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Q 2.33: 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 (~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 constants?

The above equation consisting of basic constants of atomic physics and the gravitational constant G has the dimension of time. e = charge of Electron =  = absolute permittivity =   = Mass of the Proton =  = Mass of the Electron = c = Speed of light in vacuum = G = Universal Gravitational constant = Considering T as the age of the universe and putting the values of the constants, we...

Q 2.32: It is a well known fact that during a total solar eclipse the disk of the moon almost completely covers the disk of the Sun. From this fact and from the information you can gather from examples 2.3 and 2.4, determine the approximate diameter of the moon.

From examples 2.3 and 2.4, we have, Diameter of the Earth = Distance between the Moon and the Earth, = Distance between the Sun and the Earth,  = Diameter of the Sun,  = Let, Diameter of the Moon be     Now, During Solar eclipse, the angle subtended by Sun's diameter on Earth = angle subtended by moon's diameter                 Therefore, the diameter of the moon =   

Q 2.31: The farthest objects in our Universe discovered by modern astronomers are so distant that 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 satisfactorily explained. What is the distance in km of a quasar from which light takes 3.0 billion years to reach us?

Let the distance of a quasar from Earth be D km. We know , Speed of light =  And, time taken by light to reach us , t = 3.0 billion years =   D =Speed of Light x t        =         = 

Q 2.30: 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 77.0 s. What is the distance of the enemy submarine? (Speed of sound in water = 1450\ m s^{-1}).

Given, 77.0 s is the total time between the generation of a probe wave and the reception of its echo after reflection.  Time taken by sound to reach the enemy submarine = Half of the total time =  The distance of enemy ship = Speed of sound x Time taken to reach the submarine = 

Q 2.29: 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 ?

2.56 s is the total time taken by the LASER to reach Moon and again back Earth.  Time taken by LASER to reach Moon =  We know, Speed of light   The radius of the lunar orbit around the Earth = distance between Earth and Moon =  = Speed of light x Time taken by laser one-way = 

Q 2.28: The unit of length convenient on the nuclear scale is a fermi : 1 f = 10^{-15} m. Nuclear sizes obey roughly the following empirical relation :r = r_{o} A^{1/3} where r is the radius of the nucleus, A its mass number, and r_{o} is a constant equal to about, 1.2 f. Show that the rule implies that nuclear mass density is nearly constant for different nuclei. Estimate the mass density of sodium nucleus. Compare it with the average mass density of a sodium atom obtained in Exercise. 2.27.

The equation for the radius of the nucleus is given by, The volume of the nucleus using the above relation, We know, Mass = Mass number× Mass of single Nucleus = (given)  Nuclear mass Density = Mass of nucleus/ Volume of nucleus =    The derived density formula contains only one variable, and is independent of mass number A. Since  is constant, hence nuclear mass density is nearly constant...

Q 2.27: Estimate the average mass density of a sodium atom assuming its size to be about 2.5 Å. (Use the known values of Avogadro’s number and the atomic mass of sodium). Compare it with the mass density of sodium in its crystalline phase : 970 \ kg\ m^{-3}. Are the two densities of the same order of magnitude? If so, why?

Estimate the average mass density of a sodium atom assuming its size to be about 2.5 Å. (Use the known values of Avogadro’s number and the atomic mass of sodium). Compare it with the mass density of sodium in its crystalline phase: Are the two densities of the same order of magnitude? If so, why? Given, Diameter of sodium = 2.5  Radius, r =   Now, Volume occupied by each atom   We know, One...

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

In terms of seconds, 100 years =  Given, Difference between the two clocks after 100 years = 0.02 s  In 1 s,  the time difference      Accuracy in measuring a time interval of 1 s =   Accuracy of 1 part in   

Q 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.

Given, The distance of Jupiter, D = Angular diameter,          () Let diameter of Jupiter = d km    

Q 2.23: The Sun is a hot plasma (ionized matter) with its inner core at a temperature exceeding 10^7 K, 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 = 2.0 \times 10^{30} kg, radius of the Sun = 7.0 \times 10^8 m

Given, Mass of the Sun, m = The radius of the Sun, r =  Volume V =  =    Density = Mass/Volume  =  Therefore, the density of the sun is in the range of solids and liquids and not gases. This high density arises due to inward gravitational attraction on outer layers due to inner layers of the Sun. (Imagine layers and layers of gases stacking up like a pile!)

Q 2.22: Just as precise measurements are necessary in science, it is equally important to be able to make rough estimates 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.

(a) Height of water column during monsoon is recorded as 215 cm. H = 215 cm = 2.15 m Area of the country, Volume of water column, V = AH  V = Mass of the rain-bearing clouds over India during the Monsoon, m = Volume x Density  m =  =         (Density of water = 103 kg m-3 ) b) Consider a large solid cube of known density having a density less than water. Measure the volume of water displaced...

