Samacheer Class 11 Physics - Nature of Physical World and Measurement

Physical quantities are classified into two types. There are fundamental and derived quantities. Fundamental or base quantities are quantities which cannot be expressed in terms of any other physical quantities. These are length, mass, time, electric current, temperature, luminous intensity, and amount of substance.
Quantities that can be expressed in terms of fundamental quantities are called derived quantities. For example, area, volume, velocity, acceleration, force.

In order to determine the diameter of the moon, initially, a distance of the moon is calculated using the parallax method. Let D be the distance of the moon from the earth. Let d be the diameter of the moon. Let ∝ be the angular size of the angular diameter of the moon (ie) the angle subtended by d at the earth.
We have ∝ = d/D
d = ∝ D
The angle ∝ can be measured from the same location on the earth. When two diametrically opposite points of the moon are viewed through a telescope, the angle between the two directions gives the angular size or angular diameter. Since D is the known to size or diameter d of the moon can be determined.

(1) All non zero digits are significant
Example: 1342 has 4 significant figures
(2) All zeros between two non-zero digits are significant
Example: 2008 has four significant figures
(3) All zeros to the right of non-zero digit but to the left of the decimal point are significant.
Example: 3070.00 has 4 significant figures.
(4) The trailing zeros are not significant, ie in the number without a decimal point. All zeros are significant if they come from the measurement.
Example: 4000 has one significant figure.
(5) If a number is less than 1, the zero (s) on the right of the decimal point but to the left of the first non-zero digit are not significant.
Example: 0.0034 has 2 significant figures.
(6) All zeros to the right of the decimal point and to the right of non zero digits are significant
Example: 40.00 has four significant figures.
(7) The number of significant figures does not depend on the system of units used.
Example: 1.53cm, 0.0150cm, 0.0000153 Km all have three significant figures.
(8) The power of 10 is irrelevant to the determination of significant figures
Example: 5.7 x 10 2 cm has two significant figures.

(1) This method gives no information about the dimensionless constants in the formula. Like 1, 2,7i, e etc. ie they can not be determined using this analysis.
(2) This method can not decide whether the given quantity is a scalar or a vector.
(3) Using this method one cannot derive relations involving trigonometric, exponential and logarithmic functions.
(4) It cannot be applied to an equation involving more than three physical quantities.
(5) It can be used to check whether a given physical relation is dimensionally correct or not. The physical correctness can not be checked using this
For example:
s = ut + 1/3 at² is dimensionally correct were as physically not correct, as the correct equation is s = ut + 1/2at².

The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. The precision of measurement is the closeness of two or more measured values to each other.
The true value of a certain length is near 5.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 5.5 cm. In another experiment using a measuring instrument of greater resolution, say 0.01 cm, the length is found to be 5.38 cm. We find that the first measurement is more accurate as it is closer to the true value, but it has lesser precision. On the contrary, the second measurement is less accurate, but it is more precise.
III. Long Answer Questions:

Speed of light in vacuum, c = 1 new unit of length s -1
t = 8 min. 20 sec, = 500 s
x = ct= 1 new unit of length s -1 × 500s
x = 500 new unit of length
Conceptual Questions:

One light-year = 9.46 x 10 15 m = 9.46 x 10 12 Km. As the distance of stars is extraordinarily large, so it is convenient to express them in light-year rather than in meters or in kilometers.

The device that has a minimum least count is said to be more precise.
In case of screw gauge: Least count
= 0.05 mm
Out of this screw, gauges is having the minimum least count. So screw gauge is more precise.

It became necessary to redefine all units in atomic standards because the prototype offers the following difficulties.
* It is difficult to preserve prototype models.
* It is difficult to produce replicas of prototypes for their use in different countries.
* The techniques used for producing replicas are not of very high accuracy.
* Atomic standard units can be reproduced anywhere and at any time.
* It is variant in time and space.
* It is unaffected by environmental conditions like temperature, pressure, etc.
* It has an accuracy of 1 part in 10 9.

