In the previous section, we saw the combination of errors. We saw some solved examples also. In this section, we will see dimensions of physical quantities
1. At the beginning of this chapter we saw some details about the seven base quantities and derived quantities (details here)
• We also saw base units and derived units
2. All derived quantities can be expressed in terms of a 'combination'
• Each of the derived quantities will have a definite combination
3. That combination will contain two or more of the seven base quantities
• The seven base quantities are called the seven dimensions of the physical world
4. We represent dimension by writing the quantity within square brackets
♦ [M] represent the dimension mass
♦ [L] represent the dimension length
♦ [T] represent the dimension time
♦ [A] represent the dimension electric current
♦ [K] represent the dimension thermodynamic temperature
♦ [cd] represent the dimension luminous intensity
♦ [mol] represent the dimension amount of substance
5. Each of the dimensions in a combination will have to be raised to a particular power
• When all the members in the combination are raised in this way, we get the physical quantity represented by that combination
6. Let us see an example:
(i) The physical quantity ‘speed’ is given by $\mathbf\small{\text{Speed}=\frac{\text{Distance}}{\text{Time}}}$
(ii) Distance has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
• Time has the dimension [T]
♦ It occurs only once. So it is raised to the power ‘1’
(iii) Thus we write:
• Dimensions of speed are:
$\mathbf\small{\frac{[L]}{[T]}=[L]\times [T]^{-1}=\left[LT^{-1}\right]}$
(iv) We can write:
• Speed has:
♦ 1 dimension in length
♦ -1 dimension in time
7. Another example:
(i) The physical quantity ‘momentum’ is given by Momentum = Mass × Velocity
• But 'velocity' is given by $\mathbf\small{\text{Velocity}=\frac{\text{Displacement}}{\text{Time}}}$
• So $\mathbf\small{\text{Momentum}=\text{Mass}\times\frac{\text{Displacement}}{\text{Time}}}$
(ii) Mass has the dimension [M]
♦ It occurs only once. So it is raised to the power ‘1’
• Displacement has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
• Time has the dimension [T]
♦ It occurs only once. So it is raised to the power ‘1’
(iii) Thus we write:
• Dimensions of momentum are:
$\mathbf\small{[M]\times\frac{[L]}{[T]}=[M]\times[L]\times [T]^{-1}=\left[MLT^{-1}\right]}$
(iv) We can write:
• Momentum has:
♦ 1 dimension in mass
♦ 1 dimension in length
♦ -1 dimension in time
8. Another example:
(i) The physical quantity ‘volume’ is given by:
Volume = Length × width × height
(ii) Length has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
• Width also has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
• Height also has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
(iii) Thus we write:
• Dimensions of volume are:
$\mathbf\small{[L]\times[L]\times[L]\times=[L]^3=\left[L^3\right]}$
(iv) We can write:
• Volume has:
♦ 3 dimensions in length
♦ 1 dimension in length
♦ -1 dimension in time
• We can modify the above information as:
Speed has:
♦ 0 dimension in mass
♦ 1 dimension in length
♦ -1 dimension in time
• So the dimensions of speed can be written as: $\mathbf\small{\left[M^0LT^{-1}\right]}$
1. Consider a ‘change in velocity’
• It is given by: Final velocity – Initial velocity’
• So when we say ‘change in velocity’, the ‘magnitude of the change’ is important
♦ In other words, ‘how much change’ occurred is important
2. But ‘how much change’ is not important when we consider dimensions
• It is the ‘type of physical quantity in which the change occurred’ that is important
3. In our present case, the 'type of physical quantity in which the change occurred’ is the velocity
• So we can write:
♦ 'Velocity' has the dimensions $\mathbf\small{\left[M^0LT^{-1}\right]}$
♦ 'Change in velocity' also has the dimensions $\mathbf\small{\left[M^0LT^{-1}\right]}$
4. In the same way, we can write:
♦ 'Average velocity' also has the dimensions $\mathbf\small{\left[M^0LT^{-1}\right]}$
♦ 'Speed' also has the dimensions $\mathbf\small{\left[M^0LT^{-1}\right]}$
(i) When the members of the combination are raised to the appropriate powers, we can get an idea about how the base quantities represent a physical quantity
(ii) Also, we can get an idea about which base quantities represent a physical quantity
■ The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity
• So we can write:
♦ Dimensional formula of velocity is $\mathbf\small{\left[LT^{-1}\right]}$
♦ Dimensional formula of momentum is $\mathbf\small{\left[MLT^{-1}\right]}$
