physics class 11 chapter 2 Units And Measurements

Note : In stating numerical answers, take care of significant figures.
2.1 Fill in the blanks(a) The volume of a cube of side 1 cm is equal to .....m3(b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to...(mm)2(c) A vehicle moving with a speed of 18 km h–1 covers....m in 1 s(d) The relative density of lead is 11.3. Its density is ....g cm–3 or ....kg m–3.


2.2 Fill in the blanks by suitable conversion of units(a) 1 kg m2 s–2 = ....g cm2 s–2(b) 1 m = ..... ly(c) 3.0 m s–2 = .... km h–2(d) G = 6.67 × 10–11 N m2 (kg)–2 = .... (cm)3 s–2 g–1.

2.3 A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m2 s–2.Suppose we employ a system of units in which the unit of mass equals α kg, the unitof length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2α–1 β –2 γ 2 in terms of the new units.

2.4 Explain this statement clearly :“To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying astandard for comparison”. In view of this, reframe the following statements wherevernecessary :(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.

2.5 A new unit of length is chosen such that the speed of light in vacuum is unity. Whatis the distance between the Sun and the Earth in terms of the new unit if light takes8 min and 20 s to cover this distance ?

2.6 Which of the following is the most precise device for measuring length :(a) a vernier callipers 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 light ?

2.7 A student measures the thickness of a human hair by looking at it through amicroscope of magnification 100. He makes 20 observations and finds that the averagewidth of the hair in the field of view of the microscope is 3.5 mm. What is theestimate on the thickness of hair ?

2.8 Answer the following :(a)You are given a thread and a metre scale. How will you estimate the diameter ofthe thread ?(b)A screw gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Doyou think it is possible to increase the accuracy of the screw gauge arbitrarily byincreasing the number of divisions on the circular scale ?(c) The mean diameter of a thin brass rod is to be measured by vernier callipers. Whyis a set of 100 measurements of the diameter expected to yield a more reliableestimate than a set of 5 measurements only ?

2.9 The photograph of a house occupies an area of 1.75 cm2 on a 35 mm slide. The slideis projected on to a screen, and the area of the house on the screen is 1.55 m2. Whatis the linear magnification of the projector-screen arrangement.

2.10 State the number of significant figures in the following :(a) 0.007 m2(b) 2.64 × 1024 kg(d) 6.320 J(e) 6.032 N m–2(f) 0.0006032 m2

2.11 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 significantfigures.

2.12 The mass of a box measured by a grocer’s balance is 2.300 kg. Two gold pieces ofmasses 20.15 g and 20.17 g are added to the box. What is (a) the total mass of thebox, (b) the difference in the masses of the pieces to correct significant figures ?

2.13 A physical quantity P is related to four observables a, b, c and d as follows :P = a3b2/ ( c d )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 ? If the value of P calculatedusing the above relation turns out to be 3.763, to what value should you round offthe result ?

2.14 A book with many printing errors contains four different formulas for the displacementy of a particle undergoing a certain periodic motion :(a) y = a sin 2π t/T(b) y = a sin vt(c) y = (a/T) sin t/a(d) y = (a 2) (sin 2πt / T + cos 2πt / T )(a = maximum displacement of the particle, v = speed of the particle. T = time-periodof motion). Rule out the wrong formulas on dimensional grounds.

2.15 A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ mo of aparticle in terms of its speed v and the speed of light, c. (This relation first arose asa consequence of special relativity due to Albert Einstein). A boy recalls the relationalmost correctly but forgets where to put the constant c. He writes :( ) mm1 v= 0− 2 1/2 .Guess where to put the missing c.

2.16 The unit of length convenient on the atomic scale is known as an angstrom and isdenoted by Å: 1 Å = 10–10 m. The size of a hydrogen atom is about 0.5 Å. What is thetotal atomic volume in m3 of a mole of hydrogen atoms ?

2.17 One mole of an ideal gas at standard temperature and pressure occupies 22.4 L(molar volume). What is the ratio of molar volume to the atomic volume of a mole ofhydrogen ? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratioso large ?

2.18 Explain this common observation clearly : If you look out of the window of a fastmoving train, the nearby trees, houses etc. seem to move rapidly in a direction oppositeto the train’s motion, but the distant objects (hill tops, the Moon, the stars etc.)seem to be stationary. (In fact, since you are aware that you are moving, thesedistant objects seem to move with you).

