Measurement Techniques in Physics

In physics, we often deal with both large and small objects and distances. For instance, molecules, atoms, protons, neutrons, electrons, and bacteria constitute the microcosm—where both objects and distances are small. Conversely, larger objects like stars, galaxies, and planets belong to the macrocosm.

Direct Methods of Measurement

Distances ranging from \(10^{-5} \, \text{m}\) to \(10^2 \, \text{m}\) can be measured directly:

• A metre scale measures distances from \(10^{-3} \, \text{m}\) to \(1 \, \text{m}\).
• Vernier calipers measure up to \(10^{-4} \, \text{m}\).
• A screw gauge measures up to \(10^{-5} \, \text{m}\).

However, atomic and astronomical distances require indirect measurement methods.

Indirect Methods of Measurement

To measure very small and very large distances, indirect methods are used. The prefixes for various powers of 10 are shown in Table 1.

Multiple	Prefix	Symbol	Submultiple	Prefix	Symbol
101	deca	da	10-1	deci	d
102	hecto	h	10-2	centi	c
103	kilo	k	10-3	milli	m
106	mega	M	10-6	micro	μ
109	giga	G	10-9	nano	n
1012	tera	T	10-12	pico	p
1015	peta	P	10-15	femto	f
1018	exa	E	10-18	atto	a
1021	zetta	Z	10-21	zepto	z
1024	yotta	Y	10-24	yocto	y

Derived Quantities

In 1995, the supplementary quantities of plane and solid angle were converted into derived quantities.

1. The Radian (rad): One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.
2. The Steradian (sr): One steradian is the solid angle subtended at the center of a sphere by a surface equal in area to the square of the radius of the sphere.

Relationships

Note: π radians = 180°

1 radian = \( \frac{180}{\pi} \approx 57.27° \)

1° = 60' (minutes of arc)

1' = 60'' (seconds of arc)

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 the metre. Objects of interest vary widely in size, from large galaxies and stars to small atoms and bacteria.

Units and Conversions

Relationships between radian, degree, and minute:

\( 1° = \frac{\pi}{180} \text{ rad} \approx 1.745 \times 10^{-2} \text{ rad} \)

\( 1' = \frac{1}{60} \text{ degree} \approx 2.908 \times 10^{-4} \text{ rad} \)

\( 1'' = \frac{1}{3600} \text{ degree} \approx 4.85 \times 10^{-6} \text{ rad} \)

Instruments for Measuring Small Distances

Screw Gauge

The screw gauge is an instrument used for measuring dimensions up to a maximum of about 50 mm with high precision. It utilizes the principle of magnifying linear motion via 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 used to measure the dimensions of an object, such as the diameter or depth of a hole. The least count of the vernier caliper is 0.1 mm.

Example Readings

Screw Gauge:

PSR = 6 mm ; HSC=40 divisions

Reading = [6 mm + (40 x 0.01 mm)] = 6.40 mm

Vernier Caliper:

MSR = 2.2 cm ; VSC = 4 divisions

Reading = [2.2 cm + (4 x 0.01 cm)] = 2.24 cm

Measurement of Large Distances

Triangulation Method

This method is used to measure large distances such as the height of a tree or the distance to the Moon. The method involves measuring the angle of elevation and the distance from the observation point to the base of the object.

Example: Measuring the Height of a Tree

Problem: From a point on the ground, the top of a tree is seen to have an angle of elevation of 60°. The distance between the tree and the observation point is 50 m. Calculate the height of the tree.

Solution:

Angle θ = 60°

The distance between the tree and the observation point x = 50 m

Height of the tree (h) = ?

Using triangulation method:

\( \tan \theta = \frac{h}{x} \)

\( h = x \tan \theta \)

\( h = 50 \times \tan 60° \)

\( h = 50 \times 1.732 \)

Height of the tree: \( h = 86.6 \, \text{m} \)

Parallax Method

The parallax method measures very large distances, such as the distance to a planet or star. Parallax is the apparent shift in the position of an object when viewed from two different positions.

Determining the Distance to the Moon

Using the parallax method, we can determine the distance from the Earth to the Moon. The angle subtended by the Moon's parallax and the baseline (diameter of the Earth) are measured.

Example: Distance to the Moon

Problem: The Moon subtends an angle of 1° 55' at the baseline equal to the diameter of the Earth. What is the distance of the Moon from the Earth? (Radius of the Earth is 6.4 x 10⁶ m)

Solution:

The angle (θ) = 1° 55' = 1 + (55/60)° = 1.9167°

Convert to radians: \( θ = 1.9167 \times \frac{\pi}{180} \approx 0.0335 \, \text{rad} \)

The baseline (b) = 2 x radius of Earth = \( 2 \times 6.4 \times 10^6 \, \text{m} \)

Distance to the Moon (d) = \( \frac{b}{θ} = \frac{2 \times 6.4 \times 10^6}{0.0335} \approx 3.83 \times 10^8 \, \text{m} \)

RADAR Method

The word RADAR stands for Radio Detection and Ranging. A radar can be used to accurately measure the distance of nearby planets such as Mars. In this method, radio waves are transmitted from a source and, after reflecting off the planet, are detected by the receiver.

By measuring the time interval (\( t \)) between the transmission and reception of the radio waves, the distance to the planet can be determined using the formula:

Distance (\( d \)) = Speed of radio waves (\( v \)) × Time taken (\( t \)) / 2

Here, \( v \) is the speed of the radio waves. Since the time measured is for the radio waves traveling to the planet and back, it is divided by 2.

Example

Let's calculate the distance to Mars if the time interval (\( t \)) measured is 8.0 minutes (480 seconds). The speed of radio waves (\( v \)) is approximately \( 3 \times 10^8 \) m/s.

Distance (\( d \)) = \(\frac{v \times t}{2}\)
Distance (\( d \)) = \(\frac{3 \times 10^8 \, \text{m/s} \times 480 \, \text{s}}{2}\)
Distance (\( d \)) = \( 7.2 \times 10^{10} \, \text{m} \)

Thus, the distance to Mars is \( 7.2 \times 10^{10} \) meters.

Table of Range and Order of Lengths

Size of Objects and Distances	Length (m)
Distance to the boundary of observable universe	1026
Distance to the Andromeda galaxy	1022
Size of our galaxy	1021
Distance from Earth to the nearest star (other than the Sun)	1016
Average radius of Pluto’s orbit	1012
Distance of the Sun from the Earth	1011
Distance of Moon from the Earth	108
Radius of the Earth	107
Height of Mount Everest above sea level	104
Length of a football field	102
Thickness of a paper	10-4
Diameter of a red blood cell	10-5
Wavelength of light	10-7
Length of a typical virus	10-8
Diameter of the hydrogen atom	10-10
Size of atomic nucleus	10-14
Diameter of a proton	10-15

Physics Practice Questions