Chapter 2 Unit and measurment

a)

Length of edge  = 1 cm 
Volume of the cube = 1 cm3

But, 

1 cm3 = 1 cm × 1 cm × 1 cm = (1/100)m x (1/100)m x (1/100)m 

∴1 cm3 = 10–6 m3

Hence, the volume of a cube of side 1 cm is equal to 10–6 m3.

 

(b) The total surface area of a cylinder of radius r and height h is S = 2πr (r + h).

Given that, r = 2 cm = 2 × 1 cm = 2 × 10 mm = 20 mm

h = 10 cm = 10 × 10 mm = 100 mm = 15072 = 1.5 × 10⁴ mm²

 

(c) Using the conversion, 1 km/h = 5/18 m/s

18km/h =  18 x  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/cm³

Again, 1g = 1/1000 kg

1 cm³ = 10^–6 m³

1 g/cm³ = 10^-3 / 10^-6 kg/m³ = 10³ kg/m³

∴ 11.3 g/cm³ = 11.3 × 10³ kg/m³

(a) 1 kg = 10³ g

1 m² = 10⁴ cm²

1 kg m² s^–2 = 1 kg × 1 m² × 1 s^–2

=10³ g × 10⁴ cm² × 1 s^–2 = 10⁷ g cm² s^–2

1 kg m²s^–2 = 10⁷ g cm² s^–2

 

(b) Distance = Speed × Time

Speed of light = 3 × 10⁸ m/s

Time = 1 year = 365 days = 365 × 24 hours = 365 × 24 × 60 × 60 sec

Putting these values in above formula we get

1 light year distance = (3 × 10⁸ m/s) × (365 × 24 × 60 × 60 s) = 9.46 × 1015 m 9.46 × 10¹⁵ m = 1 ly

So that 1 m = 1/ 9.46 × 10¹⁵ ly = 1.06 × 10^-16 ly

 

(c) 1 hour = 3600 sec so that 1 sec = 1/3600 hour

1 km = 1000 m so that 1 m = 1/1000 km

3.0 m s^–2 = 3.0 (1/1000 km)( 1/3600 hour)^-2 = 3.0 × 10^–3 km × ((1/3600)^-2 h^–2)

= 3.0 × 10^–3 km × (3600)² h^–2 = 3.88 × 10⁴ km h^–2 3.0 m s^–2

= 3.88 × 10⁴ km h^–2

 

(d) Given,

G= 6.67 × 10^–11 N m² (kg)^–2

We know that

1 N = 1 kg m s^–2

1 kg = 10³ g

1 m = 100 cm = 10² cm

Putting above values,we get

6.67 × 10^–11 N m² kg^–2 = 6.67 × 10^–11 × (1 kg m s^–2) (1 m²) (1Kg^–2)

Solve and cancel out the units we get

⇒ 6.67 × 10^–11 × (1 kg^–1 × 1 m³ × 1 s^–2)

Putting above values to convert Kg to g and m to cm

⇒ 6.67 × 10^–11 × (10³ g)^-1 × (10² cm)³ × (1 s^–2)

⇒ 6.67 × 10^–11 × 10^-3 g^-1 × 10⁶ cm³ × (1 s^–2)

⇒ 6.67 × 10^–8 cm3 s^–2 g^–1

G = 6.67 × 10^–11 N m² (kg)^–2

= 6.67 × 10^–8 (cm)3s^–2 g–1.

Given that, 1 Calorie=4.2 J = 4.2 Kg m² s^-2 ...... (i)

As new unit of mass = α Kg

∴ 1 Kg = 1/α new unit of mass

Similarly,

1 m = β^{– 1} new unit of length

1 s = γ^-1 new unit of time

Putting these values in (i), we get

1 calorie = 4.β (α^-1 new unit of mass) (β^{– 1} new unit of length)β ( γ^-1 new unit of time)^-2

= 4.β α^-1 β^{– 2} γ² new unit of energy (Proved)

The given statement is true because a dimensionless quantity may be large or small in comparision to some standard reference. For example, the coefficient of friction is dimensionless. The coefficient of sliding friction is greater than the coefficient of rolling friction, but less than static friction.

