## Lecture 12 Chapter 2 Units and Measurements

Exercise 2.16 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 m3 of a mole of hydrogen atoms ?

Exercise 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 ?

Exercise 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).

Exercise 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 × 1011 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 ?

Exercise 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 ?

Exercise 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.

Exercise 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.

Exercise 2.23 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.

Exercise 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.

## Lecture 11 Chapter 2 Units & Measurement Class 11

Exercise 2.7 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 ?

Exercise 2.8 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 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 callipers. Why is a set of 100 measurements of the diameter expected to yield a more reliable estimate than a set of 5 measurements only ?

Exercise 2.9 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.

Exercise 2.10 State the number of significant figures in the following :
(a) 0.007 m2
(b) 2.64 × 1024 kg
(c) 0.2370 g cm–3
(d) 6.320 J
(e) 6.032 N m–2
(f) 0.0006032 m2

Exercise 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 significant figures.

Exercise 2.12 The mass of a box measured by a grocer’s balance is 2.30 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, (b) the difference in the masses of the pieces to correct significant figures ?

Exercise 2.13 A physical quantity P is related to four observables a, b, c and d as follows : $P = \frac{a^3 b^2}{\sqrt{c} d}$ 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 calculated using the above relation turns out to be 3.763, to what value should you round off the result ?

Exercise 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 2π 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.

Exercise 2.15 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 forgets where to put the constant c. He writes : $m =\frac{m_0}{(1-v^2)^{\frac{1}{2}}}$ Guess where to put the missing c.

## Lecture 10 Exercise Chapter 2 Class 11 Physics

Note : In stating numerical answers, take care of significant figures.

Exercise 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

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

Exercise 2.3 A calorie is a unit of heat (energy in transit) 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 unit of length equals β m, the unit of time is γ s.  Show that a calorie has a magnitude 4.2 α-1β-2γ2 in terms of the new units.

Exercise 2.4 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 speed of sound is much smaller than the speed of light.

Exercise 2.5 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 ?

Exercise 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 ?

## Chapter 2 Units and Measurement Lecture 9

Example 2.9 The resistance R = V/I where V = (100 ± 5)V and I = (10 ± 0.2)A. Find the percentage error in R.

Example 2.10 Two resistors of resistances R1 = 100±3 ohm and R2 = 200 ± 4 ohm are connected
(
a) in series,
(b) in parallel.
Find the equivalent resistance of the
(a) series combination,
(b) parallel combination.
Use for (a) the relation R = R1 + R2 and for (b) $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$ and $\frac{\Delta R}{R`^{2}} = \frac{\Delta R_1}{{R_1}^2} + \frac{\Delta R_2}{{R_2}^2}$

Example 2.11 Find the relative error in Z, if $Z = \frac{A^4 B^{\frac{1}{3}}}{C D^{\frac{3}{2}}}$.

Example 2.12 The period of oscillation of a simple pendulum is $T = 2 \pi \sqrt{\frac{L}{g}}$
Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of g ?​

## Lecture 8 Chapter 2 Units and Measurement

Example 2.6 Two clocks are being tested against a standard clock located in a national laboratory. At 12:00:00 noon by the standard clock, the readings of the two clocks are :

 MondayTuesdayWednesday Thursday FridaySaturday Sunday Clock 112:00:0512:01:15 11:59:08 12:01:50 11:59:15 12:01:30 12:01:19 Clock 2 10:15:0610:14:59 10:15:18 10:15:07 10:14:53 10:15:24 10:15:11

If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer ?

# Errors in Sum, Difference, Product, Divide & Power

Example 2.8 The temperatures of two bodies measured by a thermometer are t1 = 200 C ± 0.50 C and t2 = 500C ± 0.50C. Calculate the temperature difference and the error theirin.

# Measurement Errors & Types of Errors

Example 2.7 We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s and 2.80 s. Calculate the absolute errors, relative error or percentage error.

# Parallax Method and Range of Length

Example 2.2 A man wishes to estimate the distance of a nearby tower from him. He stands at a point A in front of the tower C and spots a very distant object O in line with AC. He then walks perpendicular to AC up to B, a distance of 100 m, and looks at O and C again. Since O is very distant, the direction BO  is practically the same as AO; but he finds the line of sight of C shifted from the original line of sight by an angle θ = 400 (θ is known as ‘parallax’) estimate the distance of the tower C from his original position A.

Example 2.3 The moon is observed from two diametrically opposite points A and B on Earth. The angle θ subtended at the moon by the two directions of observation is 1o54′ . Given the diameter of the Earth to be about 1.276 ×107 m, compute the
distance of the moon from the Earth.

Example 2.4 The Sun’s angular diameter is measured to be 1920′′. The distance D of the Sun from the Earth is 1.496 × 1011 m. What is the diameter of the Sun ?