Physical Quantities
Measurement
The process of comparing an unknown physical quantity with a known fixed unit quantity is called measurement.
Physical Quantity
Those quantity which can be measured and can be expressed in numerical value are called physical quantity. For example: Mass, Length, Time, etc.
The physical quantity can be divided into two parts:
- Fundamental Quantity
- Derived Quantity
Fundamental Quantity
The basic physical quantity which is taken as standard to measure other physical quantities is known as fundamental quantity.
For example: Length, mass, time, temperature, electric current, luminous intensity, amount of chemical substance.
Derived Quantity
A quantity obtained from fundamental quantities is called a derived quantity. For example: Area, volume, density, speed, electric intensity, etc.
Unit
The standard quantity in terms of which the given physical quantities can be compared is called unit. For example: kg, second, Newton, etc.
There are two types of unit:
- Fundamental Unit
- Derived Unit
Fundamental Unit
The units of fundamental quantities are called fundamental units. For example: kg, m, s, K, etc.
Derived Unit
The units of derived quantities are called derived units. For example: m/s, m/s2, etc.
System of Units
(i) F.P.S. System
In this system length is measured in foot, mass in pound and time in second. This is British system of units. Here, F → foot, P → pound, S → second.
(ii) MKS System
In this system, length is measured in metre, mass in kg, and time in second. This system was developed in France. Here, M → metre, K → kg and S → second.
(iii) CGS System
In this system, length is measured in centimetre, mass in gram and time in second. This system was developed in France. Here, C → centimetre, G → gram and S → second.
Characteristics of a Standard Unit
- It should be well defined and of a suitable size.
- It should be easily available, so that it can be easily reproduced in laboratories.
- It should not change with time and place.
- It should not change with change in physical condition.
- It should be universally agreed, so that result of measurement at different places can be compared.
SI System
The internationally standardised unit of measurement is called SI system.
In SI system of unit, 7 fundamental units and 2 supplementary units were proposed. These units are written below:
| Fundamental Quantities | S.I. Units | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Temperature | kelvin | K |
| Luminous Intensity | candela | cd |
| Electric Current | ampere | A |
| Amount of Chemical Substance | mole | mol |
Supplementary Quantities
| Supplementary Quantity | S.I. Unit | Symbol |
|---|---|---|
| Plane angle | radian | rad |
| Solid angle | steradian | sr |
Dimensions of Physical Quantity
The power raised to the fundamental quantities which are involved in derived physical quantities is called dimension of physical quantity.
Dimensional Formula of Different Physical Quantities
1) Speed
Speed = distance / time
= [L] / [T]
= [LT−1]
∴ The dimensional formula of speed = [M0LT−1]
2) Velocity
Velocity = displacement / time
= [L] / [T]
= [LT−1]
∴ The dimensional formula of velocity = [M0LT−1]
Dimensional Equation
An equation containing physical quantities, each quantity is represented by its dimensional formula, the resulting equation is known as dimensional equation.
Next definition: When the dimension of a quantity is found and expressed in the form of an equation, the equation is called the dimensional equation.
Principle of Homogeneity of Dimension
The physical relations must obey the principle of homogeneity. According to this concept:
Suppose in physical relation, s = ut + ½at2, there are three terms: s, ut and ½at2. All the terms must have the same dimension, i.e.
to obey the principle of homogeneity.
Table of Units and Dimensions of Fundamental Physical Quantities
| S.N. | Fundamental Physical Quantity | Dimensional Formula | SI Unit |
|---|---|---|---|
| 1 | Mass | [M] | Kilogram |
| 2 | Length | [L] | metre |
| 3 | Time | [T] | second |
| 4 | Temperature | [K] or [θ] | kelvin |
| 5 | Electric Current | [I] or [A] | ampere |
| 6 | Amount of Substance | [N] | mole (mol) |
| 7 | Luminous Intensity | [J] | candela (cd) |
A + B = C + D / or, A + B − C − D = 0
According to homogeneity:
dim. of A = dim. of B = dim. of C = dim. of D.