♦ Dimensional formula of volume is $\mathbf\small{\left[M^0L^3T^0\right]}$
♦ Dimensional formula of acceleration is $\mathbf\small{\left[M^0 LT^{-2}\right]}$
♦ Dimensional formula of force is $\mathbf\small{\left[MLT^{-2}\right]}$
♦ Dimensional formula of density is $\mathbf\small{\left[ML^{-3}T^0\right]}$
■ Thus, the dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities
• Some examples are given below:
♦ Dimensional equation of velocity [v] is:
$\mathbf\small{[v]=\left[LT^{-1}\right]}$
♦ Dimensional equation of momentum [p] is:
$\mathbf\small{[p]=\left[MLT^{-1}\right]}$
♦ Dimensional equation of volume [V] is:
$\mathbf\small{[V]=\left[M^0L^3T^0\right]}$
♦ Dimensional equation of acceleration [a] is:
$\mathbf\small{[a]=\left[M^0 LT^{-2}\right]}$
♦ Dimensional equation of force [F] is:
$\mathbf\small{[F]=\left[MLT^{-2}\right]}$
♦ Dimensional equation of density [ρ] is
$\mathbf\small{[\rho]=\left[ML^{-3}T^0\right]}$
An example:
(i) We have dimensional equation for force: $\mathbf\small{[F]=\left[MLT^{-2}\right]}$
(ii) It is obtained using the equation: Force = Mass × Acceleration
• This equation represent the relation between the three physical quantities:
Force, mass and acceleration
Another example:
(i) We have dimensional equation for density: $\mathbf\small{[\rho]=\left[ML^{-3}T^0\right]}$
(ii) It is obtained using the equation: $\mathbf\small{\text{Density}=\frac{\text{Mass}}{\text{Volume}}}$
• This equation represent the relation between the three physical quantities:
Density, mass and volume
1. At the beginning of this chapter we saw some details about the seven base quantities and derived quantities (details here)
• We also saw base units and derived units
2. All derived quantities can be expressed in terms of a 'combination'
• Each of the derived quantities will have a definite combination
3. That combination will contain two or more of the seven base quantities
• The seven base quantities are called the seven dimensions of the physical world
4. We represent dimension by writing the quantity within square brackets
♦ [M] represent the dimension mass
♦ [L] represent the dimension length
♦ [T] represent the dimension time
♦ [A] represent the dimension electric current
♦ [K] represent the dimension thermodynamic temperature
♦ [cd] represent the dimension luminous intensity
♦ [mol] represent the dimension amount of substance
5. Each of the dimensions in a combination will have to be raised to a particular power
• When all the members in the combination are raised in this way, we get the physical quantity represented by that combination
6. Let us see an example:
(i) The physical quantity ‘speed’ is given by $\mathbf\small{\text{Speed}=\frac{\text{Distance}}{\text{Time}}}$
(ii) Distance has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
• Time has the dimension [T]
♦ It occurs only once. So it is raised to the power ‘1’
(iii) Thus we write:
• Dimensions of speed are:
$\mathbf\small{\frac{[L]}{[T]}=[L]\times [T]^{-1}=\left[LT^{-1}\right]}$
(iv) We can write:
• Speed has:
♦ 1 dimension in length
♦ -1 dimension in time
7. Another example:
(i) The physical quantity ‘momentum’ is given by Momentum = Mass × Velocity
• But 'velocity' is given by $\mathbf\small{\text{Velocity}=\frac{\text{Displacement}}{\text{Time}}}$
• So $\mathbf\small{\text{Momentum}=\text{Mass}\times\frac{\text{Displacement}}{\text{Time}}}$
(ii) Mass has the dimension [M]
♦ It occurs only once. So it is raised to the power ‘1’
• Displacement has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
• Time has the dimension [T]
♦ It occurs only once. So it is raised to the power ‘1’
(iii) Thus we write:
• Dimensions of momentum are:
$\mathbf\small{[M]\times\frac{[L]}{[T]}=[M]\times[L]\times [T]^{-1}=\left[MLT^{-1}\right]}$
(iv) We can write:
• Momentum has:
♦ 1 dimension in mass
♦ 1 dimension in length
♦ -1 dimension in time
8. Another example:
(i) The physical quantity ‘volume’ is given by:
Volume = Length × width × height
(ii) Length has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
• Width also has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
• Height also has the dimension [L]
♦ It occurs only once. So it is raised to the power ‘1’
(iii) Thus we write:
• Dimensions of volume are:
$\mathbf\small{[L]\times[L]\times[L]\times=[L]^3=\left[L^3\right]}$
(iv) We can write:
• Volume has:
♦ 3 dimensions in length
• Mechanics is a branch of physics
• All the physical quantities that we encounter in this branch can be written in terms of the dimensions [L], [M] and [T].
