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**Q.2.1.****Fill in the blanks****(a) The volume of a cube of side 1 cm is equal to ______ m ^{3}(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}.**

Volume of the cube = 1 cm

But, 1 cm

= (1/100) m × (1/100) m × (1/100)m

∴ 1 cm

Hence, the volume of a cube of side 1 cm is equal to 10

Given that,

= 2 × 10 mm = 20 mm

= 100 mm

∴ S = 2 ×

= 15072 = 1.5 × 10

1 km/h = (5/18) m/s

18 km/h = 18 × (5/18) = 5 m/s

Therefore, distance can be obtained using the relation:

Distance = Speed × Time = 5 × 1 = 5 m

Hence, the vehicle covers 5 m in 1 s.**(d)** Relative density of a substance is given by the relation,

Relative density = (Density of substance/Density of water)

Density of water = 1 g/cm^{3}

Density of lead = Relative density of lead x Density of water = 11.3 × 1 = 11.3 g/cm^{3}

Again,

1 cm^{3} = 10^{–6} m^{3}

1 g/cm^{3} = (10^{-3}/10^{-6})kg/m^{3}

= 10^{3} kg/m^{3}

∴ 11.3 g/cm^{3} = 11.3 × 10^{3} kg/m^{3}**Q.2.2. ****Fill in the blanks by suitable conversion of units:****(a) 1 kg m ^{2}s^{–2}= ______ g cm^{2} s^{–2}**

(b) 1 m =______ ly

(c) 3.0 m s^{–2}=______ km h^{–2}

(d) G= 6.67 × 10^{–11} N m^{2} (kg)^{–2}=______ (cm)^{3}s^{–2} g^{–1}.

1 m

1 kg m

=10

**(b)** Light year is the total distance travelled by light in one year.

1 ly = Speed of light × One year

= (3 × 10^{8} m/s) × (365 × 24 × 60 × 60 s)

= 9.46 × 10^{15} m

∴ 1 m = (1/9.46 × 10^{15})

= 1.057 × 10^{-16} ly

**(c)** 1 m = 10^{–3} km

Again,

1 s^{–1} = 3600 h^{–1}

1 s^{–2} = (3600)^{2} h^{–2}

∴ 3 m s^{–2} = (3 × 10^{–3} km) × ((3600)^{2} h^{–2})

= 3.88 × 10^{4} km h^{–2}

**(d)** 1 N = 1 kg m s^{–2}

1 kg = 10^{–3} g^{–1}

1 m^{3} = 10^{6} cm^{3}

∴ 6.67 × 10^{–11} N m^{2} kg^{–2}

= 6.67 × 10^{–11} × (1 kg m s^{–2}) (1 m^{2}) (1 s^{–2})

= 6.67 × 10^{–11} × (1 kg × 1 m^{3} × 1 s^{–2})

= 6.67 × 10^{–11} × (10^{–3} g^{–1}) × (10^{6} cm^{3}) × (1 s^{–2})

= 6.67 × 10^{–8} cm^{3} s^{–2} g^{–1}**Q.2.3. ****A calorie is a unit of heat or energy and it equals about 4.2 J where 1 J = 1 kg m ^{2}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^{–1} β^{–2} γ^{2 }in terms of the new units.**

1 calorie = 4.2 (1 kg) (1 m

New unit of mass = α kg

Hence, in terms of the new unit, 1 kg = (1/α) = α

In terms of the new unit of length,

1 m = 1/β = β

And, in terms of the new unit of time,

1s = 1/γ = γ

1s

1s

∴ 1 calorie = 4.2 (1 α

= 4.2 α

(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.

= Speed of light × Time taken by light to cover the distance

Given that in the new unit, speed of light = 1 unit

Time taken,

∴ Distance between the Sun and the Earth = 1 × 500 = 500 units

(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 ?

= 1 standard division (SD) – 1 vernier division (VD)

= 1 - (9/10) = (1/10) = 0.01 cm

**(b)** Least count of screw gauge = (Pitch/Number of divisions)

= (1/1000) = 0.001 cm

**(c)** Least count of an optical device = Wavelength of light ∼ 10^{–5} cm = 0.00001 cm

Hence, it can be inferred that an optical instrument is the most suitable device to measure length.**Q.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?****Ans. **Magnification of the microscope = 100

Average width of the hair in the field of view of the microscope = 3.5 mm

∴ Actual thickness of the hair is 3.5/100 = 0.035 mm.**Q.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?****Ans.****(a) **Wrap the thread on a uniform smooth rod in such a way that the coils thus formed are very close to each other. Measure the length of the thread using a metre scale. The diameter of the thread is given by the relation,

Diameter = Length of thread/Number of turns

**(b)** It is not possible to increase the accuracy of a screw gauge by increasing the number of divisions of the circular scale. Increasing the number divisions of the circular scale will increase its accuracy to a certain extent only.

