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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).
Solution :
Step 1: Dimensional Analysis of the Given Expression
To determine the placement of the speed of light ($c$), we must analyze the dimensional consistency of the boy's given formula: $m = \frac{m_{0}}{(1 - v^{2})^{1/2}}$.
- The left-hand side (LHS) is mass ($m$), which has the dimension $[M^1 L^0 T^0]$.
- The numerator on the right-hand side (RHS) is rest mass ($m_0$), which also has the dimension $[M^1 L^0 T^0]$.
- For the equation to be dimensionally homogeneous, the denominator term $(1 - v^{2})^{1/2}$ must be dimensionless.
- Currently, the term $v^{2}$ has the dimensions of velocity squared, which is $[L^1 T^{-1}]^2 = [L^2 T^{-2}]$. Since the number $1$ is dimensionless, subtracting $v^{2}$ from $1$ is dimensionally invalid [Per the Principle of Homogeneity of Dimensions].
Step 2: Correcting the Dimensional Discrepancy
To make the term $v^{2}$ dimensionless, it must be divided by another quantity with the same dimensions of velocity squared. The speed of light ($c$) is the only relevant constant in this context with the dimensions $[L^1 T^{-1}]$.
Therefore, we introduce $c$ such that the term becomes $\frac{v^{2}}{c^{2}}$. Now, the dimension of this ratio is:
$\frac{[L^2 T^{-2}]}{[L^2 T^{-2}]} = [M^0 L^0 T^0]$ (Dimensionless).
Step 3: Validating the Einsteinian Relation
Substituting the ratio into the boy's original expression, we get:
$m = \frac{m_{0}}{(1 - \frac{v^{2}}{c^{2}})^{1/2}}$
Step 4: Theoretical Justification
This relation originates from the Lorentz transformation factor, often denoted by the Greek letter gamma ($\gamma$):
$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$
Thus, the complete expression for relativistic mass is $m = \gamma m_0$. When $v \ll c$, the term $\frac{v^2}{c^2}$ approaches zero, and $m \approx m_0$, consistent with Newtonian mechanics [Per the Correspondence Principle].
Final Solution: The constant $c$ must be placed in the denominator such that the expression becomes $m = \frac{m_{0}}{(1 - \frac{v^{2}}{c^{2}})^{1/2}}$.
More Questions from Class 11 Physics Units and Measurements EXERCISES
- 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 $c$.
- Q2.16: 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.
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I have also earned the prestigious Top Tutor badge, reflecting consistently positive feedback from students and parents, strong teaching performance, and a proven track record of delivering excellent learning outcomes. The appreciation I receive from parents and students consistently highlights my ability to simplify difficult concepts, maintain engaging classes, encourage students to ask questions freely, and build their confidence in Mathematics and Science. I also prepare students for: • Board Examinations • School Assessments • MCQ Practice I teach students from: • CBSE • ICSE • Gujarat Board and other State Boards My online classes are attended by students from India as well as families living abroad who are looking for structured, high-quality academic support. --------------My Teaching Methodology ------------ Every class follows a systematic approach that focuses on learning, application, and continuous improvement. Step 1 – Concept Clarity Every topic is explained from the basics using digital whiteboards, visual illustrations, practical examples, and simple language. I ensure students understand the logic behind every concept before moving to problem-solving. Step 2 – Guided Practice Students actively solve problems during the class. Rather than simply providing answers, I encourage them to think critically, apply concepts, and develop independent problem-solving skills while receiving immediate feedback. Step 3 – Assessment & Reinforcement Learning is strengthened through: • Chapter-wise assignments • Regular practice worksheets • Weekly assessments • MCQ tests • Previous-year board questions • Doubt-solving sessions • Revision tests before examinations Regular performance analysis helps identify learning gaps early so that improvement is continuous. ------------Why Students & Parents Choose Me-------------- ✔ More than 17 years of teaching experience ✔ Recognized among Urban Pro's Top 10 Tutors (2024 & 2025) ✔ Urban Pro Super Tutor ✔ Concept-based teaching methodology ✔ Interactive live online classes ✔ Small batches for individual attention ✔ Personalized doubt-solving support ✔ Digital notes and study material ✔ Regular tests and performance