Chapter 1: Units and Measurements

Fill in the blanks.

(a) The volume of a cube of side 1 cm is equal to …..m³
(b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to…(mm)²
(c) A vehicle moving with a speed of 18 km h^–1 covers….m in 1 s

(d) The relative density of lead is 11.3. Its density is ….g cm^–3 or ….kg m^–3.

Fill in the blanks by suitable conversion of units.

(a) 1 kg m² s^–2 = ….g cm² s^–2

(b) 1 m = ….. ly
(c) 3.0 m s^–2 = …. km h^–2
(d) G = 6.67 × 10^–11 N m² (kg)^–2 = …. (cm)3s^–2 g^–1
Answer:

(a) 1 kg m² s^–2 = ….g cm² s^–2

1 kg m² s^-2 = 1kg x 1m² x 1s ^-2

We know that,

1kg = 10³

1m = 100cm = 10²cm

When the values are put together, we get

1kg x 1m² x 1s^-2 = 10³g x (10²cm)² x 1s^-2  = 10³g x 10⁴ cm² x 1s^-2  = 10⁷ gcm²s^-2

=>1kg m² s^-2 = 10⁷ gcm²s^-2

(b) 1 m = ….. ly

Using the formula,

Distance = speed x time

Speed of light = 3 x 10⁸ m/s

Time = 1 yr = 365 days = 365 x 24 hr = 365 x 24 x 60 x 60 sec

Put these values in the formula mentioned above, and we get

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

9.46 x 10¹⁵ m = 1ly

So that, 1m = 1/9.46 x 10¹⁵ly

=> 1.06 x 10^-16ly

=>1 meter = 1.06 x 10^-16ly

(c) 3.0 m s^–2 = …. km h^–2

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

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

= 3 x 10^-3km x (3600)² hr^-2 = 3.88 x 10⁴ km h^-2

=> 3.0 m s^-2 = 3.88 x 10⁴ km h­^-2

(d) G = 6.67 × 10^–11 N m² (kg)^–2 = …. (cm)3s^–2 g^–1

G = 6.67 x 10^-11 N m² (kg)^-2

We know that,

1N = 1kg m s^-2

1 kg = 10³ g

1m = 100cm= 10² cm

Put the values together, we get

=> 6.67 x 10^-11 Nm² kg^-2 = 6.67 x 10^-11 x (1kg m s ^-2) (1m²) (1kg^-2)

Solve the following, and cancelling out the units, we get

=> 6.67 x 10^-11 x (1kg ^-1 x 1m³ x 1s^-2)

Put the above values together to convert kg to g and m to cm.

=> 6.67 x 10^-11 x (10³g)^-1 x (10² cm)³ x (1s^-2)

=> 6.67 x 10^-8 cm³ s^-2 g ^-1

=>G = 6.67 x 10^-11 Nm²(kg)^-2 = 6.67 x 10^-8  (cm)³ s^-2 g ^-1

2.3 A calorie is a unit of heat (energy in transit), which equals about 4.2 J, where 1J =1 kg m² s^–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, and the unit of time is γ s. Show that a calorie has a magnitude of 4.2 α^–1 β^–2 γ² in terms of the new units.

Explain this statement clearly:
“To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever necessary.
(a) atoms are very small objects
(b) a jet plane moves with great speed
(c) the mass of Jupiter is very large
(d) the air inside this room contains a large number of molecules
(e) a proton is much more massive than an electron
(f) the speed of sound is much smaller than the speed of light.

(a) Atoms are small object

A new unit of length is chosen such that the speed of light in a 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?

Which of the following is the most precise device for measuring length?
(a) a vernier callipers with 20 divisions on the sliding scale
(b) a screw gauge of pitch 1 mm and 100 divisions on the circular scale
(c) an optical instrument that can measure length to within a wavelength of light

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 the hair?

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 arbitrarily increase the accuracy of the screw gauge 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?

The photograph of a house occupies an area of 1.75 cm² on a 35 mm slide. The slide is projected onto a screen, and the area of the house on the screen is 1.55 m². What is the linear magnification of the projector-screen arrangement?

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

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.