The dimensional analysis is not applicable on more than 3 physical quantities because the equating powers of M, L & T, we get three unknowns. Similar constraints are present for electrical or other non-mechanical quantities also

The principle of homogeneity of dimensions states that the dimensions of all the term in the physical expression should be same. This principle is used to check the correctness of the equation
For example
V 2 = U 2 + 2as
Writing dimensions on both sides
[LT -1 ]² = [LT -1 ]² + [LT -1 ]²
[L 2 T 2 ] = [L 2 T 2 ] + [L 2 T -2 ]
Here the dimensions of all the terms in the expression are same and equal to[L 2 T -2 ]
So the equation is dimensionally correct.

A number of measured quantities may be involved in the final calculation of an experiment. Different types of instruments might have been used for observation.
So the errors in the final result depends on –
(i) The error in individual measurements.
(ii) On the nature of mathematical operations.
The various possibilities of the propagation or combination of errors in different arithmetical operations are called propagation of errors.
Error in addition (or) sum of two quantities:
Let ∆A and ∆B be the absolute errors in measuring two quantities A and B respectively. Then
Measured value of A = A ± AA
Measured value of B = B ± AB
Consider sum A + B = Z
The error ∆Z in Z is given by
Z + ∆Z = (A ± ∆A) + (B ± ∆B).
= (A + B ) ± (∆A + ∆B)
Z + ∆Z = Z ± (∆A + ∆B)
∆Z = ∆A + ∆B
The maximum possible error in the sum of two quantities is equal to the sum of the absolute errors in the individual quantities.
Errors in multiplication (or) product of two quantities:
Let ∆A and ∆B be the absolute errors in the two quantities A and B respectively. Consider the product Z = AB
The error ∆Z in z given by
Z ± ∆Z = (A ± ∆A) (B ± ∆B)
= (AB) ± (A ∆ B) ± (B ∆ A) ± (∆A. ∆B)
Z ± ∆ Z = Z ± (A ∆ B) (B ∆ A) + (∆A. ∆B)
LHS by Z & RHS by AB = Z.
1± \(\frac { ∆ Z }{ Z }\) = 1 ± \(\frac { ∆ B }{ B }\) ± \(\frac { ∆ A }{ A }\) ± \(\frac { ∆A.∆B }{ AB }\)
Here \(\frac { ∆A.∆B }{ AB }\) can be neglected as \(\frac { ∆A}{ A }\) & \(\frac { ∆B }{ B }\) are small
∴ The maximum fractional error in Z is
\(\frac { ∆ Z }{ Z }\) = ±\(\frac { ∆A }{ A }\) x \(\frac { ∆B }{ B }\)
∴ The maximum fractional error in the product of two quantities is equal to the sum of fractional errors is the individual quantities.

