frac{hc}{G} = [ML^{3}T^{-2}]/[M^{-1}L^{3}T^{-2}]
M= \sqrt{\frac{hc}{G}}
= [h^{\frac{1}{2}} c^{\frac{1}{2}}G^{\frac{1}{2}}]
\frac{h}{c} = [ML^{2}T^{-1}]/[LT^{-1}]
= [ML]
=\sqrt{\frac{hc}{G}}\times L
L=\frac{h}{c}\times \sqrt{\frac{G}{hc}}
= [G^{\frac{1}{2}} h^{\frac{1}{2}} c^{\frac{-3}{2}}]
Also, c = [LT^{-1}]
= [G^{\frac{1}{2}} h^{\frac{1}{2}} c^{\frac{-3}{2}}T^{-1}]
& T= [G^{\frac{1}{2}} h^{\frac{1}{2}} c^{\frac{-3}{2}-1}]
T= [G^{\frac{1}{2}} h^{\frac{1}{2}} c^{\frac{-5}{2}}]
Therefore, in terms of chosen fundamental quantities, the physical quantities a, b & d can be used to represent L, M & T.


Which of the following ratios express pressure?
a) Force/area
b) Energy/volume
c) Energy/area
d) Force/volume


The answer is the option (a) Force/Area and (d) Force/Volume
Explanation: Let us start with calculating the dimension of pressure,
We know that Pressure(P) =\frac{F}{A}
= \frac{[MLT^{-2}]}{[L^{2}]}
..... this also verifies the opt (a).
Now let us compare it with other options-
= Dimesions of P
Thus, opt (b) is verified as well.
=[ML^{0}T^{-2}] ...............viz., not equal to Dimensions of P
=[ML^{-2}T^{-2}] .................viz., again not equal to dimensions of P
Thus, opt (c) & (d) are discarded.


Which of the following are not a unit of time?
a) second
b) parsec
c) year
d) light year


The answer is the option (b) Parsec and (d) Light year
Explanation: Parsec & Light year are units to measure distance while second & Year are units to measure time.

NCERT Exemplar Class 11 Physics Solutions Chapter 2: Very Short Answer


Why do we have different units for the same physical quantity?


(i) Different orders of quantity are measured by the same physical quantities.
(ii)The radius of a nucleus is measured in fermi. The distance between stars is measured in light-years. The length of cloth is measured in meter while the distance between two cities is measured in Kilometer or miles. Depending upon the physical quantity to be measured the units also vary.


Name the device used for measuring the mass of atoms and molecules.


The mass spectrograph is the device used for measuring the mass of atoms and molecules.

Name the device used for measuring the mass of atoms and molecules.


Express unified atomic mass unit in kg.


We know that the unified atomic mass unit or amu or u= \frac{1}{12} the mass of one c^{12} atom
Mass of 6.023\times10^{23} carbon atoms (_{6}C^{12})= 12\; gm
Thus, the mass of one _{6}C^{12} atom = \frac{12}{6.023}\times 10^{-23}gm
Now , 1 amu
= \frac{1}{12}\times \frac{12}{6.023}\times 10^{-23}\; gm
=1.66\times 10^{-24}\; gm
Thus, 1 amu = 1.66\times10^{-27}kg


A function f(\theta ) is defined as
f(\theta )= 1-\theta +\frac{\theta ^{2}}{2!}-\frac{\theta ^{3}}{3!}+\frac{\theta ^{4}}{4!}+.........
Why is it necessary for \theta to be a dimensionless quantity ?


(i) \theta is a dimensionless physical quantity since it is represented by an angle viz. equal to arc/radius.
(ii) As \theta is dimensionless, all its powers will also be dimensionless. Here the first term is 1 viz. also dimensionless.
Hence, the L.H.S. f(\theta) must be dimensionless so that all the terms in R.H.S. are also dimensionless.


Why length, mass and time are chosen as base quantities in mechanics?


Length, mass & time is chosen as base quantities in mechanics because unlike other physical quantities, they cannot be derived from any other physical quantities.

NCERT Exemplar Class 11 Physics Solutions Chapter 2: Short Answer


(a) The earth-moon distance is about 60 earth radius. What will be the diameter of the earth (approximately in degrees) as seen from the moon?
(b) Moon is seen to be of \left ( \frac{1}{2} \right )^{o} diameter from the earth. What must be the relative size compared to the earth?
(c) From parallax measurement, the sun is found to be at a distance of about 400 times the earth-moon distance. Estimate the ratio of sun-earth diameters.