2.19 The principle of ‘parallax’ in section 2.3.1 is used in the determination of distancesof very distant stars. The baseline AB is the line joining the Earth’s two locations sixmonths apart in its orbit around the Sun. That is, the baseline is about the diameterof the Earth’s orbit ≈ 3 × 1011m. However, even the nearest stars are so distant thatwith such a long baseline, they show parallax only of the order of 1” (second) of arcor so. A parsec is a convenient unit of length on the astronomical scale. It is thedistance of an object that will show a parallax of 1” (second) of arc from oppositeends of a baseline equal to the distance from the Earth to the Sun. How much is aparsec in terms of metres ?

2.20 The nearest star to our solar system is 4.29 light years away. How much is thisdistance in terms of parsecs? How much parallax would this star (named AlphaCentauri) show when viewed from two locations of the Earth six months apart in itsorbit around the Sun ?

2.21 Precise measurements of physical quantities are a need of science. For example, toascertain the speed of an aircraft, one must have an accurate method to find itspositions at closely separated instants of time. This was the actual motivation behindthe discovery of radar in World War II. Think of different examples in modern sciencewhere precise measurements of length, time, mass etc. are needed. Also, whereveryou can, give a quantitative idea of the precision needed.

 2.22 Just as precise measurements are necessary in science, it is equally important to beable to make rough estimates of quantities using rudimentary ideas and commonobservations. Think of ways by which you can estimate the following (where anestimate 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.

2.23 The Sun is a hot plasma (ionized matter) with its inner core at a temperature exceeding107 K, and its outer surface at a temperature of about 6000 K. At these hightemperatures, no substance remains in a solid or liquid phase. In what range do youexpect the mass density of the Sun to be, in the range of densities of solids andliquids or gases ? Check if your guess is correct from the following data : mass of theSun = 2.0 × 1030 kg, radius of the Sun = 7.0 × 108 m.

2.24 When the planet Jupiter is at a distance of 824.7 million kilometers from the Earth,its angular diameter is measured to be 35.72” of arc.

Calculate the diameter of Jupiter. Additional Exercises 

2.25 A man walking briskly in rain with speed v must slant his umbrella forward makingan angle θ with the vertical. A student derives the following relation between θ andv : tan θ = v and checks that the relation has a correct limit: as v → 0, θ →0, asexpected. (We are assuming there is no strong wind and that the rain falls verticallyfor a stationary man). Do you think this relation can be correct ? If not, guess thecorrect relation.

2.26 It is claimed that two cesium clocks, if allowed to run for 100 years, free from anydisturbance, may differ by only about 0.02 s. What does this imply for the accuracyof the standard cesium clock in measuring a time-interval of 1 s ?

2.27 Estimate the average mass density of a sodium atom assuming its size to be about2.5 Å. (Use the known values of Avogadro’s number and the atomic mass of sodium).Compare it with the density of sodium in its crystalline phase : 970 kg m–3. Are thetwo densities of the same order of magnitude ? If so, why ?

2.28 The unit of length convenient on the nuclear scale is a fermi : 1 f = 10–15 m. Nuclearsizes obey roughly the following empirical relation :r = r0 A1/3where r is the radius of the nucleus, A its mass number, and ro is a constant equal toabout, 1.2 f. Show that the rule implies that nuclear mass density is nearly constantfor different nuclei. Estimate the mass density of sodium nucleus. Compare it withthe average mass density of a sodium atom obtained in Exercise. 2.27.

2.29 A LASER is a source of very intense, monochromatic, and unidirectional beam oflight. 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 preciselyusing 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 lunarorbit around the Earth ?

 2.30 A SONAR (sound navigation and ranging) uses ultrasonic waves to detect and locateobjects under water. In a submarine equipped with a SONAR the time delay betweengeneration of a probe wave and the reception of its echo after reflection from anenemy submarine is found to be 77.0 s. What is the distance of the enemy submarine?(Speed of sound in water = 1450 m s–1).

2.31 The farthest objects in our Universe discovered by modern astronomers are so distantthat light emitted by them takes billions of years to reach the Earth. These objects(known as quasars) have many puzzling features, which have not yet been satisfactorilyexplained. What is the distance in km of a quasar from which light takes 3.0 billionyears to reach us ? 

2.32 It is a well known fact that during a total solar eclipse the disk of the moon almostcompletely covers the disk of the Sun. From this fact and from the information youcan gather from examples 2.3 and 2.4, determine the approximate diameter of themoon.

2.33 A great physicist of this century (P.A.M. Dirac) loved playing with numerical values ofFundamental constants of nature. This led him to an interesting observation. Diracfound that from the basic constants of atomic physics (c, e, mass of electron, mass ofproton) and the gravitational constant G, he could arrive at a number with thedimension of time. Further, it was a very large number, its magnitude being close tothe present estimate on the age of the universe (~15 billion years). From the table offundamental 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 ofthe universe were significant, what would this imply for the constancy of fundamentalconstants ?