**(c)** A set of 100 measurements is more reliable than a set of 5 measurements because random errors involved in the former are very less as compared to the latter.**Q.2.9. ****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?**

Area of the image of the house formed on the screen = 1.55 m

= 1.55 × 10

Arial magnification,

∴ Linear magnifications,

The given quantity is 0.007 m

If the number is less than one, then all zeros on the right of the decimal point (but left to the first non-zero) are insignificant. This means that here, two zeros after the decimal are not significant. Hence, only 7 is a significant figure in this quantity.

**(b)** 3

The given quantity is 2.64 × 10^{24} kg.

Here, the power of 10 is irrelevant for the determination of significant figures. Hence, all digits i.e., 2, 6 and 4 are significant figures.

**(c)** 4

The given quantity is 0.2370 g cm^{–3}.

For a number with decimals, the trailing zeroes are significant. Hence, besides digits 2, 3 and 7, 0 that appears after the decimal point is also a significant figure.

**(d)** 4

The given quantity is 6.320 J.

For a number with decimals, the trailing zeroes are significant. Hence, all four digits appearing in the given quantity are significant figures.

**(e)** 4

The given quantity is 6.032 Nm^{–2}.

All zeroes between two non-zero digits are always significant.

**(f)** 4

The given quantity is 0.0006032 m^{2}.

If the number is less than one, then the zeroes on the right of the decimal point (but left to the first non-zero) are insignificant. Hence, all three zeroes appearing before 6 are not significant figures. All zeros between two non-zero digits are always significant. Hence, the remaining four digits are significant figures.**Q.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.****Ans.**

Length of sheet, *l *= 4.234 m

Breadth of sheet, *b *= 1.005 m

Thickness of sheet, *h *= 2.01 cm = 0.0201 m

The given table lists the respective significant figures:

Hence, area and volume both must have least significant figures i.e., 3.

Surface area of the sheet = 2 ((*l* × *b) *+(*b *× *h)+*(*h* × *l)*)

= 2((4.234 × 1.005)+ (1.005 × 0.0201)+(0.0201 × 4.234))

= 2 (4.25517 + 0.02620 + 0.08510)

= 2 × 4.360

= 8.72 m^{2}

Volume of the sheet = *l *× *b* × *h*

= 4.234 × 1.005 × 0.0201

= 0.0855 m^{3}

This number has only 3 significant figures i.e., 8, 5, and 5.**Q. 2.12. ****The mass of a box measured by a grocer’s balance is 2.300 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?****Ans.**

Mass of grocer’s box = 2.300 kg

Mass of gold piece **I** = 20.15 g = 0.02015 kg

Mass of gold piece **II** = 20.17 g = 0.02017 kg**(a)** Total mass of the box = 2.3 0.02015 0.02017 = 2.34032 kg

In addition, the final result should retain as many decimal places as there are in the number with the least decimal places. Hence, the total mass of the box is 2.3 kg.**(b)** Difference in masses = 20.17 – 20.15 = 0.02 g

In subtraction, the final result should retain as many decimal places as there are in the number with the least decimal places.**Q. 2.13.** **A physical quantity P is related to four observables a, b, c and d as follows:**

**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?**

3 × 1 + 2 × 3 + (1/2) × 4 + 2

= 3 + 6 + 2 + 2 = 13%

Percentage error in *P* = 13 %

Value of *P *is given as 3.763.

By rounding off the given value to the first decimal place, we get *P* = 3.8.**Q. 2.14**: **A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion:**

(b)

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

(d) y = (a√2) (sin(2πt/T) + cos(2πt/T))

y = a sin(2πt/T)

Dimension of

Dimension of

Dimension of sin (2πt/T)= M

Dimension of L.H.S = Dimension of R.H.S

Hence, the given formula is dimensionally correct.

**(b)** Incorrect*y* = *a* sin *vt*

Dimension of *y* = M^{0} L^{1} T^{0}

Dimension of *a* = M^{0} L^{1} T^{0}

Dimension of *vt* = M^{0} L^{1} T^{–1} × M^{0} L^{0} T^{1 }= M^{0} L^{1} T^{0}

But the argument of the trigonometric function must be dimensionless, which is not so in the given case. Hence, the given formula is dimensionally incorrect.

**(c)** Incorrect

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

Dimension of *y* = M^{0}L^{1}T^{0}

Dimension of a/T = M^{0}L^{1}T^{–1}

Dimension of t/a = M^{0} L^{–1} T^{1}

But the argument of the trigonometric function must be dimensionless, which is not so in the given case. Hence, the formula is dimensionally incorrect.