Q 2.21: 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.

The statement "Precise measurements of physical quantities are a need of science" is indeed true. In Space explorations, very precise measurement of time in microsecond range is needed. In determining the half-life of radioactive material, very precise value of mass of nuclear particles is required. Similarly, in Spectroscopy precise value of the length in Angstroms is required.

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

Given, Distance of the star from the solar system = 4.29 ly (light years) 1 light year is the distance travelled by light in one year. (Note: Light year is a measurement of distance and not time!) (a)  4.29 ly = We know, 1 parsec = 4.29 ly = = 1.32 parsec Now, (b)  & d = ; D =  Also, We know

Q 2.25: A man walking briskly in rain with speed v must slant his umbrella forward making an angle \Theta with the vertical. A student derives the following relation between \Theta and v: tan\Theta = v and checks that the relation has a correct limit: as v \rightarrow 0, \Theta \rightarrow 0, as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not, guess the correct relation.

The derived formula  is dimensionally incorrect. We know, Trigonometric functions are dimensionless.  Hence , [ ] =  and [v] = .  To make it dimensionally correct, we can divide v by (where  is the speed of rain) Thus, L.H.S and R.H.S are both dimensionless and hence dimensionally satisfied. The new formula is : 

2.19 The principle of ‘parallax’ in section 2.3.1 is used in the determination of distances of very distant stars. The baseline AB is the line joining the Earth’s two locations six months apart in its orbit around the Sun. That is, the baseline is about the diameter of the Earth’s orbit ≈ 3 \times 10^{11} m. However, even the nearest stars are so distant that with such a long baseline, they show parallax only of the order of  1{}'' (second) of arc or so. 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 a parsec in terms of metres?

The diameter of Earth’s orbit =  The radius of Earth’s orbit, r = Let the distance parallax angle be = . Let the distance between earth and star be R. (Parsec is the distance at which average radius of earth’s orbit subtends an angle of .) We have         (Analogous to a circle, R here is the radius, r is the arc length and  is the angle covered ! ) Hence, 1 parsec .

2.18 Explain this common observation clearly : If you look out of the window of a fast moving train, the nearby trees, houses etc. seem to move rapidly in a direction opposite to 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, these distant objects seem to move with you).

Our eyes detect angular velocity, not absolute velocity. An object far away makes a lesser angle than an object which is close. That's why the moon (which is so far away!) does not seem to move at all angularly and thus seems to follow you while driving.  In other words, while in a moving train, or for that matter in any moving vehicle, a nearby object moves in the opposite direction while the...

Q 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 of hydrogen? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratio so large?

Radius of hydrogen atom = 0.5 = 0.5 x 10-10 m  (Size here refers to Diameter!) Volume occupied by the hydrogen atom=  =  = 1 mole of hydrogen contains 6.023 x 1023 hydrogen atoms. Volume of 1 mole of hydrogen atom = 6.023 x 1023 x 0.524 x 10-30 = 3.16 x 10-7 m3 The molar volume is  times greater than the atomic volume. Hence, intermolecular separation in gas is much larger than the size of...

Q 2.16: The unit of length convenient on the atomic scale is known as an angstrom and is denoted by \AA : 1\ \AA = 10^{-10} m. The size of a hydrogen atom is about 0.5 \AA. What is the total atomic volume in m^3 of a mole of hydrogen atoms?

Radius of an Hydrogen atom = 0.5 = 0.5 x 10-10 m Volume =  =  = 1 hydrogen mole contains  hydrogen atoms. The volume of 1 mole of hydrogen atom = = .

Q 2.15: A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ m_{o} of a particle in terms of its speed v and the speed of light, c. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes : m = \frac{m_{o}}{(1-v^2)^{1/2}}

Guess where to put the missing c.

The relation given is  Divide both sides by ;   L.H.S becomes  which is dimensionless. Hence, R.H.S must be dimensionless too. (After Dividing by  !)  can be dimensionless only when  Therefore, the dimensional equation is 

2.14 A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion :

(a) y = a sin 2\pi t/T

(b) y = a sin vt

(c) y = (a/T) sin\ t/a

(d) y = ( a\sqrt{2}) (sin 2\pi t / T+ cos 2\pi t / T )

(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.

Ground rules: [y] = L  () [a] = L  [v] = [t/T] is Dimenionless. (a) The dimensions on both sides are equal, the formula is dimensionally correct. (b)  [vt] = ()(T) = L ( is not dimensionless)         The formula is dimensionally incorrect (c) [a/T] = (L)/(T)     It is dimensionally incorrect, as the dimensions on both sides are not equal. (d) The dimensions on both sides are equal, the...