Question

In the expression $R{{V}_{ext}}=ma$, R stands for
(A). Resistance
(B). Rate of fuel consumption
(C). Rate of change of velocity
(D). Rate of change of mass

Answer 1

Hint: Try to understand the concepts of dimensional analysis. Express the derived physical quantities in terms of fundamental quantities. After finding the dimension of each quantity try to equate them to make the equation correct. Then you will find the dimension of the unknown quantity.

Complete step by step answer:
If we want to find an unknown quantity, we can check it from the dimensional formula of the equation. All equations should be always dimensionally correct.
To check if the equation is dimensionally correct, we should check the dimension of each quantity involved in the equation.
In the equation $R{{V}_{ext}}=ma$, the known quantities are ${{V}_{ext}}$, m and a, where ${{V}_{ext}}$ is the velocity, m is the mass and a is the acceleration of the object. The only unknown quantity is R.
Now, Dimension of ${{V}_{ext}}$ is = $\left[ {{M}^{0}}{{L}^{1}}{{T}^{-1}} \right]$
Dimension of m is = $\left[ {{M}^{1}}{{L}^{0}}{{T}^{0}} \right]$
Dimension of acceleration is = $\left[ {{M}^{0}}{{L}^{1}}{{T}^{-2}} \right]$
Now, to check if the unknown quantity,
$\begin{align}
  & \left[ R \right]\times \left[ {{M}^{0}}{{L}^{1}}{{T}^{-1}} \right]=\left[ {{M}^{1}}{{L}^{0}}{{T}^{0}} \right]\otimes \left[ {{M}^{0}}{{L}^{1}}{{T}^{-2}} \right] \\
 & \left[ R \right]\times \left[ {{M}^{0}}{{L}^{1}}{{T}^{-1}} \right]=\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right] \\
 & \left[ R \right]=\dfrac{\left[ {{M}^{1}}{{L}^{1}}{{T}^{-2}} \right]}{\left[ {{M}^{0}}{{L}^{1}}{{T}^{-1}} \right]} \\
 & \left[ R \right]=\left[ {{M}^{1}}{{L}^{0}}{{T}^{-1}} \right] \\
\end{align}$
Hence, the unit of R should be mass per unit time (or Kg per second)
Hence, R stands for rate of change of mass
The correct option is (D)
Additional information: With the help of the dimensional formula we can either check the correctness of an equation or we can find some unknown quantity in the equation.

Note: Never try to remember the dimensional formula of a physical quantity. Always express them in terms of the fundamental quantities. Then you can find the dimensions of a physical quantity easily.