(a) Here we know that, \theta = arc/radius
= \frac{R_{e}}{60R_{e}}
= \frac{1}{60}rad

Now, the angle from the moon to the diameter of the earth will be-
2\theta = 2\left ( \frac{1}{60} \right )\left ( \frac{180}{\pi } \right )
= \frac{6^{o}}{\pi }
\cong \frac{6}{3.14}\cong 2^{o}
(b) If the moon is seen from the earth diametrically, the angle will be \frac{1}{2^{o}}
& if the earth is seen from the moon, the angle will be 2^{o}
Thus, the size of the moon/ size of the earth = \frac{\frac{1}{2^{o}}}{2^{o}}= \frac{1}{4}
Therefore, as compared to the earth, the size of the moon is \frac{1}{4} the size(diameter) of the earth.
(c)Let rem = x m be the distance between the earth and the moon
Now, the distance between the sun and the earth (rse) = 400xm
On a solar eclipse, the sun is completely covered by the moon; also the angle formed by the diameter of the sun, moon and the earth are equal.
Thus, \theta _{m}= \theta _{s}
......... ( the angle from the earth to the moon is \theta {_{m}}^{o} & angle from sun to earth is \theta {_{s}}^{o}
Now since \theta _{m}= \theta _{s}
\frac{D_{m}}{r_{em}}= \frac{D_{s}}{r_{x}}
\frac{D_{m}}{x}= \frac{D_{s}}{400 x}
\frac{D_{s}}{D_{M}}= 400
Thus, 4D_{m}= D_{e}or\; D_{m}= \frac{D_{e}}{4}
Thus, \frac{D_{s}}{\frac{D_{e}}{4}}= 400
\frac{4D_{s}}{D_{e}}= 400
\frac{D_{s}}{D_{e}}= \frac{400}{4}= 100
Therefore, D_{s}= 100\; D_{e}


Which of the following time measuring devices is most precise?
(a) A wall clock.
(b) A stopwatch.
(c) A digital watch.
(d) An atomic clock.
Give reason for your answer.


The least counts of the devices are as follows-
Wall clock = 1 sec
Stopwatch = \frac{1}{10} sec
Digital watch = \frac{1}{100} sec &
Atomic clock = \frac{1}{10^{13}} sec.
Hence, the atomic clock is the most precise.


The radius of atom is of the order of 1\; \AA and radius of nucleus is of the order of fermi. How many magnitudes higher is the volume of atom as compared to the volume of nucleus?


Let us convert these different units into a common one, i.e., ‘m’.
Thus, Radius of an atom (R)= 1\AA
= 10^{-10}m
& Radius of a nucleus = 1 fermi
= 10^{-15}m
Now, let us calculate the ratio of the volumes of an atom to a nucleus-
\frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^{3}}= R^{3}/r^{3}
= (10^{-10}/10^{-15})^3
= (10^{5})^{3}= 10^{15}


The distance of a galaxy is of the order of 1025 m. Calculate the order of magnitude of time taken by light to reach us from the galaxy.


It is given that the distance travelled by light from galaxy to earth = 10^{25}m.
We know that speed of light = 3\times10^{8}m/s
Thus, the required time = distance / speed
= \frac{10^{25}}{3}\times10^{-8}\; sec
= \frac{1}{3}\times 10^{17}sec
= 3.3\bar{3}\times 10^{16}sec


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 mm, calculate the minimum inaccuracy in the measurement of distance.


Let ‘n’ be the no. Of parts on the Vernier scale, n = 50
Now, the no. Of parts of M.S. coinciding with n parts of vernier scale = (n-1)
Now, Least count of an instrument = L.C. of Mainscale/n
= 0.5mm/50
Thus, the minimum accuracy will be 0.01 mm.


During a total solar eclipse the moon almost entirely covers the sphere of the sun. Write the relation between the distances and sizes of the sun and moon.


Thus, \Omega _{s}= \Omega _{m}
Thus, \frac{\pi (\frac{D_{s}}{2})^{2}}{r{_{s}}^{2}}= \frac{\pi (\frac{D_{m}}{2})^{2}}{r{_{m}}^{2}}
i.e., \frac{D_{s}^{2}}{4r_{s}^{2}}= \frac{D_{m}^{2}}{4r_{m}^{2}}
Thus, \frac{4R_{s}^{2}}{4r_{s}^{2}}= \frac{4R_{m}^{2}}{4r_{m}^{2}}
Taking square roots,
Therefore, the ratio of the size of the sun to the moon is equal to the distances from the sun to the moon from earth.


