length, mass, Time intervals | Solved Example Problems - Measurement of Basic Quantities | 11th Physics : UNIT 1 : Nature of Physical World and Measurement
Chapter: 11th Physics : UNIT 1 : Nature of Physical World and Measurement
Measurement of Basic Quantities
MEASUREMENT OF BASIC QUANTITIES
Measurement
of length
The
concept of length in physics is related to the concept of distance in everyday
life. Length is defined as the distance between any two points in space. The SI
unit of length is metre. The objects of our interest vary widely in sizes. For
example, large objects like the galaxy, stars, Sun, Earth, Moon etc., and their
distances constitute a macrocosm. It refers to a large
world, in which both objects and distances are large. On the contrary, objects
like molecules, atoms, proton, neutron, electron, bacteria etc., and their
distances constitute microcosm, which means a small world
in which both objects and distances are small-sized.
The
Radian (rad): One radian is the angle subtended at
the centre of a circle by an arc equal in length to the radius of the circle.
The
Steradian (sr): One steradian is the solid angle
subtended at the centre of a sphere, by that surface of the sphere, which is
equal in area, to the square of radius of the sphere
Distances
ranging from 10−5 m to 102m can be measured by direct
methods. For example, a metre scale
can be used to measure the distance from 10−3 m to 1 m, vernier
calipers up to 10−4m, a screw gauge up to 10−5 m and so
on. The atomic and astronomical distances cannot be measured by any of the
above mentioned direct methods. Hence, to measure the very small and the very
large distances, indirect methods have to be devised and used. In Table 1.4, a
list of powers of 10 (both positive and negative powers) is given. Prefixes for
each power are also mentioned. These prefixes are used along with units of
length, and of mass.
i. Measurement of small distances:
screw gauge and vernier caliper Screw gauge:
The screw gauge is an
instrument used for measuring accurately the dimensions of objects up to a
maximum of about 50 mm. The principle of the instrument is the magnification of
linear motion using the circular motion of a screw. The least count of the
screw gauge is 0.01 mm Vernier caliper: A
vernier caliper is a versatile
instrument for measuring the dimensions of an object namely diameter of a hole,
or a depth of a hole. The least count of the vernier caliper is 0.1 mm
ii. Measurement of large distances
For
measuring larger distances such as the height of a tree, distance of the Moon
or a planet from the Earth, some special methods are adopted. Triangulation
method, parallax method and radar method are used to determine very large
distances.
Triangulation method for the height of an accessible object
Let
AB = h be the height of the tree or tower to be measured. Let C be the point of
observation at distance x from B.
Place a range finder at C and measure the angle of elevation, ∠ACB = θ as shown in Figure 1.3.
height h = x tan θ
Knowing
the distance x, the height h can be
determined.
Range
and order of lengths of various objects are listed in Table 1.5
Measurement of mass
Mass
is a property of matter. It does not depend on temperature, pressure and
location of the body in space. Mass of a
body is defined as the quantity of
matter contained in a body. The SI unit of mass is kilogram (kg). The masses of objects which we
shall study in this course vary over a wide range. These may vary from a tiny
mass of electron (9.11×10−31kg) to the huge mass of the known
universe (=1055 kg). The order of
masses of various objects is shown in Table 1.6.
Ordinarily, the mass of an object is determined in kilograms using a common balance like the one used in a grocery shop. For measuring larger masses like that of planets, stars etc., we make use of gravitational methods.
For measurement
of small masses of atomic/subatomic particles etc., we make use of a mass
spectrograph.
Some
of the weighing balances commonly used are common balance, spring balance,
electronic balance, etc.
Measurement
of Time intervals
A
clock is used to measure the time interval. An atomic standard of time, is
based on the periodic vibration produced in a Cesium atom. Some of the clocks
developed later are electric oscillators, electronic oscillators, solar clock,
quartz crystal clock, atomic clock, decay of elementary particles, radioactive
dating etc. The order of time intervals are tabulated in Table 1.7.
Solved Example Problems for Measurement of length
Example
1.1
From a point on the ground, the top of a tree is seen to have an
angle of elevation 60°. The distance between
the tree and a point is 50 m. Calculate the height of the tree?
Solution
Angle θ = 60°
The distance between the tree and a point x = 50 m
Height of the tree (h) = ?
For triangulation method tan
h = x tan θ
= 50 × tan 60°
= 50 × 1.732
h = 86.6 m
The height of the tree is 86.6 m.
Parallax method
Very large distances, such as the distance of a planet or a star
from the Earth can be measured by the parallax method. Parallax is the name given to
the apparent change in the position of an object with respect to the
background, when the object is seen from two different positions. The
distance between the two positions
(i.e., points of observation) is called the basis (b). For example, consider
Figure 1.4., an observer is specified by the position O. The observer is
holding a pen before him, against the background of a wall. When the pen is looked
at first by our left eye L (closing the right eye) and then by our right eye R
(closing the left eye), the position of the pen changes with respect to the
back ground of the wall. The shift in the
position of an object (say, a pen)
when viewed with two eyes, keeping one eye closed at a time is known as Parallax. The distance between the left eye (L) and the right eye (R)
in this case is the basis.
∠LOR is called the parallax
angle or parallactic angle.
Taking LR as an arc of length b and radius LO = RO = x
we get θ = b/x, b-basis, x-unknown
distance
Knowing ‘b’ and measuring θ, we can calculate x.
Determination of distance of Moon from Earth
In Figure 1.5, C is the centre of the Earth. A and B are two
diametrically opposite places on the surface of the Earth. From A and B, the
parallaxes θ1 and θ2 respectively of Moon
M with respect to some distant star are determined with the help of an
astronomical telescope. Thus, the total parallax of the Moon subtended on Earth
∠AMB =
θ1 + θ2 =
θ.
If θ
is measured in
radians, then
Knowing the values of AB
and θ,
we can calculate the distance MC of Moon from the Earth.
Example
1.2
The Moon subtends an angle of 1° 55’ at the base line
equal to the diameter of the Earth. What is the distance of the Moon from the
Earth? (Radius of the Earth is 6.4 × 106 m)
Solution
Radius of the Earth = 6.4 × 106 m
From the Figure 1.5 AB is the diameter of the Earth (b)= 2 × 6.4 × 106 m Distance of the Moon from the Earth x = ?
RADAR method
The word RADAR stands for radio detection and ranging. A radar
can be used to measure accurately the distance of a nearby planet such as Mars.
In this method, radio waves are sent from transmitters which, after reflection
from the planet, are detected by the receiver. By measuring, the time interval
(t) between the instants the radio waves are sent and received, the distance of
the planet can be determined as
where v is the speed of the radio wave. As the time taken (t) is for the distance covered during the forward and backward path of the radio waves, it is divided by 2 to get the actual distance of the object. This method can also be used to determine the height, at which an aeroplane flies from the ground.
Example
1.3
A RADAR signal is beamed towards a planet and its echo is
received 7 minutes later. If the distance between the planet and the Earth is
6.3 × 1010 m. Calculate the speed of the
signal?
Solution
The distance of the planet from the Earth d = 6.3 × 1010 m
The speed of signal
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