tracking ✔ Board exam and competitive exam preparation ✔ Friendly, patient, and motivating teaching style My Teaching Philosophy I believe every student has the ability to excel when taught with the right approach. Success is not achieved through rote learning but through understanding concepts, practicing consistently, and developing confidence. My objective is to create independent learners who enjoy learning and can confidently tackle academic challenges. Book a Free Demo Class A free demo session allows students and parents to experience my teaching methodology before enrolling. During the demo session, I will: • Assess the student's current academic level. • Teach a topic through an interactive live class. • Identify conceptual strengths and learning gaps. • Suggest a personalized learning plan based on the student's goals. Whether your child needs help strengthening fundamentals, improving school performance, preparing for board examinations, or building confidence in Mathematics and Science, I am committed to providing structured, result-oriented, and engaging online learning. Let's build strong concepts today for a brighter academic future tomorrow. WHAT PARENTS & STUDENTS SAY ---------- Nothing reflects a teacher's impact better than the words of students and parents. Here's what some of them have shared about their learning experience with me on UrbanPro: "The teaching methodology is excellent. He explains the concepts with multiple real and live examples. My son liked his teaching very much." — Parent of a CBSE Class 7 student "The classes have been superb. Topics are being made easy to understand, maths is fun to study now. No matter how difficult the topic, Sir makes sure to make it simple and easy to understand." — CBSE Class 12 student "Ashish is an outstanding online Maths tutor. His teaching style is clear, patient and engaging. My child's grades and understanding of Mathematics have improved significantly." — Parent of a Class 8 student "He is very dedicated. My daughter is now able to understand the concepts easily." — Parent of a CBSE Class 10 student "Ashish Sir is very dedicated and always on time. He makes my son practice more problems during class. My son is now in the top five in his class." — Parent of a CBSE Class 11 student These reviews consistently highlight the qualities I strive to bring into every class—concept clarity, patience, interactive teaching, structured practice, and a genuine commitment to helping every student achieve their best academically.
Ashish Bhatia is very dedicated and always on time. He makes my son practice more problems during class, and because of this we have seen a lot of improvement in his performance. My son is now in the top five in his class.
Top Tutor
Rajajinagar Mahaganapathi Nagar, Bangalore
15 yrs of
Exp
52 students
5141 hrs of classes completed
400
per hour
Online Classes
Tutor's home
I have done Post Graduation in Mathematics, Presidency college, Chennai University. I was a topper and a gold medalist in MSc Math Presidency College. I have been teaching math from last 15 years. I have taken tuitions for 11th Maths for students across different boards like CBSE, ICSE, IGCSE, ISC and State Board. I have an expertise in Math with an excellent track record. My focus is students' success. Conceptually , I make students strong so that they can handle any question in the exam. I give challenging problems in the class so that you can become competetive. Your future is very promising with me.
Mala ma'am makes me understand concepts with clarity and clears all my doubts with lots of patience making to solve problems with ease. Thank you ma'am for always being there to help and support me.
Top Tutor
Vaishali Nagar, Jaipur
9 yrs of
Exp
9 students
3087 hrs of classes completed
600
per hour
Online Classes
As a certified and top-ranked tutor on UrbanPro with over 9 years of teaching experience, I specialize in guiding Class 11 students as they transition into the commerce stream. My strength lies in simplifying complex Accountancy concepts and building a strong foundational understanding. Backed by 7+ years of industry experience in Accounting, Auditing, and Taxation, I effectively bridge theoretical learning with real-world application. My goal is to make learning engaging, practical, and confidence-boosting, helping students not only excel in exams but also prepare for higher studies in commerce.
He is good teacher for study. He teach detailly about subject. His teach style is strict. But sometimes this style is good for students.
