Question

The division of energy by time is X. The dimensional formula of X is same as that of

A. momentum

B. power

C. torque

D. electric field

Answer 1

Find out the dimensional formula for the energy first by either using the potential or kinetic or any other energy formula. Then by dividing it with the dimensional formula of time, we get a dimensional formula of X. Then, we can find out the dimensional formula for the options given and match it with that of X to get the final answer.
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Complete step by step solution:**

We will be using the dimensional notations for length, mass and time as L, M and T respectively.

We have been given that, $X=\dfrac{\text{energy}}{\text{time}}$ ………. (i)

So, we first need to find out the dimensional formula of energy, which we can find by using the formula of kinetic energy, $E_K= \dfrac{mv^2}{2}$ ………. (ii)

Dimensional formula for mass (m) is given by $[M^1 L^0 T^0]$ ………. (iii)

And, the dimensional formula for velocity (v) is $[M^0 L^1 T^{-1}]$ ………. (iv)

Substituting, equations (iii) and (iv) in equation (ii), we get the dimensional formula of energy, $E_K=[M^1 L^0 T^0]\times [M^0 L^1 T^{-1}]^2 = [M^1 L^2 T^{-2}]$ ………. (v)

Since the dimensional formula of time is, $[M^0 L^0 T^1]$ ………. (vi)

Substituting equations (v) and (vi) in equation (i), we get the dimensional formula for,

$X=[M^1 L^2 T^{-2}]\times [M^0 L^0 T^1]^{-1}=[M^1 L^2 T^{-3}]$

Now, we can find the dimensional formulae of the options given and then can match with the dimensional formula of X.

For momentum,

The formula for momentum is given by $p=mv$. Thus, by using equations (iii) and (iv), we get the dimensional formula of momentum as, $p=[M^1 L^0 T^0]\times [M^0 L^1 T^{-1}]=[M^1 L^1 T^{-1}]$, which is not same as of X.

For power,

Power is given by the work done per unit time. And we know that the unit of energy and work done is the same.

Therefore, dimensional formula for power we can get by using the equations (v) and (vi), that is $P=[M^1 L^2 T^{-2}]\times [M^0 L^0 T^1]^{-1}=[M^1 L^2 T^{-1}]$, which is same as the dimensional formula of X.

Hence, option (B) is the correct answer.

Note: Another direct approach can be by using the definition of power. We know that the basic definition of power is the work done per unit time and the work done is always in the form of either energy dissipation or energy absorption. So ultimately, power will be the energy per unit time and thus will be the same as X.