Unit and Dimension Exercise

UNIT AND DIMENSION Exercise - I UNSOLVED PROBLEMS Q.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 2.0cm, 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 1s (d) The relative density of lead is 11.3. Its density is ..... g cm–3 or ..... kg m–3. Q.2 Fill in blanks by suitable conversion of units. (a) 1 kg m2 s –2 = .... g cm2 s –2 (b) 1 m = ..... 1 y (c) 3 m s–2 = ..... km h–2 (d) G = 6.67 × 10–11 N m2 (kg)–2 = .... (cm)3 s –2 g–1. Q.3 A calorie is a unit of heat or energy and it equals about 4.2 J where 1 J = 1 kg m2 s–2 . 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 4.2 –1  –2 2 in terms of the new units. Q.4 Explain this statement clearly : "To call a dimensional quantity ''large' or 'small' is meaningless without specifying a standard for comparison ". 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 sped of sound is much smaller than the speed of light. Q.5 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 ? Q.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 with within a wavelength of light ? Q.7 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 ? Q.8 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 of 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 daimeter of a thin brass rod is to be measured by vernier callipers. Why is a set of 100 measurements of the diameter expected yield a more reliable estimate than a set of 5 measurements only Q.9 The photograph of a house occupation area of 1.75 cm2 on a 35 mm slide. The slide is projected onto a screen, and the area of the house on the screen is 1.55 m 2. What is the linear magnification of the projector-screen arrangement. Q.10 State the number of significant figures in the following : (a) 0.007 m 2 (b) 2.64 × 1024 kg –3 (c) 0.2370 g cm (d) 6.320 J (e) 6.032 N m–2 (f) 0.0006032 m2 Q.11 The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m, and 2.01 cm respectively. Given the area and volume of the sheet to correct significant figures. Q.12 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 (1) the total mass of the box, (b) the difference in the masses of the pieces to correct significant figures Q.13 A physical quantity related to four observables a, b, c and d as follows : P = a3 b2 / ( cd ) 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 ? Q.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  t/T (b) y = a sin vt (c) y = (a/T) sin t/a (d) y = (a 2 ) (sin 2  t/T + cos 2  t/T) UNIT AND DIMENSION (a) a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds. Q.15 A famous relation in physics relates 'moving mass' m to the 'rest mass' m0 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 forget where to put the m0 constant c. He writes : m = G u e s s (1  v 2 )1/ 2 where to put the missing c. Q.16 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 v and checks that the relation has a correct limit : as v  0,(  0, as expected. (We are assuming there is not 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. Q.17 It is claimed that two cesium 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 cesium clock in measuring a time-interval of 1 s ? Q.18 The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å : 1 Å = 10–10 m. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m 3 of a mole of hydrogen atoms ? Q.19 O ne mo le o f an ide al gas at stan da rd 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 ? Q.20 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 density of sodium in its crystalline phase : 970 kg m–3. Are the two densities of the same order of magnitude ? If so, why ? Q.21 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 = r0 A1/3 where r is the radius of the nucleus, A its mass number, and r0 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 2.20. Q.22 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 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 distance object seem to move with you.). Q.23 The principle of 'parallax' in section 2.3.1 is used in the determination of distances of very distant stars. The base line AB is the line joining the Earth's two locations six months apart in its orbit around the Sun. That is, the base line is about the diameter of the Earth's orbit  3 × 1011 m. However, even the nearest stars are so distant that with such a long base line, 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 Q.24 The nearest star tour 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 ? Q.25 A L A S E R i s a s o u r c e o f v e r y i n t e ns e , 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 2.46 s to return after reflection at the Moon's surface. How much is the radius of the lunar orbit around the Earth ? Q.26 A SONAR (sound navigation and ranging) used 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). UNIT AND DIMENSION Q.27 T h e f a r t h e s t o b j e c t s i n o u r U n i v e r s e 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 quasar from which light takes 3.0 billion years to reach us ? Q.28 Precise measurements of physical quantities are a need of modern times. For example, to ascertain the speed of an enemy fighter plane, one must have an accurate method to find its positions at closely separated instants of time. Only then we can hope to shell it by an antiaircraft gun. This was the actual motivation behind the discovery of radar in World War II. Think of different examples in modern science where precision measurements of length, time, mass etc. are needed Also, wherever you can, give a quantitative idea of the precision needed. Q.29 Just as precise measurements are necessary in science, it is equally important to be able to m a k e r o u g h e s t i m a t e s o f q u an t i t i e s u s i n g 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 stands of hair on your head (e) the number of air molecules in your classroom. Q.30 The Sun is a hot plasma (ionized matter) with its inner core at a temperature exceeding 107 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 × 1030 kg, radius of the sun = 7.0 × 108 m. Q.31 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. 1 Q.32 Assuming that the orbit of the planet Mercury around the Sun to be a circle, copernicus determined the orbital radius to be 0.38 AU. From this, determine the angle of maximum elongation for Mercury and its distance from the Earth when the elongation is maximum. Q.33 Suppose there existed a planet that went around the Sun twice as fast as the Earth. What would be its orbital size as compared to that of the Earth ? Q.34 Io, one of the satellites of Jupiter has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 108 m. Show that the mass of Jupiter is about one-thousandth that the Sun. Q.35 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 the fact and from the information you can gather from examples 2.1 and 2.2, determine the approximate diameter of the moon. Q.36 Suppose the Sunsharnk from its present size so that its radius is halved. What would be the change in tis gravitational potential energy (Calculate the actual number in joules). Q.37 Let us assume that our galaxy consists of 2.5 × 1011 stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution ? Take the diameter of the Milky Way to be 105 ly. Q.38 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 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 ?