(a) An atom is a very small object in comparison to a soccer ball.

(b) A jet plane moves with a speed greater than that of a bicycle.

(c) Mass of Jupiter is very large as compared to the mass of a cricket ball.

(d) The air inside this room contains a large number of molecules as compared to that present in a geometry box.

(e) A proton is more massive than an electron.

(f) Speed of sound is less than the speed of light.

Distance between the Sun and the Earth:

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

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

Time taken, t  = 8 min 20 s = 500 s 

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

A device with minimum count is the most suitable to measure length.

(a) Least count of vernier callipers

= 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.

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.

(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.

Area of the house on the slide = 1.75 cm²

Area of the image of the house formed on the screen = 1.55 m²

= 1.55 × 10⁴ cm²

Arial magnification, m_a =  Area of Image / Area of Object = (1.55 / 1.75) x 10⁴

∴Linear magnifications, m_l = underroot m_a

(a) Answer: 1

The given quantity is 0.007 m².

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) Answer: 3

The given quantity is 2.64 x 10²⁴ 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) Answer: 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) Answer: 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) Answer: 4

The given quantity is 6.032 Nm^-2.

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

 

(f) Answer: 4

The given quantity is 0.0006032 m².

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.

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:

Quantity	Number	Significant Figure
l	4.234	4
b	1.005	4
h	0.0201	3

 

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 + 0.0202005 + 0.0851034)

= 2 × 4.36 = 8.72 m²

Volume of the sheet = l × b × h

= 4.234 × 1.005 × 0.0201

= 0.0855 m³

This number has only 3 significant figures i.e., 8, 5, and 5.

Mass of grocer's box = 2.300 kg

Mass of gold piece I = 20.15g = 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.

= 3x1 +2x3 + ½ x 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.

The displacement y has the dimension of length, therefore, the formula for it should also have the dimension of length. Trigonometric functions are dimensionless and their arguments are also dimensionless. Based on these considerations now check each formula dimensionally.

The formulas in (ii) and (iii) are dimensionally wrong.

Given the relation,

m  =  m₀  /  (1-v²)^½

Dimension of m = M¹ L⁰ T⁰

Dimension of = M¹ L⁰ T⁰

Dimension of v = M⁰ L1 T^-1

Dimension of v² = M⁰ L2 T^-2

Dimension of c = M⁰ L1 T^-1

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-v²)^½ is dimensionless i.e., (1 - v²) is dimensionless. This is only possible if v² is divided by c². Hence, the correct relation is

m  =  m₀  /  (1-v²/c²)^½ .

Radius of hydrogen atom, r = 0.5 Å = 0.5 × 10^-10 m

Volume of hydrogen atom = 4/3 π r³

= 4/3 x 22/7 x (0.5 x 10^-10)³

= 0.524 x 10³⁰ m³

1 mole of hydrogen contains 6.023 × 10²³ hydrogen atoms.

∴ Volume of 1 mole of hydrogen atoms = 6.023 × 10²³ × 0.524 × 10^-30

= 3.16 × 10^-7 m³

Radius of hydrogen atom, r = 0.5 Å = 0.5 × 10^-10 m

Volume of hydrogen atom = 4/3 π r³

= 4/3 x 22/7 x (0.5 x 10^-10)³

= 0.524 x 10³⁰ m³

1 mole of hydrogen contains 6.023 × 10²³ hydrogen atoms.

∴ Volume of 1 mole of hydrogen atoms = 6.023 × 10²³ × 0.524 × 10^-30

= 3.16 × 10^-7 m³

Molar volume of 1 mole of hydrogen atoms at STP,

V_m = 22.4 L = 22.4 × 10^-3 m³

∴ V_m / V_a  =  22.4 x 10^-3  /   3.16 × 10^-7  =  7.08 × 10⁴

Hence, the molar volume is 7.08 × 10⁴ 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.

