Dimension of physical quantities and how to write the dimension of a given physical quantity
Dimension of physical quantity simply means the relationship between the physical quantity and the fundamental quantities (example of fundamental quantity is mass, length, time etc.). That is, it tries to point out the various ways through which the physical quantity of interest is related, connected or associated with the fundamental quantities.
Importance of Concept of Dimension in Physics
1. It is used to check or verify the correctness of a physical equation
2. It is also used to derive the unit of a physical quantity
3. It is used to show the relationship between physical quantities
We will illustrate the importance of dimension as we progress through examples so sit tight. For now, let’s see the basic things to know before you can write the dimension of any physical quantity.
Representation of fundamental quantities in dimension
In dimension analysis, fundamental quantities are not represented by their units rather, they are represented by a letter which is frequently their initials.
1. Mass is represented by M
2. Length is represented by L
3. Time is represented by T
4. Current is represented by A
5. Temperature is represented by K
6. Luminous intensity is represented by II
Steps on how to write the dimension of any physical quantity
1. Write down the formula of the physical quantity
2. Replace the variables in the formula with their dimensions as listed above
You can see that writing the dimension of a physical quantity also follows the same steps as that of deriving the units for it.
Example 1
Derive the dimension for the following physical quantities
1. Area
2. Volume
3. Speed
4. Velocity
5. Linear acceleration
6. Momentum
7. Force
8. Density
9. Work
10. Energy
11. Moment of inertia
12. Velocity gradient
13. Young’s modulus
14. Electric Charge
15. Electric potential or potential difference or voltage
16. Capacitance of a capacitor
17. Strain
Wow! What a long list!! Now let’s get to work. It is very important to note that a little knowledge of indices is required in order to be able to solve this problems. So you can quickly run through your multiplication, division and power law in indices
Solution:
1. Dimension of Area
Step 1: write down the formula
Area = Length X Breadth
Step 2: replace variables in the formula with their dimensions (see dimensions on the list above)
Area = L X L
apply multiplication law of indices
Area = L2
2. Dimension of volume
Step 1: write down the formula
Volume = Length X Breadth X Height
Step 2: replace variables in the formula with their dimensions (see dimensions on the list above)
= L X L X L
apply multiplication law of indices
= L3
3. Speed = $\frac{distance}{time}$
= \[\frac{L}{\begin{align}
& T \\
& L{{T}^{-1}} \\
\end{align}}\]
4. Velocity = $\frac{displacement}{\begin{align}
& time \\
& \frac{L}{\begin{align}
& T \\
& L{{T}^{-1}} \\
\end{align}} \\
\end{align}}$
5. Linear acceleration = $\frac{Velocity}{\begin{align}
& time \\
& \frac{L{{T}^{-1}}}{\begin{align}
& T \\
& =L{{T}^{-2}} \\
\end{align}} \\
\end{align}}$.