| S.N. | Physical Quantity | Formula / Relation | Dimensional Formula | SI Unit |
|---|---|---|---|---|
| 1 | Density | mass / volume | [M L−3 T0] | kg m−3 |
| 2 | Speed or Velocity | distance / time | [M0 L T−1] | m/s |
| 3 | Acceleration | velocity / time | [M0 L T−2] | m/s2 |
| 4 | Momentum | mass × velocity | [M L T−1] | kg m s−1 |
| 5 | Force | mass × acceleration | [M L T−2] | N (Newton) |
| 6 | Pressure | Force / area | [M L−1 T−2] | N m−2 or Pa |
| 7 | Work | force × displacement | [M L2 T−2] | J (joule) |
| 8 | Energy | Work (E = mc2) | [M L2 T−2] | J |
| 9 | Gravitational Constant (G) | force × d2 / (mass)2 | [M−1 L3 T−2] | N m2 kg−2 |
| 10 | Surface Tension | force / length | [M L0 T−2] | N m−1 |
| 11 | Moment of Inertia | mass × (distance)2 | [M L2 T0] | kg m2 |
| 12 | Angular Momentum | moment of inertia × angular velocity | [M L2 T−1] | kg m2 s−1 |
| 13 | Torque or Couple | force × perpendicular distance | [M L2 T−2] | N m |
| 14 | Frequency | 1 / second | [T−1] | Hz |
| 15 | Angular Velocity (ω) | velocity / radius | [T−1] | s−1 |
| 16 | Specific Heat | energy / (mass × temperature) | [M0 L2 T−2 K−1] | J kg−1 °C−1 |
| 17 | Stress | force / area | [M L−1 T−2] | N/m2 |
| 18 | Strain | Δl / l | [L0] dimensionless | Dimensionless |
| 19 | Refractive Index | V1 / V2 | Dimensionless | — |
| 20 | Mechanical Advantage | Load / Effort | Dimensionless | — |
| 21 | Electric Charge | I × t | [M0 L0 T I] | Amp-sec or Coulomb |
| 22 | Electric Resistance | V / I | [M L2 T−3 I−2] | ohm (Ω) or volt/amp |
| 23 | Boltzmann’s Constant | energy / temperature | [M L2 T−2 K−1] | J/K |
| 24 | Planck’s Constant (h) | E = hf | [M L2 T−1] | J·s or eV·s |
| 25 | Power of Lens (P) | P = 1/f | [L−1] | dioptre |
| 26 | Angle | — | Dimensionless | rad |
1) To Check the Correctness of a Physical Equation
Example: Check whether the physical equation v2 = u2 + 2as is dimensionally correct or not.
Given formula: v2 = u2 + 2as
[L.H.S.] = [v2] = [LT−1]2 = [L2T−2] …(i)
[R.H.S.] = [u2 + 2as]
= [LT−1]2 + [LT−2·L]
= [L2T−2] + [L2T−2]
Since the number 2 is dimensionless,
[R.H.S.] = [L2T−2] + [L2T−2] …(ii)
From equation (i) and (ii), [L.H.S.] = [R.H.S.]
Hence, the given equation is dimensionally correct.
2) To Derive the Relationship Between Different Physical Quantities
Example: Derive the dimensional relation of time period of pendulum with mass, length and acceleration due to gravity.
T ∝ lagb
or, T = k lagb …(i)
where k is dimensionless constant.
[T] = [L]a[LT−2]b
[M0L0T1] = [M0La+bT−2b]
Equating dimensions on both sides:
(i) 1 = −2b ⇒ b = −½
(ii) a + b = 0 ⇒ a = ½
Putting the value of a and b in equation (i):
T = k l1/2g−1/2
T = k √(l/g)
3) To Convert a Unit From One System Into Another
Example: Convert 10 dyne into newton.
Let 10 dyne = N2 newton.
Here, dyne is the unit of force in CGS and newton in SI.
Dimensional formula of force = [MLT−2]
∴ a = 1, b = 1 and c = −2 in mass, length and time respectively.
| CGS System | SI System |
|---|---|
| N1 = 10 dyne | N2 = ? |
| M1 = 1 g | M2 = 1 kg |
| L1 = 1 cm | L2 = 1 m |
| T1 = 1 s | T2 = 1 s |
According to the conversion formula:
N2 = N1 (M1/M2)a (L1/L2)b (T1/T2)c
N2 = 10 × (1g/1kg)1 × (1cm/1m)1 × (1s/1s)−2
= 10 × 10−3 × 10−2 × 1
10 dyne = 10−4 N
4) To Determine the Dimension of a Constant
Example: Determine the dimension of universal gravitational constant (G).
We have, F = Gm1m2 / r2
or, G = Fr2 / (m1m2)
= [MLT−2][L2] / ([M][M])
[G] = [M−1L3T−2]
Hence, the dimensions of G are −1 in mass, 3 in length and −2 in time.
Example: Find the dimensional formula of coefficient of viscosity.
We have, F = 6πηrv
η = F / (6πrv)
= [MLT−2] / ([L][LT−1])
[η] = [ML−1T−1]
Hence, the dimensional formula of coefficient of viscosity is [ML−1T−1].
5) To Establish Relationship Between Different Physical Quantities
Example: The centripetal force depends upon mass, velocity and radius of circular path.
Then, F ∝ mavbrc
F = k mavbrc …(i)
where k is dimensionless constant.
