## Concept of Length

### Definition

Length of an object can be defined as the distance of separation between any two points at the extreme ends of the object.

### Introduction

The term ‘length’ in physics is considered as ‘distance’. The most common unit of length is ‘meter’. Originally one meter was defined as one time millionth of the distance from the North Pole to the equator of earth. But this was not a convenient definition for practical usage and therefore came another definition.

In year 1899, the ‘General conference of weights and measurements’ modified the definition of ‘meter’ as:

‘Meter’ is the length between two bars on a platinum-iridium bar, kept at a constant temperature of 273. 16k and at 1 bar pressure.

In the year 1960, the above definition of ‘meter’ was again modified to:

“The standard ‘meter’ is exactly equal to 1650763.73 wavelengths in vacuum, of the radians from krypton isotope of mass ‘86’ ”

Yet another definition of meter was adopted in 1983 at the 17th general conference of weights and measures, (assuming velocity of light in vacuum C=299,792,458 m/s)
One meter is the length of the path travelled by light in vacuum in 1/299,792,458 of a second.

### Measurement of length

There are two methods of measuring length

• Direct method
• Indirect method

a) ‘Direct method’ involves comparison of distance or length to be measured with the chosen standard of length. The direct method for the measurement of length involves the use of

1. A meter scale for distances from 10-3 m to 10-2 m.
2. A Vernier caliper for distances up to 10-4 m.
3. A screw gauge and a spherometer for distances up to 10-5 m.

For measuring large distances from a few hundred meters to those of astronomical objects, the following indirect methods may be used.

• Echo (reflection) Method
• LASER Method
• SONAR Method
• Parallax Method
• Spectroscopic Method
• By Kepler’s third law
• Copernicus Method

#### Indirect methods for measuring small distances

1. Avogadro’s Method (for radius of atom)
2. Size of Molecule of Oleic Acid
3. Rutherford Method

Echo Method

This method may be used to find the distance of a hill from same point of observation. To do so, a gun/shot is fired towards the hill and the time interval between the instant of firing the gun and the instant of hearing the echo of the gunshot is noted . In this time interval, sound has travelled from the observer to the hill and then back to the observer. Now, if ‘V’ is the velocity of sound, ‘S’ is the distance of hill from the observer and ‘t’ is the total time taken then

LASER Method

LASER is an abbreviation for Light Amplification by Stimulated Emission of Radiations. It is very intense, monochromatic and unidirectional beam of light. The distance of moon from earth has been calculated by this method, using the same principle of ‘Echo’. The laser beam transmitted from earth is received back on earth after reflection from the moon. The time interval ‘t’ between transmission and reception of the beam is measured accurately. If ‘C’ is the velocity of laser beam in air or vacuum, then distance ‘S’ between earth and moon can be calculated as :

RADAR is an abbreviation for RAdio Detection And Ranging. Radio waves are sent into space from a transmitter. If an airplane comes between the path of radio waves, they are reflected back and detected by a detector. In this equipment, radio waves with traveling speed of c=3X108m/s are used. Measuring time interval between transmission and detection, the distance of the airplane can be computed by the formula

SONAR METHOD

SONAR is an abbreviation for SOund NAgivation and Ranging. This method is used for detecting an object under water and measuring its distance using ultrasonics. Here again, the same reflection method is used. The distance ‘S’ of the sea rock from any observation point is given by

where ‘V’ is velocity of ultrasonic waves in sea water and ‘t’ is the time interval between transmission and reception of ultrasonic signal.

Distance measurement using SONAR

Parallax Method

This method is used for measuring distances of nearby stars to planet earth. The diameter AB of earth’s orbit around the sun (S) is chosen as the base linc, M is the nearby star whose distance (d) from earth is to be measured. F is far off star whose direction is taken practically some at all the positions of earth in its orbital motion.

Parallax method

Suppose ‘A’ is the position of Earth in its orbital motion at any time. Using an astronomical telescope, it is measured that, (See in the above figure).

It is clear that,

After six months, when earth is at B, the position diametrically opposite to the position A on the orbit.

Again, from the above figure

Clearly,

This is an angle which nearby subtends on the orbital diameter of earth.

As

Clearly,

AB = 2SB = 2AS = 2AV

Hence, if (θ1 + θ2) is known (in radian) then AN can be obtained.

#### Indirect Method for measuring very small distances

The atoms are spherical in shape. When a large number of atoms are put together, the empty space are left between them. According to Avogadro’s hypothesis, volume of all atoms in any mass of the substance is 2/3 of the volume occupied by that mass of the substance.

Suppose

m = mass of a substance
V = Volume occupied by the substance
S = Density of the substance.

Let ‘m’ be the atomic weight of the substance and ‘N’ be the Avogadro number.

Therefore, Number of atoms in mass ‘m’ of the substance

= Nm/M

If ‘r’ be the radius of atom, then volume of each atom

= 4/3 πr3

Volume of all the atoms in the substance

= 4/3 πr3 × Nm/M

Now, according to Avogadro’s hypothesis,

Volume of all the atoms = 2/3( Volume of substance)

The value of ‘r’ so calculated is of the order of 10-10 m.

### Reference

• Pradeep’s Fundamental Physics (XI)
• Modern’s Abc of Physics (XI)