**(d)** Correct

y = (a√2)(sin2π(t/T) + cos2π(t/T)

Dimension of *y* = M^{0} L^{1} T^{0}

Dimension of *a* = M^{0} L^{1} T^{0}

Dimension of t/T = M^{0} L^{0} T^{0}

Since the argument of the trigonometric function must be dimensionless (which is true in the given case), the dimensions of *y* and *a* are the same. Hence, the given formula is dimensionally correct.**Q.2.15. ****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 = (m

Given the relation,

m = (m

Dimension of

Dimension of m

Dimension of

Dimension of

Dimension of

The given formula will be dimensionally correct only when the dimension of L.H.S is the same as that of R.H.S. This is only possible when the factor,(1-v

**Q.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 0.5Å . what is the total atomic volume in m^{3} of a mole of hydrogen atoms?**

Radius of hydrogen atom,

Volume of hydrogen atom = (4/3)πr

= (4/3 × 22/7) × (0.5 × 10

= 0.524 × 10

1 mole of hydrogen contains 6.023 × 10

∴ Volume of 1 mole of hydrogen atoms = 6.023 × 10

= 3.16 × 10

Volume of hydrogen atom = (4/3)πr

= (4/3 × 22/7) × (0.5

= 0.524 x 10

Now, 1 mole of hydrogen contains 6.023 × 10^{23} hydrogen atoms.

∴ Volume of 1 mole of hydrogen atoms, *V*_{a} = 6.023 × 10^{23} × 0.524 × 10^{–30}

= 3.16 × 10^{–7} m^{3}

Molar volume of 1 mole of hydrogen atoms at STP,*V*_{m} = 22.4 L = 22.4 × 10^{–3} m^{3}

∴ V_{m}/V_{a} = (22.4 × 10^{-3}/3.16 × 10^{-7}) = 7.08* ×* 10^{4}

Hence, the molar volume is 7.08 × 10^{4} times higher than the atomic volume. For this reason, the inter-atomic separation in hydrogen gas is much larger than the size of a hydrogen atom.**Q.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).****Ans. **Line of sight is defined as an imaginary line joining an object and an observer’s eye. When we observe nearby stationary objects such as trees, houses, etc. while sitting in a moving train, they appear to move rapidly in the opposite direction because the line of sight changes very rapidly. On the other hand, distant objects such as trees, stars, etc. appear stationary because of the large distance. As a result, the line of sight does not change its direction rapidly.**Q.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 × 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 meters?**

Radius of Earth’s orbit,

Let the distance parallax angle be = 4.847 × 10

Let the distance of the star be

Parsec is defined as the distance at which the average radius of the Earth’s orbit subtends an angle of.

∴ We have θ = (r/D)

D = r/θ = (1.5 × 10

= 0.309 × 10

Hence, 1 parsec ≈ 3.09 × 10

Distance of the star from the solar system = 4.29 ly

1 light year is the distance travelled by light in one year.

1 light year = Speed of light × 1 year

= 3 × 10

∴ 4.29 ly = 405868.32 × 10

1 parsec = 3.08 × 10

∴ 4.29 ly = (405868.32 × 10

Using the relation,

θ = d/ D

where,

Diameter of Earth`s orbit d = 3 × 10

Distance of the star from the Earth, D = 405868.32 × 10

∴ θ = (3 × 10

= 7.39 × 10

But, 1 sec = 4.85 × 10

∴ 7.39 × 10

= (7.39 × 10

X-ray spectroscopy is used to determine the inter-atomic separation or inter-planer spacing.

The development of mass spectrometer makes it possible to measure the mass of atoms precisely.

Area of country,

Hence, volume of rain water,

= 7.09 × 10

Density of water,

Hence, mass of rain water =

= 7.09 × 10

Hence, the total mass of rain-bearing clouds over India is approximately 7.09 × 10

**(b)** Consider a ship of known base area floating in the sea. Measure its depth in sea (say *d*_{1}).

Volume of water displaced by the ship, *V*_{b} = *A* *d*_{1}

Now, move an elephant on the ship and measure the depth of the ship (*d*_{2}) in this case.

Volume of water displaced by the ship with the elephant on board, *V*_{be}= *Ad*_{2}

Volume of water displaced by the elephant = *Ad*_{2} – *Ad*_{1}

Density of water = *D*

Mass of elephant = *AD* (*d*_{2} – *d*_{1})

**(c)** Wind speed during a storm can be measured by an anemometer. As wind blows, it rotates. The rotation made by the anemometer in one second gives the value of wind speed.

**(d)** Area of the head surface carrying hair = *A*

With the help of a screw gauge, the diameter and hence, the radius of a hair can be determined. Let it be *r.*

∴ Area of one hair = π*r*^{2}

Number of strands of hair ≈ (Total surface area/Area of one hair π) = A/r^{2}

**(e)** Let the volume of the room be *V.*

One mole of air at NTP occupies 22.4 l

i.e., 22.4 × 10^{–3} m^{3} volume.