1. We saw the dimensions of speed: $\mathbf\small{\left[LT^{-1}\right]}$
• We wrote that, speed has:
• All the physical quantities that we encounter in this branch can be written in terms of the dimensions [L], [M] and [T].
1. We saw the dimensions of speed: $\mathbf\small{\left[LT^{-1}\right]}$
• We wrote that, speed has:
♦ -1 dimension in time
• We can modify the above information as:
Speed has:
♦ 0 dimension in mass
♦ 1 dimension in length
♦ -1 dimension in time
• So the dimensions of speed can be written as: $\mathbf\small{\left[M^0LT^{-1}\right]}$
2. We saw the dimensions of momentum: $\mathbf\small{\left[MLT^{-1}\right]}$
• All the three basic quantities have a place in the combination. No modification is needed
3. We saw the dimensions of volume: $\mathbf\small{\left[L^3\right]}$
♦ 3 dimensions in length
• We can modify the above information as:
Volume has:
♦ 0 dimension in mass
♦ 3 dimensions in length
♦ 0 dimension in time
• So the dimensions of volume can be written as: $\mathbf\small{\left[M^0L^3T^0\right]}$
• All the three basic quantities have a place in the combination. No modification is needed
3. We saw the dimensions of volume: $\mathbf\small{\left[L^3\right]}$
• We wrote that, volume has:
• We can modify the above information as:
Volume has:
♦ 0 dimension in mass
♦ 3 dimensions in length
♦ 0 dimension in time
• So the dimensions of volume can be written as: $\mathbf\small{\left[M^0L^3T^0\right]}$
• When we write dimensions, the magnitudes are not considered. It is the 'quality of the type of the physical quantity' that enters. Let us see how this information is applicable
• It is given by: Final velocity – Initial velocity’
• So when we say ‘change in velocity’, the ‘magnitude of the change’ is important
♦ In other words, ‘how much change’ occurred is important
2. But ‘how much change’ is not important when we consider dimensions
• It is the ‘type of physical quantity in which the change occurred’ that is important
3. In our present case, the 'type of physical quantity in which the change occurred’ is the velocity
• So we can write:
♦ 'Velocity' has the dimensions $\mathbf\small{\left[M^0LT^{-1}\right]}$
♦ 'Change in velocity' also has the dimensions $\mathbf\small{\left[M^0LT^{-1}\right]}$
4. In the same way, we can write:
♦ 'Average velocity' also has the dimensions $\mathbf\small{\left[M^0LT^{-1}\right]}$
♦ 'Speed' also has the dimensions $\mathbf\small{\left[M^0LT^{-1}\right]}$
Let us see an example based on the above information:
(i) The physical quantity ‘acceleration’ is given by $\mathbf\small{\text{Acceleration}=\frac{\text{Change in velocity}}{\text{Time}}}$
(ii) 'Change in velocity' and 'velocity' have the same dimensions: $\mathbf\small{\left[M^0LT^{-1}\right]}$
(iii) Time has the dimension [T]
♦ It occurs only once. So it is raised to the power ‘1’
(iv) Thus we write:
• Dimensions of acceleration are:
$\mathbf\small{\frac{[LT^{-1}]}{[T]}=[LT^{-1}]\times [T]^{-1}=\left[LT^{-2}\right]}$
(v) We can write:
• Acceleration has:
♦ 0 dimension in mass
♦ 1 dimension in length
♦ -1 dimension in time
(vi) So the dimensions of acceleration are: $\mathbf\small{\left[M^0 LT^{-2}\right]}$
Another example:
(i) The physical quantity ‘force’ is given by Force = Mass × Acceleration
(ii) Mass has the dimension [M]
♦ It occurs only once. So it is raised to the power ‘1’
(iii) Acceleration has the dimension $\mathbf\small{\left[M^0 LT^{-2}\right]}$
(iv) Thus we write:
• Dimensions of force are:
$\mathbf\small{[M]\times \left[LT^{-2}\right]=\left[MLT^{-2}\right]}$
(v) We can write:
Momentum has:
♦ 1 dimension in mass
♦ 1 dimension in length
♦ -2 dimension in time
(i) The physical quantity ‘acceleration’ is given by $\mathbf\small{\text{Acceleration}=\frac{\text{Change in velocity}}{\text{Time}}}$
(ii) 'Change in velocity' and 'velocity' have the same dimensions: $\mathbf\small{\left[M^0LT^{-1}\right]}$
(iii) Time has the dimension [T]
♦ It occurs only once. So it is raised to the power ‘1’
(iv) Thus we write:
• Dimensions of acceleration are:
$\mathbf\small{\frac{[LT^{-1}]}{[T]}=[LT^{-1}]\times [T]^{-1}=\left[LT^{-2}\right]}$
(v) We can write:
• Acceleration has:
♦ 0 dimension in mass
♦ 1 dimension in length
♦ -1 dimension in time
(vi) So the dimensions of acceleration are: $\mathbf\small{\left[M^0 LT^{-2}\right]}$
Another example:
(i) The physical quantity ‘force’ is given by Force = Mass × Acceleration
(ii) Mass has the dimension [M]
♦ It occurs only once. So it is raised to the power ‘1’
(iii) Acceleration has the dimension $\mathbf\small{\left[M^0 LT^{-2}\right]}$
(iv) Thus we write:
• Dimensions of force are:
$\mathbf\small{[M]\times \left[LT^{-2}\right]=\left[MLT^{-2}\right]}$
(v) We can write:
Momentum has:
♦ 1 dimension in mass
♦ 1 dimension in length
♦ -2 dimension in time
From the above discussions, two points become clear:
(ii) Also, we can get an idea about which base quantities represent a physical quantity
■ The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity
• So we can write:
♦ Dimensional formula of velocity is $\mathbf\small{\left[LT^{-1}\right]}$
♦ Dimensional formula of momentum is $\mathbf\small{\left[MLT^{-1}\right]}$
♦ Dimensional formula of volume is $\mathbf\small{\left[M^0L^3T^0\right]}$
♦ Dimensional formula of acceleration is $\mathbf\small{\left[M^0 LT^{-2}\right]}$
♦ Dimensional formula of force is $\mathbf\small{\left[MLT^{-2}\right]}$
♦ Dimensional formula of density is $\mathbf\small{\left[ML^{-3}T^0\right]}$
■ An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity
• Some examples are given below:
♦ Dimensional equation of velocity [v] is:
$\mathbf\small{[v]=\left[LT^{-1}\right]}$
♦ Dimensional equation of momentum [p] is:
$\mathbf\small{[p]=\left[MLT^{-1}\right]}$
♦ Dimensional equation of volume [V] is:
$\mathbf\small{[V]=\left[M^0L^3T^0\right]}$
♦ Dimensional equation of acceleration [a] is:
$\mathbf\small{[a]=\left[M^0 LT^{-2}\right]}$
♦ Dimensional equation of force [F] is:
$\mathbf\small{[F]=\left[MLT^{-2}\right]}$
♦ Dimensional equation of density [ρ] is
$\mathbf\small{[\rho]=\left[ML^{-3}T^0\right]}$
• The dimensional equation can be obtained from the equation representing the relations between the physical quantities.
(i) We have dimensional equation for force: $\mathbf\small{[F]=\left[MLT^{-2}\right]}$
(ii) It is obtained using the equation: Force = Mass × Acceleration
• This equation represent the relation between the three physical quantities:
Force, mass and acceleration
Another example:
(i) We have dimensional equation for density: $\mathbf\small{[\rho]=\left[ML^{-3}T^0\right]}$
(ii) It is obtained using the equation: $\mathbf\small{\text{Density}=\frac{\text{Mass}}{\text{Volume}}}$
• This equation represent the relation between the three physical quantities:
Density, mass and volume
In the next section, we will see dimensional analysis
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