If the unit of force is 100 N, unit of length is 10 m and unit of time is 100 s, what is the unit of mass in this system of units?


The dimensions of -
Force = [M^{1}L^{1}T^{-2}]
=100 N ..........(i)
Length = [L^{1}]
= 10 m ...........(ii)
Time = [T^{1}]
= 100 sec ...........(iii)
Now let us substitute the values of eq. (ii) & (iii) in eq. (i)
We get,
Thus, M=10^{5}kg, L=10^{1}m,F=10^{2}N \; and \; T=10^{2}seconds


Give an example of
(a) a physical quantity which has a unit but no dimensions.
(b) a physical quantity which has neither unit nor dimensions.
(c) a constant which has a unit.
(d) a constant which has no unit.


(a) A plane angle is an example of a physical quantity which has unit but no dimension since, plane angle = arc/radius in radian & solid angle.
(b) Relative density is a physical quantity which neither has neither units nor dimension since µr (relative density) is a ratio of same physical quantities.
(c) Gravitational constant (G) and Dielectric constant (k) are an example of constant that have a unit.
G=6.67 \times 10^{-11}N-m^{2}-kg
k=9 \times 10^{9}N-m^{2}-C^{-2}(Vacuum)
(d) Examples of the constant that has no units are Reynolds no. & Avogadro’s no.


Calculate the length of the arc of a circle of radius 31.0 cm which \frac{\pi }{6} subtends an angle of at the centre.


Given :radius = 31 cm
Angle = \frac{\pi }{6}
Length of arc = ?
Now, we know that
Angle = length of arc/radius of arc
\frac{\pi }{6}=\frac{x}{31}
x = 31 \times \frac{\pi }{6}
= 31 \times \frac{3.14 }{6}
x = 16.22 \; cm


The displacement of a progressive wave is represented by y = A\; \sin (wt - k \; x ), where x is distance and t is time. Write the dimensional formula of (i) \omega and (ii) k.


According to the principle of homogeneity dimensional formula on L.H.S. & R.H.S. is equal.
Thus, the dimension of A \; \sin (\omega t - kx) = Dimension of y
\omega t - kx has no dimensions because it is an angle of sin.
\frac{2\pi }{T}t=Kx
[M^{0}L^{0}T^{0}] = k[L]
Thus, \omega has no dimension & dimension of k=[M^{0}L^{-1}T^{0}].


Time for 20 oscillations of a pendulum is measured as t_{1}= 39.6 \; s; t_{2} = 39.9 \; s; t_{3} = 39.5\; s. What is the precision in the measurements? What is the accuracy of the measurement?


Given data :
t_{1}= 39.6 \; s
t_{2} = 39.9 \; s
and t_{3} = 39.5\; s
L.C. of the instrument is 0.1 sec, hence Precision (LC) = 0.1 sec.
For 20 oscillations, the mean value will be
=39.7 \; s
Now the absolute errors in measurement
\left | \Delta t_{1} \right |=\left | \bar{t}- t_{1}\right |=\left | 39.7-39.6 \right |=\left | 0.1 \right |=0.1\; s
\left | \Delta t_{2} \right |=\left | \bar{t}- t_{2}\right |=\left | 39.7-39.9 \right |=\left | 0.2 \right |=0.2\; s
\left | \Delta t_{3} \right |=\left | \bar{t}- t_{3}\right |=\left | 39.7-39.5 \right |=\left | 0.2 \right |=0.2\; s
Now, Mean absolute error =\frac{0.1+0.2+0.2}{3}
=\frac{0.5}{3}\cong 0.2\; s
& Accuracy of measurement =\pm 0.2\; s

NCERT Exemplar Class 11 Physics Solutions Chapter 2: Long Answer


A new system of units is proposed in which unit of mass is \alpha kg, unit of length \beta m and unit of time γ s. How much will 5 J measure in this new system?


