Answer
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Hint: First of all, write the formula for the given quantities i.e., ${\text{force = mass}} {\times acceleration}$, ${\text{density = }}\dfrac{{{\text{mass}}}}{{{\text{volume}}}}$, ${\text{velocity = }}\dfrac{{{\text{change in displacement}}}}{{{\text{time}}}}$, ${\text{pressure = }}\dfrac{{{\text{force}}}}{{{\text{time}}}}$ and ${\text{pressure = }}\dfrac{{{\text{force}}}}{{{\text{time}}}}$and then write the dimensional formula for them.
Complete step by step solution:
${\text{density = }}\dfrac{{{\text{mass}}}}{{{\text{volume}}}}$Dimensions of a derived unit are the powers to which the fundamental units of mass (M), length (L), time (T) etc. must be raised to represent that unit.
Dimensional formula in actual is an expression that shows which fundamental units are required to represent the unit of a physical quantity.
(1) Formula for ${\text{force = mass}} {\times acceleration}$.
Dimensional formula for mass is ${\text{[}}{{\text{M}}^1}{{\text{L}}^0}{{\text{T}}^0}{\text{]}}$ and dimensional formula for acceleration is${\text{[}}{{\text{M}}^{\text{0}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$.
Thus, dimensional formula for force is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$.
SI unit of force is ${\text{kg m }}{{\text{s}}^{{\text{ - 2}}}}$.
(2) Dimensional formula for mass is ${\text{[}}{{\text{M}}^1}{{\text{L}}^0}{{\text{T}}^0}{\text{]}}$ and dimensional formula for volume is ${\text{[}}{{\text{M}}^0}{{\text{L}}^3}{{\text{T}}^0}{\text{]}}$.
Thus, dimensional formula for density is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{ - 3}}{{\text{T}}^0}{\text{]}}$.
SI unit of density is ${\text{kg }}{{\text{m}}^{{\text{ - 3}}}}$.
(3) Formula for ${\text{velocity = }}\dfrac{{{\text{change in displacement}}}}{{{\text{time}}}}$
Dimensional formula for displacement is ${\text{[}}{{\text{M}}^0}{{\text{L}}^1}{{\text{T}}^0}{\text{]}}$ and dimensional formula for time is ${\text{[}}{{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^1}{\text{]}}$.
Thus, dimensional formula for velocity is ${\text{[}}{{\text{M}}^0}{{\text{L}}^1}{{\text{T}}^{ - 1}}{\text{]}}$.
SI unit of velocity is ${\text{m }}{{\text{s}}^{{\text{ - 1}}}}$.
(4) Formula for ${\text{work = force}} {\times distance}$
Dimensional formula for force is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$ and dimensional formula for distance is ${\text{[}}{{\text{M}}^{\text{0}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{\text{0}}}{\text{]}}$.
Thus, dimensional formula for work is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^2}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$.
SI unit of work is ${\text{kg }}{{\text{m}}^2}{\text{ }}{{\text{s}}^{ - 2}}$.
(5) Formula for ${\text{pressure = }}\dfrac{{{\text{force}}}}{{{\text{time}}}}$
Dimensional formula for force is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$ and dimensional formula for time is ${\text{[}}{{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^1}{\text{]}}$.
Thus, dimensional formula for pressure is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{ - 1}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$.
SI unit of pressure is Pascal (${\text{Pa}}$).
Note: Dimensions are denoted with square brackets. In mechanics, mass, length and time are the basic quantities and the units used for the measurement of these quantities are known as fundamental units. Dimensional equation is that equation obtained by equating the physical quantity with its dimensional formula. For example, the dimensional formula for length is given as
${\text{[}}{{\text{M}}^0}{{\text{L}}^1}{{\text{T}}^0}{\text{]}}$ and the dimensional formula for area is ${\text{[}}{{\text{M}}^0}{{\text{L}}^2}{{\text{T}}^0}{\text{]}}$.
Complete step by step solution:
${\text{density = }}\dfrac{{{\text{mass}}}}{{{\text{volume}}}}$Dimensions of a derived unit are the powers to which the fundamental units of mass (M), length (L), time (T) etc. must be raised to represent that unit.
Dimensional formula in actual is an expression that shows which fundamental units are required to represent the unit of a physical quantity.
(1) Formula for ${\text{force = mass}} {\times acceleration}$.
Dimensional formula for mass is ${\text{[}}{{\text{M}}^1}{{\text{L}}^0}{{\text{T}}^0}{\text{]}}$ and dimensional formula for acceleration is${\text{[}}{{\text{M}}^{\text{0}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$.
Thus, dimensional formula for force is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$.
SI unit of force is ${\text{kg m }}{{\text{s}}^{{\text{ - 2}}}}$.
(2) Dimensional formula for mass is ${\text{[}}{{\text{M}}^1}{{\text{L}}^0}{{\text{T}}^0}{\text{]}}$ and dimensional formula for volume is ${\text{[}}{{\text{M}}^0}{{\text{L}}^3}{{\text{T}}^0}{\text{]}}$.
Thus, dimensional formula for density is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{ - 3}}{{\text{T}}^0}{\text{]}}$.
SI unit of density is ${\text{kg }}{{\text{m}}^{{\text{ - 3}}}}$.
(3) Formula for ${\text{velocity = }}\dfrac{{{\text{change in displacement}}}}{{{\text{time}}}}$
Dimensional formula for displacement is ${\text{[}}{{\text{M}}^0}{{\text{L}}^1}{{\text{T}}^0}{\text{]}}$ and dimensional formula for time is ${\text{[}}{{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^1}{\text{]}}$.
Thus, dimensional formula for velocity is ${\text{[}}{{\text{M}}^0}{{\text{L}}^1}{{\text{T}}^{ - 1}}{\text{]}}$.
SI unit of velocity is ${\text{m }}{{\text{s}}^{{\text{ - 1}}}}$.
(4) Formula for ${\text{work = force}} {\times distance}$
Dimensional formula for force is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$ and dimensional formula for distance is ${\text{[}}{{\text{M}}^{\text{0}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{\text{0}}}{\text{]}}$.
Thus, dimensional formula for work is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^2}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$.
SI unit of work is ${\text{kg }}{{\text{m}}^2}{\text{ }}{{\text{s}}^{ - 2}}$.
(5) Formula for ${\text{pressure = }}\dfrac{{{\text{force}}}}{{{\text{time}}}}$
Dimensional formula for force is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$ and dimensional formula for time is ${\text{[}}{{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^1}{\text{]}}$.
Thus, dimensional formula for pressure is ${\text{[}}{{\text{M}}^{\text{1}}}{{\text{L}}^{ - 1}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$.
SI unit of pressure is Pascal (${\text{Pa}}$).
Note: Dimensions are denoted with square brackets. In mechanics, mass, length and time are the basic quantities and the units used for the measurement of these quantities are known as fundamental units. Dimensional equation is that equation obtained by equating the physical quantity with its dimensional formula. For example, the dimensional formula for length is given as
${\text{[}}{{\text{M}}^0}{{\text{L}}^1}{{\text{T}}^0}{\text{]}}$ and the dimensional formula for area is ${\text{[}}{{\text{M}}^0}{{\text{L}}^2}{{\text{T}}^0}{\text{]}}$.
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