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Units and Measurements EXERCISES Solutions
- Q2.1(a): Fill in the blanks (a) The volume of a cube of side 1 cm is equal to .....m$^3$
- Q2.1(a): Fill in the blanks The volume of a cube of side 1 cm is equal to .....(m^{3})
- Q2.1(b): Fill in the blanks The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to ...(mm)^{2}
- Q2.1(b): Fill in the blanks (b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to ...(mm)$^2$
- Q2.1(c): Fill in the blanks (c) A vehicle moving with a speed of 18 km h$^{–1}$ covers....m in 1 s
- Q2.1(c): Fill in the blanks A vehicle moving with a speed of 18 km (h^{-1}) covers....m in 1 s
- Q2.1(d): Fill in the blanks (d) The relative density of lead is 11.3. Its density is ....g cm$^{–3}$ or ....kg m$^{–3}$.
- Q2.1(d): Fill in the blanks The relative density of lead is 11.3. Its density is ....(g cm^{-3}) or ....(kg m^{-3}).
- Q2.10(a): State the number of significant figures in the following : 0.007 (m^{2})
- Q2.10(a): State the number of significant figures in the following : (a) 0.007 m$^2$
- Q2.10(b): State the number of significant figures in the following : (2.64 \times 10^{24}) kg
- Q2.10(b): State the number of significant figures in the following : (b) $2.64 \times 10^{24}$ kg
- Q2.10(c): State the number of significant figures in the following : 0.2370 (g cm^{-3})
- Q2.10(c): State the number of significant figures in the following : (c) 0.2370 g cm$^{–3}$
- Q2.10(d): State the number of significant figures in the following : (d) 6.320 J
- Q2.10(d): State the number of significant figures in the following : 6.320 J
- Q2.10(e): State the number of significant figures in the following : (e) 6.032 N m$^{–2}$
- Q2.10(e): State the number of significant figures in the following : 6.032 (N m^{-2})
- Q2.10(f): State the number of significant figures in the following : (f) 0.0006032 m$^2$
- Q2.10(f): State the number of significant figures in the following : 0.0006032 (m^{2})
- Q2.11: 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.
- Q2.11: 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.
- Q2.12(a): 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 the total mass of the box?
- Q2.12(a): 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,
- Q2.12(b): 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 the difference in the masses of the pieces to correct significant figures ?
- Q2.12(b): 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 (b) the difference in the masses of the pieces to correct significant figures ?
- Q2.13: 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 ?
- Q2.13: 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 ?
- Q2.14(a): A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (y = a \sin \frac{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.
- Q2.14(a): A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (a) $y = a \sin \frac{2\pi t}{T}$ (a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.
- Q2.14(b): A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (y = a \sin vt) ((a) = maximum displacement of the particle, (v) = speed of the particle. (T) = time-period of motion). Rule out the wrong formulas on dimensional grounds.
- Q2.14(b): A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (b) $y = a \sin vt$ (a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.
- Q2.14(c): A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (y = (\frac{a}{T}) \sin \frac{t}{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.
- Q2.14(c): A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (c) $y = (\frac{a}{T}) \sin \frac{t}{a}$ (a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.
- Q2.14(d): A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (d) $y = (\frac{a}{\sqrt{2}}) (\sin \frac{2\pi t}{T} + \cos \frac{2\pi t}{T})$ (a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.
- Q2.14(d): A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (y = (\frac{a}{\sqrt{2}}) (\sin \frac{2\pi t}{T} + \cos \frac{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.
- Q2.15: A famous relation in physics relates ‘moving mass’ $m$ to the ‘rest mass’ $m_0$ 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 : $m = \frac{m_0}{(1 - v^2)^{1/2}}$. Guess where to put the missing $c$.
- Q2.15: A famous relation in physics relates ‘moving mass’ (m) to the ‘rest mass’ (m_{0}) 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})^{1/2}}). Guess where to put the missing (c).
- Q2.16: 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 m$^3$ of a mole of hydrogen atoms ?
- Q2.16: 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 (m^{3}) of a mole of hydrogen atoms ?
- Q2.17: 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 ?
- Q2.17: 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 ?
- Q2.18: 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).
- Q2.18: 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).
- Q2.19: 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 (\approx 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 ?
- Q2.19: 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 $\approx 3 \times 10^{11}$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 ?