The uncertainty in a measurement is called an error.
There are 3 types of errors namely –
* Random error
* Systematic error
* Gross error.
1. Systematic errors – These are reproducible inaccuracies that are consistently in the same direction. These occur often due to problem that persists throughout the experiment. Systematic errors are further classified as
* instrumental error
* imperfection in experimental technique or procedure
* personal errors
* errors due to external causes
* least count error.
(a) Instrumental error: When an instrument is not calibrated properly at the time of manufacture instrumental errors may occur.
Example: If the measurement is made with a meter scale whose end is worn out the result obtained will have errors.
Correction – These errors can be corrected by choosing the instrument carefully.
(b) Imperfections in experimental technique or procedure: These errors arise due to limitations in the experimental arrangement.
Example: While performing experiments with a calorie meter, if there is no proper insulation, there will be radiation losses. This results in an error.
Correction – Necessary steps and corrections should be applied and followed while performing experiments.
(c) personal errors: These errors are due to individuals performing the experiments., maybe due to incorrect initial setting up of the experiment or carelessness of the individuals making the observation due to improper precautions.
(d) Errors due to external causes: The change in external conditions during experiments can cause error in measurement.
Example: Changes in temperature, humidity or pressure during measurement may affect the result of the measurement.
(e) Least count error: Least count is the smallest value that can be measured by the measuring instrument and the error due to this measurement is the least count error. The instrument’s resolution is the cause of the error. The error is half of the least value measured by the device.
Correction – Least count error can be reduced by using a high precision instrument for measurement.
(2) Random errors – Random errors may arise due to random and unpredictable variations is experimental conditions like pressure, temperature voltage supply etc., Errors may also due to persona! errors by the observer. Random errors are sometimes called “Chance errors”.
Example: While measuring the thickness of a wire using a screw gauge, different readings are taken in different trails.
Correction – By taking the arithmetic mean of all readings observed may reduce the random error and the mean value is taken as best possible true value.
(3) Gross errors – The error caused due to sheer carelessness of an observer is called gross error.
Examples: Improper setting of the instrument Making wrong observations without bothering about the sources of errors and precautions. Using wrong values in calculation Recording wrong observations
Correction – This error can be minimized only when the observer is careful and mentally alert.

Measurement of small distances by screw gauge and vernier calipers 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 screw gauge is 0.01 mm.
Vernier calipers: A vernier caliper is a versatile instrument for measuring the dimensions of an object namely diameter of a hole or the depth of a hole. The least count of vernier caliper is a 0.01 cm.
Measurement of larger distances: For measuring larger distance such as height of a tree, distance of a moon or a planet from earth, the triangulation method, parallax method and radar method are used.
(a) Triangulation method for the height of an accessible object:
Let AB = h, be the height of a tree, to be measured. Let C be the point of observation at a distance X from B. Using a range finder placed
at C, ∠ACB = θ is measured.
∴ Considering ∆ ABC,
tan θ = \(\frac { AB }{ BC }\)
h = x tan θ
By knowing x, h can be calculated.
(b) Radar method:
RADAR is an acronym of Radio detection and ranging. A RADAR can be used to measure the distance of near planet, moon, enemy planes, moving as well as stationary targets etc. In this process, Radio signals are transmitted from the transmitter and after reflection from target, the radio signals are received by the receiver.
The time interval is recorded between the two instants i.e from time of transmission to time of reception. By knowing velocity and time, distance can be measured.
IV. Numerical Problems:

Time taken = 80s
Velocity of sound = V = 1460 m/s
Distance of enemy submarine d = ?
V = \(\frac{2d}{t}\)
d = \(\frac{Vt}{2}\)
= \(\frac{1460×80}{2}\)
= 1460 x 40
= 58400 m
d = 58.4k m

Given: radius: 3.12 m (Three significant figures)
Area of the circle = πr 2 = 3.14 × (3.12 m) 2 = 30.566
If the result is rounded off into three significant figure, the area of the circle = 30.6 m 2

Given Distance of Jupiter = 824.7 × 10 6 km = 8.247 × 10 11 m
angular diameter = 35.72 × 4.85 × 10 -6 rad = 173.242 × 10 -6 rad = 1.73 × 10 -4 rad
∴ Diameter of Jupiter D = D × d = 1.73 × 10 -4 rad × 8.247 × 10 11 m
= 14.267 × 1o 7 m = 1.427 × 10 8 m (or) 1.427 × 10 5 Km

l = 20 x 10 -2 m = 20 cm
∆l = 2mm = 0.2 cm
Time for 50 oscillations = 40s
Time for 1 oscillation = T = \(\frac { 40 }{ 50 }\)
= \(\frac { 4 }{ 5 }\)s
∆T = 1s.
∆T = \(\frac { 1 }{ 50 }\)S
= 1% + 5%
= 6%
11th Physics Guide Nature of Physical World and Measurement Additional Important Questions and Answers
I. Multiple choice questions: