UNITS AND MEASUREMENTS COMPLETE CLASS NOTES : All Concepts, Tricks & PYQs | NEET | MDCAT | MCAT| 11th | 12th | PHYSICS
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NEET | MDCAT |MCAT
Need for measurement: Units of measurement; systems of units; SI units, fundamental and derived units. Length, mass and time measurements; accuracy and precision of measuring instruments; errors in measurement; significant figures. Dimensions of physical quantities, dimensional analysis and its app...
1. Errors
2. Errors in Measurement
3. PHYSICAL QUANTITY
4. UNIT
5. Types of Physical Quantity
6. parallax method is used to measure distance of star from Earth
7. Dimensions
8. SOME BASIC PHYSICAL QUANTITIES WITH THEIR
DIMENSIONAL FORMULA
9. Significant Figures
10. Rounding Off
11. EXERCISE ……………
AND MORE EXAM TOPIC COVER HERE WITH MCQ……
, UNITS AND MEASUREMENT
Errors – The difference between the measured and Relative error or Fractional error
the true value of a physical quantity is called as 𝑀𝑒𝑎𝑛 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑒𝑟𝑟𝑜𝑟
=
𝛥𝑎
𝑀𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒
error. Errors are of two main types. 𝑎𝑚
Systematic error – These errors may be due to (4) Percentage error: When the relative/fractional
faulty design or calibration of measuring error is expressed in percentage, we call it
instrument, or imperfection in experimental percentage error. Thus
𝛥𝑎
procedure and personal error. These errors can be Percentage error = × 100%
𝑎𝑚
minimized by improving the experimental
techniques and better equipment.
EX-1 The period of oscillation of a simple
Random error – These errors can arise due to
pendulum in the experiment is recorded as 2.63 s,
unpredictable fluctuations in experimental
2.56 s, 2.42 s, 2.71 s and 2.80 s respectively. The
conditions, over which we don’t have control. So,
average absolute error is
these errors cannot be minimized.
2.63+2.56+2.42+2.71+2.80
Average value = 5
= 2.62 𝑠𝑒𝑐
Now |𝛥𝑇1| = 2.63 − 2.62 = 0.01s
Errors in Measurement: -
|𝛥𝑇2| = 2.62 − 2.56 = 0.06s
(1) Absolute error: The difference of the true value
|𝛥𝑇3| = 2.62 − 2.42 = 0.20s
and the measured value of the quantity is called as
|𝛥𝑇4| = 2.71 − 2.62 = 0.09s
absolute error.
|𝛥𝑇5| = 2.80 − 2.62 = 0.18s
Let a physical quantity be measured n times. Let the
Mean absolute error
measured value be a1, a2, a3, ….. an. The arithmetic
𝑎 +𝑎 +......+𝑎𝑛
mean of these value is 𝑎𝑚 = 1 2 |𝛥𝑇1 | + |𝛥𝑇2 | + |𝛥𝑇3 | + |𝛥𝑇4 | + |𝛥𝑇5|
𝑛 𝛥𝑇 =
5
Usually, am is taken as the true value of the quantity, 0.54
= = 0.108 = 0.11𝑠𝑒𝑐
if the same is unknown otherwise. 5
By definition,absolute errors in the measured
values of the quantity are (1) Error in sum of the quantities: Suppose x = a +
𝛥𝑎1 = 𝑎𝑚 − 𝑎1 b
𝛥𝑎2 = 𝑎𝑚 − 𝑎2 Let a = absolute error in measurement of a
…………. b = absolute error in measurement of b
𝛥𝑎𝑛 = 𝑎𝑚 − 𝑎𝑛 x = absolute error in calculation of x i.e. sum of a
The absolute errors may be positive or and b.
negative. The maximum absolute error in x is 𝛥𝑥 = ±(𝛥𝑎 +
𝛥𝑏)
(2) Mean absolute error: It is the arithmetic mean (𝛥𝑎+𝛥𝑏)
Percentage error in the value of 𝑥 = ×
of the magnitudes of absolute errors in all the 𝑎+𝑏
measurements of the quantity. It is represented by 100%
𝛥𝑎. Thus
|𝛥𝑎1 | + |𝛥𝑎2|+. . . . . |𝛥𝑎𝑛| Ex-2 If 𝒎𝟏(𝟐±. 𝟐)𝒌𝒈 𝒂𝒏𝒅 𝒎𝟐 = (𝟒±. 𝟏)𝒌𝒈. Find
𝛥𝑎 = the sum of the masses with in error limits
𝑛
Hence, the final result of measurement may be Solution:- 𝑚 = 𝑚1 + 𝑚2
written as 𝑎 = 𝑎𝑚 ± 𝛥𝑎 = (2 + 4) ± (. 2 + .1) = (6 ± .3)𝑘𝑔
This implies that any measurement of the quantity
is likely to lie between (𝑎𝑚 + 𝛥𝑎) and (𝑎𝑚 − 𝛥𝑎). (2) Error in difference of the quantities:
Suppose x = a – b
(3) Relative error or Fractional error: The relative Let a = absolute error in measurement of a,
error or fractional error of measurement is defined
b = absolute error in measurement of b
as the ratio of mean absolute error to the mean
x = absolute error in calculation of x i.e. difference
value of the quantity measured. Thus
of a and b.
1
, 𝛥𝑥 𝛥𝑎
The maximum absolute error in x is 𝛥𝑥 = ±(𝛥𝑎 + The maximum fractional error in x is = ±( +
𝑥 𝑎
𝛥𝑏) 𝛥𝑏
(𝛥𝑎+𝛥𝑏) )
Percentage error in the value of 𝑥 = × 𝑏
𝑎−𝑏
100% Percentage error in the value of x = (% error in value
of a) + (% error in value of b)
Ex-3 The temperature of two bodies measured by
a thermometer are 𝑻𝟏 = 𝟐𝟎 ℃ ± Ex – 5 A body travels uniformly a distance of (13.8
𝟎. 𝟓℃ 𝒂𝒏𝒅 𝑻𝟐 = 𝟓𝟎℃ ± 𝟎. 𝟓℃. Calculate the 0.2) m in a time (4.0 0.3) s. The velocity of the
temperature difference and the error limits. body within error limits is
[NCERT] Here, 𝑆 = (13.8 ± 0.2) m and 𝑡 = (4.0 ± 0.3) sec
Sol- ∆𝑻 = 𝑻𝟐 − 𝑻𝟏 Expressing it in percentage error, we have,
= (50℃ ± 0.5℃) − (20℃ ± 0.5℃) 0.2
𝑆 = 13.8 ± × 100% = 13.8 ± 1.4%
∆𝑇 = 30℃ ± 1℃ 13.8
0.3
and 𝑡 = 4.0 ± × 100% = 4 ± 7.5%
4
(3) Error in product of quantities: 𝑠 13.8 ± 1.4
∵𝑉= = = (3.45 ± 0.3) 𝑚/𝑠.
Suppose x = a b 𝑡 4 ± 7.5
Let a = absolute error in measurement of a,
b = absolute error in measurement of b (5) Error in quantity raised to some power : Suppose
𝑎𝑛
x = absolute error in calculation of x i.e. product of 𝑥=
𝑏𝑚
a and b. Let a = absolute error in measurement of a,
𝛥𝑥 𝛥𝑎
The maximum fractional error in x is
𝑥
= ±( 𝑎
+ b = absolute error in measurement of b
𝛥𝑏
) x = absolute error in calculation of x
𝑏 𝛥𝑥 𝛥𝑎
Percentage error in the value of x = (% error in value The maximum fractional error in x is = ± (𝑛 +
𝑥 𝑎
of a) + (% error in value of b) 𝛥𝑏
𝑚 )
𝑏
Percentage error in the value of x = n (% error in
Ex-4 The length and breadth of the rectangular value of a) + m (%error in value of b)
sheet is given by
𝑳 = (𝟏𝟐±. 𝟑)𝒎 • The quotient rule is not
𝒃 = (𝟒±. 𝟐)𝒎 applicable If the numerator or
Find the area of the rectangle within the error denominator are dependent
limits. on each other.
Sol. 𝐴 = 𝐿 × 𝑏 = 12 × 4 = 48 𝑚2 • Errors are always additive in
∆𝐴 ∆𝐿 ∆𝑏 nature
= +
𝐴 𝐿 𝐵
∆𝐴 . 3 . 2 Ex – 6 The percentage errors in the measurement
= +
48 12 4 of mass and speed are 2% and 3% respectively.
∆𝐴 3 How much will be the maximum error in the
=
48 40 estimation of the kinetic energy obtained by
∆𝐴 = 3.6 𝑚2 measuring mass and speed
Area =(48 ± 3.6)𝑚2 1
𝐸 = 𝑚𝑣2
2
We know,
𝑎
(4) Error in division of quantities: Suppose 𝑥 = % Error in K.E. = % error in mass + 2 × % error in
𝑏
velocity = 2 + 2 × 3 = 8 %
Let a = absolute error in measurement of a,
b = absolute error in measurement of b
x = absolute error in calculation of x i.e. division of
a and b.
2
, Ex – 7 In an experiment, the following PHYSICAL QUANTITY
observations were recorded: L = 2.820 m, M =
3.00 kg, l = 0.087 cm, Diameter D = 0.041 cm Taking All the quantities in terms of which laws of physics
𝑚 𝟒𝑴𝒈𝑳 are described and which can be measured directly
g = 9.81 𝑠2 using the formula, 𝒀 = , the
𝝅𝑫𝟐 𝒍 or indirectly are called physical quantities.
maximum permissible error in Y is
4𝑀𝑔𝐿 OR
𝑌=
𝜋𝐷2 𝑙
so maximum permissible error in Y = 𝛥𝑌 × 100 A quantity which can be measured and with the
𝑌
𝛥𝑌 𝛥𝑀 𝛥 𝛥 2𝛥𝐷 𝛥𝑙 help of which, various physical happenings can be
𝑔 𝐿
× 100 = ( + + + + ) × 100 explained and expressed in the form of laws, is
𝑌 𝑀 𝑔 𝐿 𝐷 𝑙
𝛥𝑌 1 1 1 1 1 called a physical quantity.
× 100 = ( + + +2× + )
𝑌 300 981 2820 41 87 MEASUREMENT –
× 100
𝛥𝑌 A measurement is a comparison with
× 100 = 0.065 × 100 = 6.5%
𝑌 internationally accepted standard measuring unit.
Ex–8 The period of oscillation of a simple UNIT –
𝒍 The process of measurement is a comparison
pendulum is given by 𝑻 = 𝟐𝝅 √ where l is about
𝒈
process.
100 cm and is known to have 1mm accuracy. The
period is about 2s. The time of 100 oscillations is Unit is the standard quantity used for comparison.
measured by a stop watch of least count 0.1 s. The
The chosen standard for measurement of a physical
percentage error in g is
quantity, which has the same nature as that of life
𝑙
𝑇 = 2𝜋√ ⇒ 𝑇2 = 4𝜋 𝑙 ⇒ 𝑔 = 4𝜋 𝑙
2 2 of the quantity is called the unit of that quantity.
𝑔 𝑔 𝑇2
1𝑚𝑚 0.1
Here % error in l = × 100 = × 100 =
100𝑐𝑚 100 CHARACTERISTICS OF A UNIT –
0.1%
0.1
and % error in T = × 100 = 0.05% (1) It should be suitable in size (suitable to use)
2×100
% error in g = % error in l + 2(% error in T) (2) It should be accurately defined (so that
= 0.1 + 2 × 0.05 = 0.2 % everybody understands the unit in same way)
𝟏
𝑨 𝟒𝑩𝟑
EX-9 Find the relative error in Z, if 𝒁 = 𝟑
(3) It should be easily reproducible.
𝑪𝑫𝟐
[NCERT] (4) It should not change with time.
Sol. The relative error in Z is (5) It should not change with change in physical
∆𝑍 ∆𝐴 1 ∆𝐵 ∆𝐶 3 ∆𝐷
= 4 ( ) + ( ) ( ) + ( ) + ( )( ) conditions. i.e., temperature, pressure, moisture
𝑍 𝐴 3 𝐵 𝐶 2 𝐷 etc.
(6) It should be universally acceptable.
EX-10 A physical quantity P is related to four
observables a, b, c and d as follows: System of units are classified mainly into four types:
𝑷 = 𝒂𝟑𝒃𝟐/(√𝒄𝒅)
The percentage errors of measurement in a, b, c 1. C.G.S system:
and d are 1%, 3%, 4% and 2% respectively. What
It stands for Centimetre-Gram-Second system. In
is the percentage error in the quantity P? [NCERT]
∆𝑃 ∆𝑎 ∆𝑏 this system, length, mass and time are measured in
sol. × 100 = 3 ( × 100) + 2 ( × 100)+
𝑃 𝑎 𝑏 centimetre, gram and second respectively.
1∆𝑐( ∆𝑑
2 𝑐 × 100) + ( × 100)
𝑑
1
3(1%) + 2(3%) + (4%) + (2%) = 13%
2
3
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