-
Fill in the
blanks :
(i)
The volume of a cube of side 1 cm is equal to...... 
.
(ii)
The surface area of a solid cylinder of radius 2 cm and height 10 cm is equal
to ...... (mm)2.
(iii)
A vehicle moving with a speed of 18 km/h covers ......mm 1 s.
(iv) The
relative density of lead is 11.3. Its density is 
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Fill in the blanks by suitable
conversion of units :
(i)
(ii)
(iii)
(iv)
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A calorie is a unit of heat energy and
it equals about 4.2 J, where
.
Suppose we employ a system of units in which the unit of mass equals
kg, the unit of
length equals
m, the
unit of time is
s.
Show that a calorie has a magnitude
in
terms of the new units.
-
(i) Explain the statement clearly : To call a dimensional quantity 'large' or 'small' is meaningless without specifying a standard for comparison.
(ii) In view of this, reframe the following statements, wherever necessary:
(a) Atoms are very small objects
(b) A jet plane moves with great speed
(c) The mass of Jupiter is very large
(d) The air inside this room contains a large number of molecules
(e) A proton is much more massive than an electron (f) The speed of sound is much smaller than the speed of light.
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A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the sun and the earth in terms of the new unit if light takes 8 min and 20 s to cover this distance ?
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Which of the following is the most precise device for measuring length:
(a) a vernier calliper with 20 divisions on the sliding scale,
(b) a screw gauge of pitch 1 mm and 100 divisions on the circular scale
(c) an optical instrument that can measure length to within a wavelength of visible light ?
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A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What is the estimate on the thickness of hair ?
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Answer the following :
(a) You are given a thread and a metre scale. How will you estimate the diameter of the thread ?
(b) A screw gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Do you think it is possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular scale ?
(c) The mean diameter of a thin brass rod is to be measured by vernier calipers. Why is a set of 100 measurements of the diameter expected to yield a more reliable estimate than a set of 5 measurements only ?
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The photograph of a house occupies an area of 1.75 cm2 on a 35 mm slide. The slide is projected on to a screen and the area of the house on the screen is 1.55 m2. What is the linear magnification of the projector-screen arrangement ?
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State the number of significant figures
in the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
-
The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m and 2.01 cm, respectively.
Give the area and volume of the sheet to correct significant figures.
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The mass of a box measured by a grocer's balance is 2.3 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is (a) the total mass of the box and (b) the difference in the masses of the pieces to correct significant figures?
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A physical
quantity P is related to four observations:
a, b,
c and d as follows : 
The percentage
errors of measurement in a, b, c and d are 1%, 3%, 4% and 2% respectively. What
is the percentage error in the quantity P Rs If the value of P calculated using
the above relation turns out to be 3.763 , to what value should you round off
the result ?
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A book with many printing errors
contains four different formulae for the displacement y of a particle undergoing
a certain periodic motion :
(i)
(ii)
(iii)
(iv)
(a = maximum displacement of the
particle,
speed of the particle,
T= time-period of motion). Rule out the wrong formula on dimensional grounds.
-
A book with many printing errors
contains four different formulae for the displacement y of a particle undergoing
a certain periodic motion :
(i)
(ii)
(iii)
(iv)
(a = maximum displacement of the
particle,
speed of the particle,
T= time-period of motion). Rule out the wrong formula on dimensional grounds.
-
A famous relation in physics relates
'moving mass' m to the 'rest mass'
of
a particle in terms of its speed v and the speed of light c. (This relation
first arose as a consequence of special relatively due to Albert Einstein). A
boy recalls the relation almost correctly but for gets where to put the
constant c.
He writes:
Guess where to put the missing c.
-
The radius of a hydrogen atom is about
0.5 A. What is the total atomic volume in
of a mole of
hydrogen atoms ?
-
One mole of an ideal gas at S.T.P. occupies 22.4 L.
What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? Why is this ratio so large ? Take the radius of hydrogen molecule to be 1 A.
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Explain why on looking through the window of a fast moving train, the nearby trees and electric poles etc. appear to run in direction opposite to that of motion of the train, while far off houses, hilltops, Moon, stars etc. appear stationary.
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A parsec is a convenient unit of length on the astronomical scale. It is the distance of an object that will show a parallax of 1" (second) of arc from opposite ends of a baseline equal to the distance from the earth to the sun. How much is parsec in terms of metres ?
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The nearest star (Alpha Centauri) to our solar system is 4.29 light years away. How much is this distance in terms of parsec ? How much parallax would this star show when viewed from two locations of the earth six months apart in its orbit around the sun ?
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Precise measurements of physical quantities are a need of science. For example, to ascertain the speed of an aircraft, one must have an accurate method to find its positions at closely separated instants of time. This was the actual motivation behind the discovery of radar in World War II. Think of different examples in modern science where precise measurements of length, time, mass etc. are needed. Also, wherever you can, give a quantitative idea of the precision needed.
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Just as precise measurements are necessary in science, it is equally important to be able to make rough estiamtes of quantities using rudimentary ideas and common observations. Think of ways by which you can estimate the following (where an estimate is difficult to obtain, try to get an upper bound on the quantity):
(a) the total mass of rain-bearing clouds over India during the Monsoon
(b) the mass of an elephant
(c) the wind speed during a storm
(d) the number of strands of hair on your head
(e) the number of air molecules in your classroom.
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The sun is a hot plasma (ionized matter)
with its inner core at a temperature exceeding
and its outer
surface at a temperature of about 6000 K. At these high temperatures no substance
remains in a solid or liquid phase. In what range do you expect the mass
density of the sun to be ? In the range of densities of solids and liquids or
gases ? Check if your guess is correct from the following data: mass of the sun
radius of the
sun
.
-
When the planet Jupiter is at a distance of 824.7 million kilometres from the earth, its angular diameter is measured to be 35.72 s of arc. Calculate the diameter of Jupiter.
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A man walking briskly in rain with speed
v must slant his umbrella forward making an angle
with the
vertical. A student derives the following relation between
and
and checks that
the relation has a correct limit :as
as
expected. (We are assuming there is no strong wind and that the rainfalls
vertically for a stationary man). Do you think this relation can be correct ?
If not, guess at the correct relation.
-
A laser is a source of very intense, monochromatic, and unidirectional beam of light. These properties of a laser light can be exploited to measure long distances. The distance of the Moon from the Earth has been already determined very precisely using a laser as a source of light. A laser light beamed at the moon takes 2.56 s to return after reflection at the moon's surface. How much is the radius of the lunar orbit around the earth ?
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A SONAR (Sound Navigation and Ranging) uses ultrasonic waves to detect and locate objects under water. 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 s. What is the distance of the enemy submarine ?
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The farthest objects (known as quasers) in our universe are so distant that light emitted by them takes billion of years to reach the earth. What is the distance in km of a quaser from which light takes 3.0 billion years to reach us ?
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It is a well known fact that during a
solar eclipse the disc of the moon almost completely covers the disc of the
sun.
From this fact and from the information
that sun's angular distance a is measured to be 1920", determine the
approximate diameter of the moon. Given earth-moon distance
.
-
A great physicist of this century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics (c, e, mass of electron, mass of proton) and the gravitational constant G, he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe (w 15 billion years).
From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constant ?
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question_answer32)
The number of significant figure in
0.069000 is
(a) 5 (b)
4
(c) 2 (d)
3
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question_answer33)
The sum of number 436.32, 227.2 and 0.301
in appropriate significant figure is
(a) 663.821 (b)
664
(c) 663.8 (d)
663.82
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question_answer34)
The mass and volume of a body are 4.237 g
and \[2.5\,c{{m}^{3}}\], respectively. The density of the material of the body
in correct significant figure is
(a) \[1.6048\,g\,c{{m}^{-3}}\] (b) \[1.69\,g\,c{{m}^{-3}}\]
(c) \[1.7\,g\,c{{m}^{-3}}\] (d)
\[1.695\,g\,c{{m}^{-3}}\]
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question_answer35)
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
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question_answer36)
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) \[164\pm 3\,c{{m}^{2}}\] (b)
\[163.62\pm 2.6\,\,c{{m}^{2}}\]
(c) \[163.6\pm 2.6\,c{{m}^{2}}\] (d)163.62
± 3 cm2
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question_answer37)
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.
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question_answer38)
Measure of two quantities along with the
precision of respective measuring instrument is
\[A=2.5\,m\,{{s}^{-1}}\pm
0.5\,m\,{{s}^{-1}}\]
\[B=0.10\,s\pm 0.01\,s\]
The value of A B will be
(a) \[(0.25\pm 0.08)\,m\] (b) \[(0.25\pm
0.5)\,m\]
(c) \[(0.25\pm 0.05)\,m\] (d)
\[(0.25\pm 0.135)\,m\]
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question_answer39)
You measure two quantities as \[A=1.0\,m\pm
0.5\,m,\] \[B=2.0\,m\pm 0.2\,m\].
We should report correct value for \[\sqrt{AB}\]
as:
(a) \[1.4\text{ }m\pm 0.4\text{ }m\] (b)
\[1.41\text{ }m\pm 0.15\text{ }m\]
(c) \[1.4\text{ }m\pm 0.3\text{ }m\] (d)
\[1.4\text{ }m\pm 0.2\text{ }m\]
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question_answer40)
Which of the following measurement
is most precise?
(a) 5.00 mm (b) 5.00
cm
(c) 5.00 m (d)
5.4 cm
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question_answer41)
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
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question_answer42)
Young?s modulus of steel is \[1.9\times
{{10}^{11}}N/{{m}^{2}}\]. When expressed in CGS units of dynes/\[c{{m}^{2}}\],
it will be equal to \[(1N={{10}^{5}}\,\,\text{dyne},\text{
}1{{m}^{2}}=104\text{ }c{{m}^{2}})\]
(a) \[1.9\times {{10}^{10}}\] (b)
\[1.9\times {{10}^{11}}\]
(c) \[1.9\times {{10}^{12}}\] (d)
\[1.9\times {{10}^{13}}\]
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question_answer43)
If momentum (P), area (A) and time (T) are
taken to be fundamental quantities, then energy has the dimensional formula
(a) \[({{P}^{1}}\,{{A}^{-1}}\,{{T}^{1}})\] (b)
\[({{P}^{2}}\,{{A}^{1}}\,{{T}^{1}})\]
(c) \[({{P}^{1}}\,{{A}^{-1/2}}\,{{T}^{1}})\]
\[({{P}^{1}}\,{{A}^{1/2}}\,{{T}^{-1}})\]
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question_answer44)
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) \[y=a\sin \,2\,\,\pi t/T\]
(b) \[y=a\sin \,\upsilon t\]
(c) \[y=a\sqrt{2}\left( \sin \frac{2\pi
t}{T}-\cos \frac{2\pi t}{T} \right)\]
(d) \[y=a\sqrt{2}\left( \sin \frac{2\pi
t}{T}-\cos \frac{2\pi t}{T} \right)\]
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question_answer45)
If P, Q, R are physical quantities, having
different dimensions, which of the following combinations can never be a
meaningful quantity?
(a) \[(P-Q)/R\] (b) \[PQ-R\]
(c) \[PQ/R\] (d) \[(PR-{{Q}^{2}})/R\]
(e) \[(R+Q)/P\]
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question_answer46)
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
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question_answer47)
If Planck?s constant (h) and speed
of light in vacuum (c) are taken as two fundamental quantities, which
one 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 \[({{m}_{e}})\]
(b) Universal gravitational constant (G)
(c) Charge of electron (e)
(d) Mass of proton \[({{m}_{p}})\]
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question_answer48)
Which of the following ratios express
pressure?
(a) Force/Area (b)
Energy/Volume
(c) Energy/Area (d)
Force/ Volume
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question_answer49)
Which of the following are not a unit of
time?
(a) Second (b)
Parsec
(c) Year (d)
Light Year
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question_answer50)
Why do we have different units for the same
physical quantity?
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question_answer51)
The radius of atom is of the order of \[1\,\overset{\text{o}}{\mathop{\text{A}}}\,\]
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?
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question_answer52)
Name the device used for measuring the mass
of atoms and molecules.
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question_answer53)
Express unified atomic mass unit in
kg.
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question_answer54)
A function \[f(\theta )\] is defined as:
\[=f(\theta )1-\theta +\frac{{{\theta
}^{2}}}{2!}-\frac{{{\theta }^{3}}}{3!}+\frac{{{\theta }^{4}}}{4!}L\]
Why is it necessary for q to be a dimensionless quantity?
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question_answer55)
Why length, mass and time are chosen as
base quantities is mechanics?
