Question

If force \[F\] , acceleration \[a\] , and time \[T\] are taken as the fundamental physical quantities, the dimension of length on this system of units:
$\left( A \right)FA{T^2}$
$\left( B \right)FAT$
$\left( C \right)FT$
$\left( D \right)A{T^2}$

Answer 1

Hint: In the CGS system we express gram as the unit of mass, centimeter as the unit of length, and the second as the unit of time. First equate the energy dimension with the dimension of force, acceleration, and time. Then, write the dimension of energy in terms of force, acceleration, and time.

Complete step by step solution:
The physical quantities are measured by these 3 systems that are the F.P.S system (foot, pound, second) C.G.S (centimeter, gram, second), and M.K.S(meter, kilogram, second).
In the CGS system, we express gram as the unit of mass, centimeter as the unit of length, and the second as the unit of time.
The powers to which a physical quantity is raised is called the dimension of a physical quantity. They represent a physical quantity.
Dimensionless quantities are the quantities without a dimensional formula. It is used to find the correctness of an equation. It determines relationships between different physical quantities. It can also find the unit of a given physical quantity.
The energy dimension,
$\Rightarrow E = K{F^a}{A^b}{T^c}$
Equating energy dimension with force, acceleration, and time.
$\Rightarrow \left[ {M{L^2}{T^{ - 2}}} \right] = {\left[ {ML{T^{ - 2}}} \right]^a}{\left[ {L{T^{ - 2}}} \right]^b}{\left[ T \right]^c}$
$\Rightarrow \left[ {M{L^2}{T^{ - 2}}} \right] = \left[ {{M^a}{L^{a + b}}{T^{ - 2a - 2b + c}}} \right]$
Now we get,
$\Rightarrow a = 1,a + b = 2 \Rightarrow b = 1$
$ \Rightarrow - 2a - 2b + c = - 2 \Rightarrow b = 1$
Hence,
$\Rightarrow E = KFA{T^2}$, K is a constant.

Hence option A is the correct option.

Note: The force which produces an acceleration $1c{m^2}$ in a body of mass of one gram, in its direction is called dyne. IN the CGS system we express gram as the unit of mass, centimeter as the unit of length, and the second as the unit of time. Dimensionless quantities are the quantities without a dimensional formula. It is used to find the correctness of an equation.