Chapter 2 - UNITS AND MEASUREMENTS

(a) How many astronomical units (A.U.) make 1 parsec?
(b) Consider a sunlike star at a distance of 2 parsecs. When it is
seen through a telescope with 100 magnification, what should
be the angular size of the star? Sun appears to be (1/2)° from
Exemplar Problems–Physics
12
the earth. Due to atmospheric fluctuations, eye can’t resolve
objects smaller than 1 arc minute.
(c) Mars has approximately half of the earth’s diameter. When it
is closest to the earth it is at about 1/2 A.U. from the earth.
Calculate what size it will appear when seen through the same
telescope.

In an experiment to estimate the size of a molecule of oleic acid
1 mL of oleic acid is dissolved in 19 mL of alcohol. Then 1 mL of
this solution is diluted to 20 mL by adding alcohol. Now 1 drop of
this diluted solution is placed on water in a shallow trough. The
solution spreads over the surface of water forming one molecule
thick layer. Now, lycopodium powder is sprinkled evenly over the
film and its diameter is measured. Knowing the volume of the drop
and area of the film we can calculate the thickness of the film which
will give us the size of oleic acid molecule.
Read the passage carefully and answer the following questions:
(a) Why do we dissolve oleic acid in alcohol?
(b) What is the role of lycopodium powder?
(c) What would be the volume of oleic acid in each mL of solution
prepared?
(d) How will you calculate the volume of n drops of this solution
of oleic acid?
(e) What will be the volume of oleic acid in one drop of this
solution?

If velocity of light c, Planck’s constant h and gravitational contant
G are taken as fundamental quantities then express mass, length
and time in terms of dimensions of these quantities.

The displacement of a progressive wave is represented by
y = A sin(wt – k x ), where x is distance and t is time. Write the
dimensional formula of (i) ω and (ii) k.

Give an example of
(a) a physical quantity which has a unit but no dimensions.
(b) a physical quantity which has neither unit nor dimensions.
(c) a constant which has a unit.
(d) a constant which has no unit.

If the unit of force is 100 N, unit of length is 10 m and unit of time
is 100 s, what is the unit of mass in this system of units?

During a total solar eclipse the moon almost entirely covers the
sphere of the sun. Write the relation between the distances and
sizes of the sun and moon.

The vernier scale of a travelling microscope has 50 divisions which
coincide with 49 main scale divisions. If each main scale division
is 0.5 mm, calculate the minimum inaccuracy in the measurement
of distance.

Which of the following time measuring devices is most precise?
(a) A wall clock.
(b) A stop watch.
(c) A digital watch.
(d) An atomic clock.
Give reason for your answer.

(a) The earth-moon distance is about 60 earth radius. What will
be the diameter of the earth (approximately in degrees) as seen
from the moon?
(b) Moon is seen to be of (½)°diameter from the earth. What must
be the relative size compared to the earth?
(c) From parallax measurement, the sun is found to be at a
distance of about 400 times the earth-moon distance. Estimate
the ratio of sun-earth diameters.

A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion:

(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.

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?

 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?

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.

(f) 0.0006032 m²

(e) 6.032 N m^-2

(d) 6.320 J

(c) 0.2370 g cm^-3

(b) 2.64 x 10²⁴ kg

State the number of significant figures in the following:

 The photograph of a house occupies an area of 1.75 cm²on a 35 mm slide. The slide is projected on to 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?

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?

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?

You are given a thread and a metre scale. How will you estimate the diameter of the thread?

 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?

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 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?

. A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m²s^–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 γ² in terms of the new units.

. Fill in the blanks by suitable conversion of units:
(a) 1 kg m²s^–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.

(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^–1covers....m in 1 s
(d) The relative density of lead is 11.3. Its density is ....g cm^–3or . ...kg m^–3.