 

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.

Diameter of Earth's orbit = 3 × 10¹¹ m

Radius of Earth's orbit, r = 1.5 × 10¹¹ m

Let the distance parallax angle be1” = 4.847 × 10^-6 rad.

Let the distance of the star be D.

Parsec is defined as the distance at which the average radius of the Earth's orbit subtends an angle of 1”.

∴ We have θ = r / D

Hence, 1 parsec  ≈ 3.09 × 10¹⁶ m.

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 × 10⁸ × 365 × 24 × 60 × 60 = 94608 × 10¹¹ m

∴ 4.29 ly = 405868.32 × 10¹¹ m

∵ 1 parsec = 3.08 × 10¹⁶ m

∴ 4.29 ly =  [405868.32 x 10¹¹] / [3.08 x 10¹⁶] = 1.32 parsec

Using the relation,

θ = d / D

where,

Diameter of earth's orbit, d = 3 x 10¹¹ m

Distance of the star from the earth , D = 405868.32 x 10¹¹ m

∴ θ = 3 x 10¹¹ m  / 405868.32 x 10¹¹ m

= 7.39 x 10-6 rad

But, 1 sec = 4.85 × 10^-6 rad

∴ 7.39 x 10^-6 rad  =   7.39 x 10^-6  /  4.85 × 10^-6  = 1.52"

It is indeed very true that precise measurements of physical quantities are essential for the development of science. For example, ultra-shot laser pulses (time interval ∠¼ 10^-15 s) are used to measure time intervals in several physical and chemical processes.

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.

(a) During monsoons, a metrologist records about 215 cm of rainfall in India i.e., the height of water column, h = 215 cm = 2.15 m

Area of country, A = 3.3 × 10¹² m²

Hence, volume of rain water, V = A × h = 7.09 × 10¹² m³

Density of water, p = 1 × 103 kg m^-3

Hence, mass of rain water = p × V = 7.09 × 10¹⁵ kg

Hence, the total mass of rain-bearing clouds over India is approximately 7.09 × 10¹⁵ kg.

 

(b) Consider a ship of known base area floating in the sea. Measure its depth in sea (say d₁).

Volume of water displaced by the ship, Vb = A d₁

Now, move an elephant on the ship and measure the depth of the ship (d₂) in this case.

Volume of water displaced by the ship with the elephant on board, V_be= Ad₂

Volume of water displaced by the elephant = Ad₂ - Ad₁

Density of water = D

Mass of elephant = AD (d₂ - d₁)

 

(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 = πr²

Number of strands of hair = Total Surface area / Area of one hair  = A / πr²

 

(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 m³ volume.

Number of molecules in one mole = 6.023 × 10²³

∴Number of molecules in room of volume V

=  6.023 × 10²³  /  22.4 × 10^-3   V = 134.915 × 10²⁶ V

= 1.35 × 10²⁸ V

Mass of the Sun, M = 2.0 × 10³⁰ kg

Radius of the Sun, R = 7.0 × 10⁸ m

Volume of the Sun, V = 4/3 πR³

= 4/3 x 22/7 x (7.0x10⁸)³

= 88/21  x 343 x 10²⁴

= 1437.3 x 10²⁴ m³

Density of the Sun = mass / Volume = 2.0x10³⁰ / 1437.3 x 10²⁴

= 1.4 x 10³ kg/m⁵

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, D = 824.7 × 10⁶ km = 824.7 × 10⁹ m

Angular diameter = 35.72"  =  35.72 x 4.874 x 10^-6 rad

Diameter of Jupiter = d

Using the relation,

θ = d / D

d  =  θD

= 824.7 x 10⁹ x 35.72 x 4.874 x 10^-6

= 143520.76 x 10³

= 1.435 x 10⁵

Incorrect;  on dimensional ground

The relation is  tan θ = v 

Dimension of R.H.S = M⁰ L¹ T^-1

Dimension of L.H.S = M⁰ L⁰ T⁰

( ∵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.