- Q2.2(a): Fill in the blanks by suitable conversion of units 1 (kg m^{2} s^{-2}) = ....(g cm^{2} s^{-2})
- Q2.2(a): Fill in the blanks by suitable conversion of units (a) 1 kg m$^2$ s$^{–2}$ = ....g cm$^2$ s$^{–2}$
- Q2.2(b): Fill in the blanks by suitable conversion of units (b) 1 m = ..... ly
- Q2.2(b): Fill in the blanks by suitable conversion of units 1 m = ..... ly
- Q2.2(c): Fill in the blanks by suitable conversion of units 3.0 (m s^{-2}) = .... (km h^{-2})
- Q2.2(c): Fill in the blanks by suitable conversion of units (c) 3.0 m s$^{–2}$ = .... km h$^{–2}$
- Q2.2(d): Fill in the blanks by suitable conversion of units (G = 6.67 \times 10^{-11} N m^{2} (kg)^{-2} = .... (cm)^{3} s^{-2} g^{-1}).
- Q2.2(d): Fill in the blanks by suitable conversion of units (d) G = $6.67 \times 10^{–11}$ N m$^2$ (kg)$^{–2}$ = .... (cm)$^3$ s$^{–2}$ g$^{–1}$.
- Q2.20: 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 ?
- Q2.20: 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 ?
- Q2.21: 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.
- Q2.21: 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.
- Q2.22(a): 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) : the total mass of rain-bearing clouds over India during the Monsoon
- Q2.22(a): 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
- Q2.22(b): 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) : (b) the mass of an elephant
- Q2.22(b): 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) : the mass of an elephant
- Q2.22(c): 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) : (c) the wind speed during a storm
- Q2.22(c): 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) : the wind speed during a storm
- Q2.22(d): 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) : (d) the number of strands of hair on your head
- Q2.22(d): 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) : the number of strands of hair on your head
- Q2.22(e): 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) : (e) the number of air molecules in your classroom.
- Q2.22(e): 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) : the number of air molecules in your classroom.
- Q2.23: 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.
- Q2.23: 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.
- Q2.24: 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.
- Q2.24: 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.
- Q2.3: A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 1 kg m$^2$ s$^{–2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha$ kg, the unit of length equals $\beta$ m, the unit of time is $\gamma$ s. Show that a calorie has a magnitude 4.2 $\alpha^{–1} \beta^{–2} \gamma^2$ in terms of the new units.
- Q2.3: A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where (1J = 1 kg m^{2} s^{-2}). Suppose we employ a system of units in which the unit of mass equals (\alpha) kg, the unit of length equals (\beta) m, the unit of time is (\gamma) s. Show that a calorie has a magnitude (4.2 \alpha^{-1} \beta^{-2} \gamma^{2}) in terms of the new units.
- Q2.4(a): 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
- Q2.4(a): 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 : atoms are very small objects
- Q2.4(b): 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 : (b) a jet plane moves with great speed
- Q2.4(b): 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 jet plane moves with great speed
- Q2.4(c): 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 : (c) the mass of Jupiter is very large
- Q2.4(c): 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 : the mass of Jupiter is very large
- Q2.4(d): 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 : (d) the air inside this room contains a large number of molecules
- Q2.4(d): 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 : the air inside this room contains a large number of molecules
- Q2.4(e): 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 : (e) a proton is much more massive than an electron
- Q2.4(e): 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 proton is much more massive than an electron
- Q2.4(f): 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 : (f) the speed of sound is much smaller than the speed of light.
- Q2.4(f): 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 : the speed of sound is much smaller than the speed of light.
- Q2.5: 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 ?
- Q2.5: 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 ?
- Q2.6: Which of the following is the most precise device for measuring length :
- Q2.6: Which of the following is the most precise device for measuring length :
- Q2.7: 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 ?
- Q2.7: 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 ?
- Q2.8(a): Answer the following : You are given a thread and a metre scale. How will you estimate the diameter of the thread ?
- Q2.8(a): Answer the following : (a)You are given a thread and a metre scale. How will you estimate the diameter of the thread ?
- Q2.8(b): Answer the following : 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 ?
- Q2.8(b): Answer the following : (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 ?
- Q2.8(c): Answer the following : 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 ?
- Q2.8(c): Answer the following : (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 ?
- Q2.9: The photograph of a house occupies an area of 1.75 (cm^{2}) 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 (m^{2}). What is the linear magnification of the projector-screen arrangement.
- Q2.9: The photograph of a house occupies an area of 1.75 cm$^2$ 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 m$^2$. What is the linear magnification of the projector-screen arrangement.
Other Exercises for CBSE - Class 11 Physics Units and Measurements
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