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Dimensional analysis is a technique used to compare the relationships between different quantities without confusing them with their units. It is applied when converting a unit between two different forms. The units must remain constant when answering mathematical problems in order to make the task simple. Dimensional analysis is done by simply reducing the typical physical properties like viscosity, density, acceleration, and energy into their fundamental dimensions of Length(L), Mass(M), Time(T), and electric current (I), as well as units of measurement like miles vs. kilometers, or pounds vs. kilograms.
Read More: Electric Current Formula
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Key Terms: Dimensional Analysis, Mass, Length, Kinetic Energy, Units, Time, Oscillation, Conversion Factors
What is Dimensional Analysis?
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Dimensional Analysis is a process of converting physically derived quantities into fundamental quantities for further observations. It is also used for the study of the relation between the dimensions and units of the quantities.
The fundamental dimensions are Length(L), Mass(M), and Time(T).
For example, acceleration (m/s²) in the form of M, L, and T can be written as L/T².
Dimensional Analysis Video Explanation
Read More: Unit of Length
Unit Conversion and Dimensional Analysis
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Dimensional analysis is also referred to as the unit factor method or factor label method. It is because during conversion a unit factor is involved.
For example, To calculate the number of meters in 5 km,
Known: 1 km = 1000 meters,
So, 5 km = 5 × 1000 = 5000 meters
Therefore, 1000 is used to convert the unit, so it can be called a conversion factor.
To use a conversion factor it is necessary that the values should represent the same quantity, like 1 hour is the same as 60 minutes, and 1000 meters is the same as 1 kilometer.
Examples of conversion factors:
- The periodic oscillation of a pendulum
- The energy of a vibrating object,
- Demand Vs Capacity for a rotating disc.
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Principle of Homogeneity
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Principle of Homogeneity says that the physical units in an equation have the same dimensions. If it's not the case, then it means that the equation does not have the same dimensions and it will not be representing a physical situation. This is the homogeneity principle. It is useful because it facilitates the conversion of units between different systems.
Example 1: In the equation, Ma Lb Tc =Mx Ly Tz
Then, according to this principle, a = x, b = y, and c = z.
Read More: Unit of Velocity
Dimensional Formula
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Dimensional Formula of any quantity is the statement indicating the powers to which the fundamental units are to be increased in order to obtain one unit of a derived quantity.
F
To Check the Accuracy of the Equation
Let’s understand this with an example,
The correct formula relation between speed, distance, and time is unknown
(i) Speed = Distance/Time is correct or
(ii) Speed =Time/Distance.
Now, dimensional analysis is employed to check whether this equation is correct or not.
By reducing both sides of the equation in its fundamental units form, we derive
(i) [L][T]-¹ = [L] / [T] (Right)
(ii) [L][T]-¹ = [T] / [L] (Wrong)
From the above example, it is evident that the dimensional formula establishes the correctness of an equation.
Read More: Dimensions of Physical Quantity
Derivation of Kinetic Energy
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The energy an object has as a result of motion is known as kinetic energy. It is described as the effort required to move a mass-determined body from rest to the indicated velocity. The SI Unit of Kinetic Energy is Joule.
Derivation: Dimensional formula of Kinetic Energy.
Kinetic energy of any object is given by,
| K.E. = ½(mass × velocity²) |
Dimensional formula of mass = M¹L0T0
Dimensional formula of velocity = m/s = L¹T-¹
On substituting the above dimensions of mass and velocity into the kinetic energy equation,
| → ½([ M¹L0T0] × [L¹T-¹]²) |
Hence,
| K.E. =[M¹L0T0] × [L¹T-¹]² |
Derived
| K.E. = [M]¹[L]²[T]-² |
Therefore, Dimensional Formula for kinetic energy is [M]¹[L]²[T]-²
Read More: Kinetic Theory
Applications of Dimensional Analysis
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Some of the applications of dimensional analysis in physics are as follows:
- Using the Homogeneity principle, the correctness of an equation or any other physical relation can be checked.
- The relation will be correct if the Left-hand side and right-hand side have identical dimensions.
- It is used to convert the unit of Physical quantity from one to another.
- It is also used to show the nature of the physical quantity.
- The dimensional expressions can be represented as algebraic quantities.
- The most important application of dimensional analysis is to derive formulas.
Read More: Units and Measurement
Limitations of Dimensional Analysis
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Dimensional Analysis has some limitations too. They are:
- Dimensional analysis does not provide any information regarding dimensional constants.
- Using Dimensional analysis cannot derive some complex functions like trigonometric functions, exponential, and logarithmic functions.