[MLT−2] = [M]a[LT−1]b[L]c
[MLT−2] = [MaLb+cT−b]
Equating dimensions on both sides:
(i) a = 1
(ii) −b = −2 ⇒ b = 2
(iii) b + c = 1 ⇒ 2 + c = 1 ⇒ c = −1
F = k(mv2/r) = mv2/r
Q.1 (A)
If v = a + bt, where v is velocity and t is time, then find dimension of ‘a’ & ‘b’.
v = a + bt
or, v − a − bt = 0
According to homogeneity:
dim. of a = dim. of v = [LT−1]
∴ dim. of a = [M0LT−1]
dim. of bt = dim. of v
b × T = LT−1
b = L/T2 = [LT−2]
∴ dim. of b = [M0LT−2]
Q.1 (B)
If x = a + c/m where x = displacement, m = mass, find the dimension of ‘a’ & ‘c’.
We have, x = a + c/m
or, x − a − c/m = 0
According to homogeneity:
Dim. of a = dim. of x = L
∴ Dim. of a = [M0LT0]
Dim. of c/m = dim. of x
c/M = L
or, c = ML
∴ Dim. of c = [MLT0]
Q.1 (C)
If μ = A + B/λ2, where μ = refractive index or dimensionless quantity and λ = wavelength (L), find the dimension of ‘A’ & ‘B’.
Given, μ − A − B/λ2 = 0
According to homogeneity:
∴ dim. of A = [M0L0T0]
dim. of B/λ2 = dim. of μ
B/[L2] = [M0L0T0]
or, B = [M0L2T0]
∴ dim. of B = [M0L2T0]
Q.1 (D)
Find dim. of x if y = tan(xt), where t is time.
Given, y = tan(xt)
dim. of xt = [M0L0T0]
x × T = [M0L0T0]
∴ x = [M0L0T−1]
Q.1 (E)
If P = a(r²x), find dim. of x, where r is radius.
Given, P = a(r²x)
We know, power → dimensionless.
i.e., dim. of r2x = [M0L0T0]
x[L2] = [M0L0T0]
or, x = [M0L−2T0]
Q.2
Find the dimensional formula of potential difference.
Potential Difference = W/Q = (F × displacement)/(I × t)
= [MLT−2][L] / ([A][T])
[V] = [ML2T−3A−1]
Q.3
Find the dimensional formula of resistance.
Resistance = V/I = W/(Q × I)
= W/(I × t × I)
= (F × displacement)/([I2][T])
[R] = [ML2T−3A−2]
- It cannot give information about dimensionless constant involved in a physical relation.
- It is very difficult to derive the dimensional relation between more than three physical quantities.
- It cannot be used to derive the relation involving trigonometric, exponential and logarithmic functions.
- It can’t tell us whether a quantity is vector or scalar.
- The dimensionally correct relation may not be physically correct.
Error
The difference between standard value and observed value of a physical quantity is called error.
Example
The measurement of resistance of a conductor is expressed as (10 ± 5%) Ω. What does it mean?
The measurement of resistance of a conductor is expressed as (10 ± 5%) Ω, which means 10 Ω is observed value and 5% is error, so the standard value of this resistance lies between (10 − 0.05) Ω to (10 + 0.05) Ω.
Types of Error
(a) Systematic Error
The error introduced due to the limitation of the formula used and fault in the instrument is called systematic error.
(b) Random Error
The error introduced due to the carelessness of the experiment and unfavourable condition of the environment such as temperature, pressure and humidity etc. is called random error.
Accurate Measurement
The measurement in which observed value of any physical quantity is closer to the standard value of that physical quantity is called accurate measurement.
Example
Standard value of g in lab is 9.8 m/s2. The measured value of g is obtained as 9.67 m/s2 and 9.79 m/s2. The value of 9.79 m/s2 is very close to the standard value, so it is more accurate than the value 9.67 m/s2.
Precise Measurement
The measurement in which the observed value of a physical quantity can be reproduced again and again by repeated experiment and procedure is called precise measurement.
If the readings are very close to each other, they are called precise readings. The thickness of a glass plate measured by spherometer are 2.43 mm, 2.44 mm, 2.42 mm and 2.44 mm. These readings are very close to each other. So they are precise measurement.
Least Count
Significant Figure
The meaningful digits of a number are called significant figure. Significant figure depends on the least count of a measuring device.
Example
Let us take the length of rod by a ruler for three times. Readings are obtained as 10.2 cm, 10.3 cm and 10.3 cm. So mean length:
Mean length = (10.2 + 10.3 + 10.3) / 3
= 10.266666 cm
Hence, all digits are not significant; only first three digits are significant. The length of rod is taken as 10.2 cm or 10.3 cm.
Question
The length of a rod is exactly 1 cm. An observer records the reading as 1.0 cm, 1.00 cm and 1.000 cm. Which is most accurate measurement?
Length = (1.0 + 1.00 + 1.000) / 3 = 3.000 / 3 = 1.000
Hence the length of rod = 1.0 or 1.00.
Discussion
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