Joules is a unit of work/energy, so let us first write the dimensions of energy-
Energy =[ML^{2}T^{-2}]
Let n_{1} be the SI system of unit & n_{2} be the new system of unit
Thus, n_{2}u_{2}=n_{1}u_{1}
M_{1}=1\; kg & M_{2}=\alpha \; kg
L_{1}= 1 \; m & L_{2}= \beta \; m
T_{1}= 1 \; second & T_{2}= \gamma \; sec
Thus, n_{2}= 5[\frac{1\; kg}{\alpha \; kg}]^{1}[\frac{1\; m}{\beta \; m}]^{2}[\frac{1\; s}{\gamma \; s}]^{-2}
=5[\alpha ^{-1}\beta ^{-2}\gamma ^{2}]
Thus, by the new system, [\alpha ^{-1}\beta ^{-2}\gamma ^{2}].


A physical quantity X is related to four measurable quantities a, b, c and d as follows: X = a^{2} b^{3} c^{\frac{5}{2}} d^{-2}. The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4%, respectively. What is the percentage error in quantity X ? If the value of X calculated on the basis of the above relation is 2.763, to what value should you round off the result.


As per the information given in the question, percentage error in a = (\frac{\Delta a}{a})(100) = 1\; ^{o}/_{o}
As per the information given in the question, Percentage error in b = (\frac{\Delta b}{b})(100) = 2\; ^{o}/_{o}
As per the information given in the question, Percentage error in c = (\frac{\Delta c}{c})(100) = 3\; ^{o}/_{o}
As per the information given in the question, Percentage error in d = (\frac{\Delta d}{d})(100) = 4\; ^{o}/_{o}
Percentage error in X = (\frac{\Delta x}{x})(100)
\\\frac{\Delta X}{X}\times 100=\pm(2\times1+3\times2+2.5\times3+2\times4)\\\pm(2+6+7.5+8) =\pm 23.5\; ^{o}/_{o}
Now, let us calculate the mean absolute error
=\pm \frac{23.5}{100}
=\pm 0.235=0.24 (after rounding up)
Therefore, we get ‘2.8’ after rounding X i.e. 2.763.


In the expression P=E\; I^{2}m^{-5}G^{-2},E,m,I\; and\; G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity.


The dimensional formula of –
Now let us find out the dimension of P
\ \ =\frac{M^3L^{6}T^{-4}}{M^3L^{6}T^{-4}}
Hence, it is clear that P is a dimensionless quantity.


If velocity of light c, Planck’s constant h and gravitational constant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.


Let us first write down the dimensions of-
= \frac{[ML^{2}T^{-2}]}{[T^{-1}]}
G = [M^{-1}L^{3}T^{-2}]
(i) Let us consider that Mass m is directly proportional to [h]^{a}[C]^{b}[G]^{c}
[M^{1}L^{0}T^{0}] = k [ML^{2}T^{-1}]^{a}[LT^{-1}]^{b}[M^{-1}L^{3}T^{-1}]^{c}
[M^{1}L^{0}T^{0}] = k [M^{a-c}L^{2a+b+3c}T^{-a-b-2c}]
Comparing the powers on R.H.S. & L.H.S.
Thus, a=c+1
2a+b+3c=0\; \; \; \; \; \; \; .....(i)
-a-b-2c=0 \; \; \; \; \; \; \; ..... (ii)
Substituting the value of a in eq (i) & (ii)
2(c+1) + b + 3c = 0
2c + 2 + b + 3c =0
b+5c = -2 \; \; \; \; \; \; \; \; ..... (iii)
-(c+1)-b-2c = 0
-b-3c=1 \; \; \; \; \; \; \; ...... (iv)
By adding (iii) & (iv), we get,
now, a = c+1
Thus, a = \frac{-1}{2}+1 = \frac{1}{2}
Substituting these values in eq. (iii)
b + 5(-1/2) = -2
b = -2 + \frac{2}{5}
b = \frac{1}{2}
Thus, m = kh^{\frac{1}{2}}C^{\frac{1}{2}}G^{\frac{-1}{2}}
Now, let L\; \alpha \; C^{a}h^{b}G^{c}
Thus, [L^{1}] = k[LT^{-1}]^{a}[ML^{2}T^{-1}]^b[M^{-1}L^{3}T^{2}]^{c}
Equating the powers on both sides
a+2b+3c = 1 \; \; \; \; \; \; \; ...... (i)
-a-b-2c = 0 \; \; \; \; \; \; .... (ii)
Now, since b=c
a+2b+3b=1 becomes
a+5b=1 \; \; \; \; \; \; .....(iii)
& eq. (ii)
a+b+2c = 0 becomes
a + 3b = 0\; \; \; \; \; \; \; \; ...... (iv)
Subtracting eq (iv) from eq (iii) we get,
Thus, c=\frac{1}{2}
And eq (iv) will be
a +3(\frac{1}{2}) = 0
Therefore, I = k\; C^{\frac{3}{2}}h^{\frac{1}{2}}G^{\frac{1}{2}}
Let us consider T\; \alpha \; G^{a}h^{b}C^{c}
Thus, [M^{0}L^{0}T^{-1}] = k[M^{-1}L^{3} T^{-2}]^{a}[ML^{2}T^{-1}][LT^{-1}]^{c}
= k[M^{-a+b} L^{3a+2b+c}T^{-2a-b-c}]
-a+b=0 \; \; \; \; \; \; \; ... (i)
3a + 2b + c = 0 \; \; \; \; \; \; ....... (ii)
-2a-b-c = 1 \; \; \; \; \; \; \; \; .... (iii)
On adding eq (ii) & (iii) we get,
a+b=1 \; \; \; \; \; \: \: \: \: \: ..... (iv)
From eq (i) &(iv)
Thus, c=\frac{-5}{2}
T = k\; G^{\frac{1}{2}}h^{\frac{1}{2}}C^{\frac{-5}{2}}
Thus, T = k\sqrt{\frac{hG}{c^{5}}}


An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that T=\frac{k}{R}\sqrt{\frac{r3}{g}} where k is a dimensionless constant and g is acceleration due to gravity.


According to Kepler’s 3rd law of planetary motion,
T^{2}\; \alpha \; \; r^{3}
T\; \alpha \; \; r^{\frac{3}{2}}
Now, R & g also affects T
T \; \; \; \alpha\; \; \; g^{a}R^{b}r^{\frac{3}{2}}
T = K[LT^{-2}]^{a}[L^{b}][L]^{\frac{3}{2}}
[T^{1}] = K[L^{a+b+\frac{3}{2}}T^{-2a}]
Now, comparing M, L & T according to the principle of homogeneity
a+b+\frac{3}{2} = 0 \; \; \; \; \; \; ..... (i)
-2a = 1 \; \; \; \; \; \; ..... (ii)
Thus, \frac{-1}{2} + b + \frac{3}{2} = 0

i.e., T=\frac{K}{R}\sqrt{\frac{r^{3}}{g}}

An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that where k is a dimensionless constant and g is acceleration due to gravity.


In an experiment to estimate the size of a molecule of oleic acid, 1mL of oleic acid is dissolved in 19mL of alcohol. Then 1mL of this solution is diluted to 20mL by adding alcohol. Now, 1 drop of this diluted solution is placed on water in a shallow trough. The solution spreads over the surface of water forming one molecule thick layer. Now, lycopodium powder is sprinkled evenly over the film we can calculate the thickness of the film which will give us the size of oleic acid molecule.
Read the passage carefully and answer the following questions:

a) Why do we dissolve oleic acid in alcohol?
b) What is the role of lycopodium powder?
c) What would be the volume of oleic acid in each mL of solution prepared?
d) How will you calculate the volume of n drops of this solution of oleic.
e) What will be the volume of oleic acid in one drop of this solution?


(a) We can reduce the concentration of oleic acid by dissolving it in a proper solvent to get a molecular level. It can be dissolved in organic solvent alcohol and not ionic solvent water since it is an organic compound.
(b) Mixing of oleic acid in water is prevented by Lycopodium when a drop of oleic acid is poured on water. Thus, if we spread lycopodium powder on the water surface, the layer of oleic acid will be formed on the surface of the powder,
(c)(\frac{1}{20})(\frac{1}{20\; V}) = \frac{V}{400} ml is the concentration of oleic acid in an alcohol solution. Its required conc. in 1 ml solution is 1/400 ml.
(d) By dropping one ml of solution drop by drop in a beaker through burette and counting its no., can be used to calculate the volume of n drop solution. If there is 1ml of n drop, then its volume will be 1/n ml.
(e) If there is 1ml of n drop, then its volume will be 1/n ml. The concentration of oleic acid in one drop of the solution will be
\frac{1}{400\; V}=\frac{1}{400}.\frac{1}{n}ml
=\frac{1}{400}ml of oleic acid.


a) How many astronomical units (AU) make 1 parsec?
b) Consider the sun like a star at a distance of 2 parsec. When it is seen through a telescope with 100 magnification, what should be the angular size of the star? Sun appears to be (1/2) degree from the earth. Due to atmospheric fluctuations, eye cannot resolve objects smaller than 1 arc minute.
c) Mars has approximately half of the earth’s diameter. When it is closer to the earth it is at about ½ AU from the earth. Calculate at what size it will disappear when seen through the same telescope.


(a)1 A.U. long arc subtends the angle of 1s or 1 arc sec at distance of 1 parsec.
Thus, angle 1 sec = \frac{1 A.U.}{1} parsec
Thus, 1 parsec = \frac{1 A.U.}{1} arc sec
Thus, 1 parsec= \frac{630(3600)}{11}
(b)Angle of the sun’s diameter \left ( \frac{1}{2} \right )^{o} is subtended by 1 A.U. since the distance from the sun increases angle subtended in the same ratio.
2\times10^{5}A.U. will form an angle of \theta =\left ( \frac{1}{4\times10^{5}} \right )^{o}, since the diameter is the same angle subtended on earth by 1 parsec will be same.
If the sunlike star is at 2 parsec the angle becomes half = (1.25 \times 10^{-6})^{o}
Thus, angle = 75 \times 10^{-6} min
When it is seen with a telescope that has a magnification of 100, the angle formed will be 7.5 \times 10^{-3} min, viz., less than a minute.
Hence, it can’t be observed by a telescope.


Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass (m) to energy (E) as E = mc2, where c is speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in MeV where 1 MeV = 1.6 \times 10^{-13}J, the masses are measured in unified equivalent of 1u is 931.5 MeV.
a) Show that the energy equivalent of 1 u is 931.5 MeV.
b) A student writes the relation as 1 u = 931.5 MeV. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.


(a) Given :
By the formula E=mc^{2}
=939.4\; Mev
\cong 931.5\; Mev
(b) 1 u mass converted into total energy will be released by 931.5 MeV.
However, 1 amu = 931.5 MeV is dimensionally incorrect.
E=mc^{2}\rightarrow 1uc^{2}\cong 931.5\; Mev, will be dimensionally correct.

Class 11 Physics NCERT Exemplar Solutions Chapter 2 Topics:

Main Subtopics in NCERT Exemplar Class 11 Physics Solutions Chapter 2 Units and Measurement

  • 2.1 Introduction
  • 2.2 The International System Of Units
  • 2.3 Measurement Of Length
  • 2.4 Measurement Of Mass
  • 2.5 Measurement Of Time
  • 2.6 Accuracy, Precision Of Instruments And Errors In Measurement
  • 2.7 Significant Figures
  • 2.8 Dimensions Of Physical Quantities
  • 2.9 Dimensional Formulae And Dimensional Equations
  • 2.10 Dimensional Analysis And Its Applications

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What will The Students Learn in NCERT Exemplar Class 11 Physics Solutions Chapter 2?

Students would be able to understand the dynamics behind the change in measuring trends, learn about the basic units, and try to bring all that knowledge to life by using these concepts in actual measuring instruments. NCERT Exemplar Class 11 Physics solutions chapter 2 would also introduce the students to utilise the system of precise measurement to answer real-life questions related to the mass of rain-bearing clouds and ascertain the speed of an aircraft through dimensional analysis and determine the distance between two stars.

NCERT Exemplar Class 11 Physics Solutions Chapter-Wise

Important Topics To cover For Exams in NCERT Exemplar Solutions For Class 11 Physics Chapter 2 Units and Measurement

· NCERT Exemplar Class 11 Physics solutions chapter 2 introduces the student to various measurement units across the world and the standard unit that is accepted internationally. It talks about measuring different physical quantities like length and mass and abstract quantities like time.

· The dimensional formulae and equations are also introduced in this chapter, which helps to utilise dimensional analysis in real-life applications. Students are also made aware of the importance of significant figures of measurement.

· NCERT Exemplar Class 11 Physics solutions chapter 2 also helps students understand the accuracy and precision in making measurements and the errors that one stumbles on along the way. It also highlights the different types of errors found in instruments and the measuring procedure to quickly identify and rectify them.

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This chapter's essential topics are the International System of Units, Measurement of quantities, accuracy and precision, dimensional formulae, and dimensional analysis. 


Get answers from students and experts

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg


An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)


Option 2)

\; K\;

Option 3)


Option 4)


In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)


Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)


Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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