6. Momentum = Mass X Velocity
= M X LT-1
= MLT-1
7. Force = Mass X Acceleration
= M X LT-2
= MLT-2
8. Density = $\begin{align}
& \frac{mass}{\begin{align}
& volume \\
& =\frac{M}{{{L}^{3}}} \\
\end{align}} \\
& =M{{L}^{-3}} \\
\end{align}$
9. Work = Force X distance
= Mass X Acceleration X Distance
= M X LT-2 X L
= ML2T-2
10. Energy = Force X distance
= MLT-2 X L
= ML2T-2
11. Moment of Inertia = mass X length2
= M X L2
= ML2
12. Velocity Gradient = $\frac{Velocity}{\begin{align}
& dis\tan ce \\
& =\frac{M{{L}^{-1}}}{L} \\
& =M{{L}^{-2}} \\
\end{align}}$
13. Young’s modulus = $\frac{force\times length}{\begin{align}
& area\times extension \\
& =\frac{ML{{T}^{-2}}\times L}{{{L}^{2}}\times L} \\
& =M{{L}^{-1}}{{T}^{-2}} \\
\end{align}}$
14. Electric charge = Current X Time
= A X T
= AT
15. Electric potential or potential difference or voltage =
$\begin{align}
& \frac{Power}{Current} \\
& =\frac{\frac{workdone}{time}}{current} \\
& =\frac{\frac{Force\times dis\tan ce}{time}}{current} \\
& =\frac{\frac{ML{{T}^{-2}}\times L}{T}}{A} \\
& =\frac{M{{L}^{2}}{{T}^{-3}}}{A} \\
& =M{{L}^{2}}{{T}^{-3}}{{A}^{-1}} \\
\end{align}$
16. Capacitance $\begin{align}
& \frac{Ch\arg e}{Potential} \\
& =\frac{AT}{M{{L}^{2}}{{T}^{-3}}{{A}^{-1}}} \\
& ={{A}^{2}}{{T}^{4}}{{M}^{-1}}{{L}^{-2}} \\
\end{align}$
17. Strain $\begin{align}
& \frac{extension}{length} \\
& =\frac{L}{L} \\
& =\dimensionless \\
\end{align}$
You can see from question 17 that not all physical quantities can be dimensioned.
Test your understanding by attempting to write the dimension of the following physical quantities
Practice Exercises:
1. Universal gravitational constant (G)
2. Momentum
3. Resistance
4. Power
5. Pressure
6. Impulse
7. Electrical force
8. Escape velocity
9. Centripetal force
10. Weight
11. Temperature
12. coefficient of friction ยต
Hint: don’t be intimidated by the exercises given above, they are all simple and they follow the same steps as those above. Just follow the steps given above and you will get it. if you still have difficulties, you can let me know through the comment section and also remember to comment your answers too.
First, write down the formula. Sometimes, it is necessary to break down the formula to the root before proceeding. For example, Force = mass X acceleration, you can breakdown acceleration to get velocity/time, you can also breakdown velocity to get displacement/time and then replace them with their dimensions. .
Dimension of physical quantity simply means the relationship between the physical quantity and the fundamental quantities (example of fundamental quantity is mass, length, time etc.). That is, it tries to point out the various ways through which the physical quantity of interest is related, connected or associated with the fundamental quantities.
Importance of Concept of Dimension in Physics
1. It is used to check or verify the correctness of a physical equation
2. It is also used to derive the unit of a physical quantity
3. It is used to show the relationship between physical quantities
We will illustrate the importance of dimension as we progress through examples so sit tight. For now, let’s see the basic things to know before you can write the dimension of any physical quantity.
Representation of fundamental quantities in dimension
In dimension analysis, fundamental quantities are not represented by their units rather, they are represented by a letter which is frequently their initials.
1. Mass is represented by M
2. Length is represented by L
3. Time is represented by T
4. Current is represented by A
5. Temperature is represented by K
6. Luminous intensity is represented by II
Steps on how to write the dimension of any physical quantity
1. Write down the formula of the physical quantity
2. Replace the variables in the formula with their dimensions as listed above
You can see that writing the dimension of a physical quantity also follows the same steps as that of deriving the units for it.
