Question

Dimensional formula of power is .
A. $\left[ {{M}^{2}}{{L}^{2}}{{T}^{-2}} \right]$
B. \[\left[ M{{L}^{2}}{{T}^{-3}} \right]\]
C. \[\left[ {{M}^{2}}L{{T}^{-2}} \right]\]
D. \[\left[ ML{{T}^{-2}} \right]\]

Answer 1

Hint: To write dimensional formulas for any quantity we need a formula for that. Then we write formulas in basic terms like mass , time, length etc.

We can define power with following formula as below:
\[Power=\dfrac{Work}{Time}\]

Complete step-by-step answer:
In general the rate of doing work is defined as power. We can write formula of power as below:
\[Power=\dfrac{Work}{Time}\]
Work is defined as force multiplied by distance.
Hence we can write formula of power as below:
\[Power=\dfrac{Force\,\times \,dis\tan ce}{Time}\]
Now we can write dimensional formulas for each quantity.
Dimensional formula for force is $\left[ ML{{T}^{-2}} \right]$ because $F=ma$ where m is mass and a is acceleration.
Dimensional formula for distance is $\left[ L \right]$
Dimensional formula for time is $\left[ T \right]$.
Now we can find dimensional formula for power is
\[\Rightarrow P=\dfrac{\left[ ML{{T}^{-2}} \right]\times \left[ L \right]}{\left[ T \right]}\]
\[\Rightarrow P=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ T \right]}\]
\[\Rightarrow P=\left[ M{{L}^{2}}{{T}^{-2-1}} \right]\]
\[\Rightarrow P=\left[ M{{L}^{2}}{{T}^{-3}} \right]\]
Hence option B is correct.

Note: To simplify dimensional formulas we can apply multiplication and division properties of exponent.
According to the multiplication property of exponent if we have exponent terms of the same base in multiplication then we can add their exponent. We can write it as below:
${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
According to the division property of exponent if we have exponent terms of the same base in division then we can subtract their exponent. We can write it as below:
${{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}$