Difference in time of caesium clocks = 0.02 s

Time required for this difference = 100 years

= 100 × 365 × 24 × 60 × 60 = 3.15 × 10⁹ s

In 3.15 × 10⁹ s,the caesium clock shows a time difference of 0.02 s.

In 1s, the clock will show a time difference of 0.02 / 3.15x10⁹ s.

Hence, the accuracy of a standard caesium clock in measuring a time interval of 1 s is 3.15 x10⁹ / 0.02 =  157.5 x 10⁹ s  = 1.5 x 10¹¹ s

Diameter of sodium atom = Size of sodium atom = 2.5 Å

Radius of sodium atom, r = 1/2 x 2.5 Å = 1.25 Å

= 1.25 × 10^-10 m

Volume of sodium atom, V = 4/3 πr³

= 4/3 x 22/7 x (1.25 × 10^-10) ³

According to the Avogadro hypothesis, one mole of sodium contains 6.023 × 10²³ atoms and has a mass of 23 g or 23 × 10^-3 kg.

∴ Mass of one atom = 23 × 10^-3  / 6.023 x 10^-3 kg

Density of sodium atom, p =

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.

Radius of nucleus r is given by the relation,

r  = r₀ A^1/3  … (i)

r₀ = 1.2 f = 1.2 × 10^-15 m

Volume of nucleus, V= 4/3 πr³

= 4/3 π( r₀ A^1/3)³  =  4/3 π r₀³ A ......(ii)

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,

p = Mass of nucleus  /  Volume of nucleus

This relation shows that nuclear mass depends only on constant r₀. Hence, the nuclear mass densities of all nuclei are nearly the same.

Density of sodium nucleus is given by,

= (4.98 / 21.71) x 10^-18 

= 2.29 x 10^-17 kg m^-3

 

 

 

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

Speed of light = 3 × 10⁸ m/s

Time taken by the laser beam to reach Moon = 2..56 x 1/2 = 1.28 s

Radius of the lunar orbit = Distance between the Earth and the Moon = 1.28 × 3 × 10⁸ = 3.84 × 10⁸ m = 3.84 × 10⁵ km

Let the distance between the ship and the enemy submarine be 'S'.

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 (2S).

Time taken for the sound to reach the submarine = 1/2 x 77 = 38.5 s

∴ Distance between the ship and the submarine (S) = 1450 × 38.5 = 55825 m = 55.8 km

Time taken by quasar light to reach Earth = 3 billion years

= 3 x 10⁹ years

= 3 x 10⁹ x 365 x 24 x 60 x 60 s

Speed of light = 3 x 10⁸ m/s

Distance between the Earth and quasar

= (3 x 10⁸) x (3 x 10⁹ x 365 x 24 x 60 x 60)

= 283824 x 10²⁰ m

= 2.8 x 10²² km

The position of the Sun, Moon, and Earth during a lunar eclipse is shown in the given figure.

Distance of the Moon from the Earth = 3.84 × 10⁸ m

Distance of the Sun from the Earth = 1.496 × 10¹¹ m

Diameter of the Sun = 1.39 × 10⁹ 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× 10⁶ m.

One relation consists of some fundamental constants that give the age of the Universe by:

 

Where,

t = Age of Universe

e = Charge of electrons = 1.6 ×10^-19 C

ε₀ = Absolute permittivity

m_p = Mass of protons = 1.67 × 10^-27 kg

m_e = Mass of electrons = 9.1 × 10^-31 kg

c = Speed of light  = 3 × 10⁸ m/s

G = Universal gravitational constant = 6.67 × 10¹¹ Nm² kg^-2

Also,1/ 4πε₀ = 9x10⁹  Nm²/C²

Substituting these values in the equation, we get