Example 1
Derive the dimension for the following physical quantities
1. Area
2. Volume
3. Speed
4. Velocity
5. Linear acceleration
6. Momentum
7. Force
8. Density
9. Work
10. Energy
11. Moment of inertia
12. Velocity gradient
13. Young’s modulus
14. Electric Charge
15. Electric potential or potential difference or voltage
16. Capacitance of a capacitor
17. Strain
Wow! What a long list!! Now let’s get to work. It is very important to note that a little knowledge of indices is required in order to be able to solve this problems. So you can quickly run through your multiplication, division and power law in indices
Solution:
1. Dimension of Area
Step 1: write down the formula
Area = Length X Breadth
Step 2: replace variables in the formula with their dimensions (see dimensions on the list above)
Area = L X L
apply multiplication law of indices
Area = L2
2. Dimension of volume
Step 1: write down the formula
Volume = Length X Breadth X Height
Step 2: replace variables in the formula with their dimensions (see dimensions on the list above)
= L X L X L
apply multiplication law of indices
= L3
3. Speed = $\frac{distance}{time}$
= \[\frac{L}{\begin{align}
& T \\
& L{{T}^{-1}} \\
\end{align}}\]
4. Velocity = $\frac{displacement}{\begin{align}
& time \\
& \frac{L}{\begin{align}
& T \\
& L{{T}^{-1}} \\
\end{align}} \\
\end{align}}$
5. Linear acceleration = $\frac{Velocity}{\begin{align}
& time \\
& \frac{L{{T}^{-1}}}{\begin{align}
& T \\
& =L{{T}^{-2}} \\
\end{align}} \\
\end{align}}$.
6. Momentum = Mass X Velocity
= M X LT-1
= MLT-1
7. Force = Mass X Acceleration
= M X LT-2
= MLT-2
8. Density = $\begin{align}
& \frac{mass}{\begin{align}
& volume \\
& =\frac{M}{{{L}^{3}}} \\
\end{align}} \\
& =M{{L}^{-3}} \\
\end{align}$
9. Work = Force X distance
= Mass X Acceleration X Distance
= M X LT-2 X L
= ML2T-2
10. Energy = Force X distance
= MLT-2 X L
= ML2T-2
11. Moment of Inertia = mass X length2
= M X L2
= ML2
12. Velocity Gradient = $\frac{Velocity}{\begin{align}
& dis\tan ce \\
& =\frac{M{{L}^{-1}}}{L} \\
& =M{{L}^{-2}} \\
\end{align}}$
13. Young’s modulus = $\frac{force\times length}{\begin{align}
& area\times extension \\
& =\frac{ML{{T}^{-2}}\times L}{{{L}^{2}}\times L} \\
& =M{{L}^{-1}}{{T}^{-2}} \\
\end{align}}$
14. Electric charge = Current X Time
= A X T
= AT
15. Electric potential or potential difference or voltage =
$\begin{align}
& \frac{Power}{Current} \\
& =\frac{\frac{workdone}{time}}{current} \\
& =\frac{\frac{Force\times dis\tan ce}{time}}{current} \\
& =\frac{\frac{ML{{T}^{-2}}\times L}{T}}{A} \\
& =\frac{M{{L}^{2}}{{T}^{-3}}}{A} \\
& =M{{L}^{2}}{{T}^{-3}}{{A}^{-1}} \\
\end{align}$
16. Capacitance $\begin{align}
& \frac{Ch\arg e}{Potential} \\
& =\frac{AT}{M{{L}^{2}}{{T}^{-3}}{{A}^{-1}}} \\
& ={{A}^{2}}{{T}^{4}}{{M}^{-1}}{{L}^{-2}} \\
\end{align}$
17. Strain $\begin{align}
& \frac{extension}{length} \\
& =\frac{L}{L} \\
& =\dimensionless \\
\end{align}$
You can see from question 17 that not all physical quantities can be dimensioned.
Test your understanding by attempting to write the dimension of the following physical quantities
Practice Exercises:
1. Universal gravitational constant (G)
2. Momentum
3. Resistance
4. Power
5. Pressure
6. Impulse
7. Electrical force
8. Escape velocity
9. Centripetal force
10. Weight
11. Temperature
12. coefficient of friction ยต
Hint: don’t be intimidated by the exercises given above, they are all simple and they follow the same steps as those above. Just follow the steps given above and you will get it. if you still have difficulties, you can let me know through the comment section and also remember to comment your answers too.
First, write down the formula. Sometimes, it is necessary to break down the formula to the root before proceeding. For example, Force = mass X acceleration, you can breakdown acceleration to get velocity/time, you can also breakdown velocity to get displacement/time and then replace them with their dimensions. .
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