Number of molecules in one mole = 6.023 × 10^{23}

∴ Number of molecules in room of volume *V*

= (6.023 × 10^{23} / 22.4 × 10^{-3}) × V

= 134.915 × 10^{26} *V*

= 1.35 × 10^{28} *V***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 × 10^{30} kg, radius of the Sun = 7.0 × 10^{8} m.**

Radius of the Sun,

Volume of the Sun,

= (4/3) × (22/7) × (7.0 × 10

= (88/21) × 343 × 10

= 1437.3 × 10

Density of the Sun = Mass/Volume = (2.0 × 10

The density of the Sun is in the density range of solids and liquids. This high density is attributed to the intense gravitational attraction of the inner layers on the outer layer of the Sun.

Distance of Jupiter from the Earth,

Angular diameter =35.72" = 35.72 × 4.874 × 10

Diameter of Jupiter =

Using the relation,

θ = d/D

d = θD = 824.7 × 10

= 143520.76 × 10

= 1.435 × 10

on dimensional ground

The relation is tan θ = v.

Dimension of R.H.S = M

Dimension of L.H.S = M

(∴ The trigonometric function is considered to be a dimensionless quantity)

Dimension of R.H.S is not equal to the dimension of L.H.S. Hence, the given relation is not correct dimensionally.

To make the given relation correct, the R.H.S should also be dimensionless. One way to achieve this is by dividing the R.H.S by the speed of rainfall v'.

Therefore, the relation reduces to tan θ = v/v'.

This relation is dimensionally correct.

Time required for this difference = 100 years

= 100 × 365 × 24 × 60 × 60 = 3.15 × 10

In 3.15 × 10

In 1s, the clock will show a time difference of (0.02/3.15 × 10

Hence, the accuracy of a standard cesium clock in measuring a time interval of 1 s is (3.15 × 10

= 157.5 × 10

Radius of sodium atom,

= 1.25 × 10

Volume of sodium atom,

= (4/3) × 3.14 × (1.25 × 10

According to the Avogadro hypothesis, one mole of sodium contains 6.023 × 10

∴ Mass of one atom = (23 × 10

Density of sodium atom,

It is given that the density of sodium in crystalline phase is 970 kg m^{–3}.

Hence, the density of sodium atom and the density of sodium in its crystalline phase are not in the same order. This is because in solid phase, atoms are closely packed. Thus, the inter-atomic separation is very small in the crystalline phase.**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_{0}A^{1/3}**

r = r

r

Volume of nucleus,

Now, the mass of a nuclei *M* is equal to its mass number i.e.,*M* = *A* amu = *A* × 1.66 × 10^{–27} kg

Density of nucleus,

ρ = Mass of nucleus/Volume of nucleus

This relation shows that nuclear mass depends only on constant r_{0}. Hence, the nuclear mass densities of all nuclei are nearly the same.

Density of sodium nucleus is given by,

**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?****Ans. **Time taken by the laser beam to return to Earth after reflection from the Moon = 2.56 s

Speed of light = 3 × 10^{8} m/s

Time taken by the laser beam to reach Moon = (1/2) × 2.56 = 1.28s

Radius of the lunar orbit = Distance between the Earth and the Moon

= 1.28 × 3 × 10^{8}

= 3.84 × 10^{8} m

= 3.84 × 10^{5} km**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}).**

Speed of sound in water = 1450 m/s

Time lag between transmission and reception of Sonar waves = 77 s

In this time lag, sound waves travel a distance which is twice the distance between the ship and the submarine (2

Time taken for the sound to reach the submarine = (1/2) × 77 = 38.5 s

∴ Distance between the ship and the submarine (

= 55825 m

= 55.8 km

= 3 × 10

= 3 × 10

Speed of light = 3 × 10

Distance between the Earth and quasar

= (3 × 10

= 283824 × 10

= 2.8 × 10

Distance of the Moon from the Earth

= 3.84 × 10^{8} m

Distance of the Sun from the Earth

= 1.496 × 10^{11} m

Diameter of the Sun = 1.39 × 10^{9} m

It can be observed that ΔTRS and ΔTPQ are similar. Hence, it can be written as:

Hence, the diameter of the Moon is 3.57 × 10^{6} m.**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?**

Where,*t* = Age of Universe

e = Charge of electrons = 1.6 ×10^{–19} C

ε_{0} = Absolute permittivity

m_{p} = Mass of protons = 1.67 × 10^{–27} kg

m_{c} = Mass of electrons = 9.1 × 10^{–31} kg

c = Speed of light = 3 × 10^{8} m/s

G = Universal gravitational constant = 6.67 × 10^{11} Nm^{2} kg^{–2}

Also,

Substituting these values in the equation, we get

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