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question_answer56)
(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 \[{{({\scriptstyle{}^{1}/{}_{2}})}^{o}}\]
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.
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question_answer57)
Which of the following time measuring
devices is most precise?
(a) A wall clock. (b)A stop
watch.
(c) A digital watch (d) An
atomic clock.
Give reason for your answer
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question_answer58)
The distance of a galaxy is of the order of
\[{{10}^{25}}\,m\]. Calculate the order of magnitude of time taken by light to
reach us from the galaxy.
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question_answer59)
The vernier scale of a traveling 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.
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question_answer60)
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 the moon.
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question_answer61)
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?
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question_answer62)
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 unit.
(d) a constant which has no unit.
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question_answer63)
Calculate the length of the arc of a circle
of radius 31.0 cm which subtends an angle of \[\frac{\pi }{6}\] at the centre.
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question_answer64)
Calculate the solid angle subtended by the
periphery of an area of \[1\,c{{m}^{2}}\] at a point situated symmetrically at
a distance of 5 cm from the area.
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question_answer65)
The displacement of a progressive wave is
represented \[y=A\sin (wt-k\chi ),\] where \[x\] is distance and t is
time. Write the dimensional formula of (i) w and (ii) k.
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question_answer66)
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 measurement? What is the accuracy of the measurement?
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question_answer67)
A new system of units is proposed in which
unit of mass is a kg, unit of length b m and unit of time \[\gamma s\]. How
much will 5J measure in this new system?
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question_answer68)
The volume of a liquid flowing out per
second of a pipe of length \[l\] and radious \[r\] is written by a student as \[V=\frac{\pi
}{8}\frac{{{\Pr }^{4}}}{\eta l}\]
Where P is the pressure difference between
the two ends of the pipe and n is coefficient of viscosity of the liquid having
dimensional formula \[M{{L}^{-1}}{{T}^{-1}}\].
Check whether the equation is dimensionally
correct.
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question_answer69)
A physical quantity \[X\] is related to
four measureable quantities a, b, c and d as
follows:
\[X={{a}^{2}}{{b}^{3}}{{c}^{{5}/{2}\;}}{{d}^{-2}}\]
The percentage error in the measurement of a,
b, c and d are 1%, 2% 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.
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question_answer70)
In the expression \[P=\text{
}E{{l}^{2}}\,\,{{m}^{-5}}{{G}^{-2}},\text{ }m,\text{ }l\] and G denote energy,
mass, angular momentum and gravitational constant, respectively. Show that P is
a dimensionless quantity.
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question_answer71)
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.
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question_answer72)
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 \[T=\,\frac{k}{R}\,\sqrt{\frac{{{r}^{3}}}{g}},\]
where k is a dimensionless constant and g is acceleration due to
gravity.
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question_answer73)
In an experiment to estimate the size of a
molecule of oleic acid 1 mL of oleic acid is dissolved in 19 mL of alcohol.
Then 1 mL of this solution is diluted to 20 mL 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 and its diameter is
measured. Knowing the valued of the drop and are of 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 power?
(c) What would be the volume of oleic acid
in each mL of solution prepare?
(d) How will you calculate the volume of n drops
of this solution of oleic acid?
(e) What will be the volume of oleic acid
in one drop of this solution?
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question_answer74)
(a) How many astronomical units (A.U.) make
1 parsec?
(b) Consider a sunlike 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 \[{{(1/2)}^{o}}\] from
the earth. Due to atmospheric fluctuations, eye can?t resolve objects smaller
than 1 arc minute.
(c) Mars has approximately half of the
earth?s diameter.
When it is closest to the earth it is a
about 1/2 A.U. from the earth. Calculate what size it will appear when seen
through the same telescope.
(Comment: This is to illustrate why
a telescope can magnify plants but not star.)
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question_answer75)
Einstein?s mass ? energy relation emerging
out of his famous theory of relativity relates mass \[(m)\] to energy \[(E)\] as
\[E=m{{c}^{2}},\] where c is speed of light in vacuum. At the nuclear
lever, 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 atomic mass unit \[(u)\] where \[1u=1.67\times
{{10}^{-27}}\,kg\].
(a)
Show that the energy equivalent of 1 u is 931.5 MeV.
(b)
A student writes the relation as 1 u is 931.5 MeV. The teacher points out that
the relation is dimensionally incorrect.
Write the correct relation.
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