- It does not specify if a physical quantity is a vector or a scalar.
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Things to Remember
- Dimensional Analysis is a process of converting the physically derived quantities into their fundamental quantities for further observations.
- The fundamental dimensions are Length(L), Mass(M), and Time(T).
- An equation is dimensionally accurate if each term on both sides has the same dimensional formula.
- Dimensional analysis is also called the unit factor method or factor label method.
- Dimensional formula cannot assess whether the physical quantity is a vector or scalar quantity.
- Kinetic energy is the energy that an object has as a result of motion.
Previous Years’ Questions
- A screw gauge has least count of 0.01 mm and there are 50 divisions … [NEET – 2020]
- Dimensions of stress are...[NEET – 2020]
- Lumen is the unit of… [UPSEE – 2019]
- A quantity z, to be estimated has a dependency on the variables… [TS EAMCET – 20219]
- A student measured the diameter of a small steel ball using… [NEET – 2018]
- A physical quantity of the dimensions of length that can be formed… [NEET – 2017]
- Planck’s constant (ℎ), speed of light in vacuum (c) and Newton”s gravitational constant… [NEET – 2016]
- Which of the following is the smallest unit… [UPSEE – 2010]
- Body weighs 22.42 g and has a measured volume of 4.7… [WBJEE – 2010]
- A student measures the distance traversed in free fall… [NEET – 2010]
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Sample Questions
Ques. What are Dimensionless quantities? (2 Marks)
Ans. Physical quantities which do not have dimensions are called dimensionless quantities.
For example angles, specific gravity, strain, etc.
A physical quantity which is also a ratio of quantities of the same dimensions is also a dimensionless quantity.
Ques. Convert 300 cm into feet. (2 Marks)
Ans. First, convert cm into inches and then inches into feet.
300 cms = 300 × 1 / 30.48
After the calculation,
300 cms = 9.48 feet
Ques. Check the correctness of the equation: s = ut + ½ at² (5 Marks)
Ans. Here in the above equation
LHS is s, where s is displacement and
RHS is ut + ½ at²,where
u= initial velocity, t= time, a= acceleration
Now, dimensions of the quantities:
s or displacement = M0L¹T0 , t or time = M0L0T¹
a or acceleration = velocity / time =m/s × s = m/s² = L¹T-²
Now equating both hand sides,
M0L¹T0 = [L¹T-²] × T²
L¹ = L¹
LHS = RHS
The equation, s = ut + ½ at² is correct.
Ques. What is a Dimensional Formula? (2 Marks)
Ans. The dimensional formula of a physical quantity is an expression telling us how and which of the fundamental quantities enter into the unit of that quantity.
It is customary to express the fundamental quantities by a capital letter, e.g., length (L), mass (AT), time (T), electric current (I), temperature (K) and luminous intensity (C). We write appropriate powers of these capital letters within square brackets to get the dimensional formula of any given physical quantity.
Ques. What are the uses of Dimensional Formulas? (3 Marks)
Ans.
(i) Checking the results obtained
(ii) Conversion from one system of units to another
(iii) Deriving relationships between physical quantities
(iv) Scaling and studying of models.
The underlying principle for these uses is the principle of homogeneity of dimensions. According to this principle, the ‘net’ dimensions of the various physical quantities on both sides of a permissible physical relation must be the same; also only dimensionally similar quantities can be added to or subtracted from each other.
Ques. What are the limitations of Dimensional Analysis? (4 Marks)
(i) by this method the value of the dimensionless constant cannot be calculated.
(ii) by this method the equation containing trigonometric, exponential, and logarithmic terms cannot be analyzed.
(iii) if a physical quantity in mechanics depends on more than three factors, then a relation among them cannot be established because we can have only three equations by equalizing the powers of M, L, and T.
(iv) it doesn’t tell whether the quantity is vector or scalar.
Ques. Define Principle of Homogeneity. (1 Mark)
Ans. According to the Principle of Homogeneity, the physical units in an equation have the same dimensions. If it's not the case, it means the equation does not have the same dimensions.
Ques. A man walking briskly in rain with speed v must slant his umbrella forward making an angle θ with the vertical. A student derives the following relation between θ and v: tanθ = v and checks that the relation has a correct limit: as v—>θ, θ —>0, as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not, guess the correct relation. (3 Marks)
Ans. According to the principle of homogeneity of dimensional equations,
Dimensions of L.H.S. = Dimensions of R.H.S.
Here, v = tan θ
i.e., [L1 T-1] = dimensionless, which is incorrect.
Correcting the L.H.S., we. get
v/u= tan θ, where u is the velocity of rain.
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