Question

If ‘Muscular strength’ times ‘Speed’ is equal to power, then the dimensional formula for ‘Muscular strength’ is:
(A) $MLT$
(B) $ML{T^{ - 2}}$
(C) $M{L^2}{T^{ - 2}}$
(D) $M{L^0}{T^{ - 2}}$

Answer 1

Dimensional formula is an expression that shows the power to which the fundamental units to be raised in order to obtain one unit of a derived quantity.
Dimensions of a physical quantity are the powers to which fundamental quantities must raise to represent the given physical quantity. For example, Speed ${\text{ = }}\dfrac{{{\text{Distance}}}}{{{\text{time}}}}$ or Speed $= \left( {{\text{Distance}}} \right) \times {\left( {{\text{time}}} \right)^{ - 1}}$.
Here dimensions of speed are 1 in distance is and -1 in time is.
The dimensional formula of the magnetic field can be calculated by using the formula Muscular strength: M $= \dfrac{{{\text{Power}}}}{{{\text{Speed}}}}$ that has come from Power: P${\text{ = Muscular Strength}} \times {\text{Speed}}$.

Complete step by step answer:
To calculate the dimensional formula of Muscular strength we take the formula
Muscular strength $= \dfrac{{{\text{Power}}}}{{{\text{Speed}}}}$…………………….. (i)
This is the Muscular strength associated with the speed and power of a moving body.
We have to write each physical quantity used in its dimensional form in equation (i).
Power: $P = \dfrac{{{\text{Work}}}}{{{\text{time}}}}$ $= \dfrac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ T \right]}}$
$\Rightarrow P = \left[ {M{L^2}{T^{ - 3}}} \right]$
Speed (V) must be taken as meter/second:
$v = \left[ {L{T^{ - 1}}} \right]$
Substituting the value of $v$ and $P$ in equation (i), we get:
Muscular strength (M):
${\text{M = }}\dfrac{{\left[ {M{L^2}{T^{ - 3}}} \right]}}{{\left[ {L{T^{ - 1}}} \right]}}$
On solving we get,
${\text{M = }}\left[ {ML{T^{ - 2}}} \right]$

The dimensional formula for ‘Muscular strength’ is $ML{T^{ - 2}}$. Hence, the correct answer is option (B).

Note:
In order to answer this type of question, the key is to know the dimensional formulas of seven fundamental quantities and should have the ability to apply them for solving the dimensional formula for derived quantities. While solving the dimensional formula questions students must note that the given physical quantities must be expressed in its absolute units only. One should notice that the dimensional formula that we get for muscular strength is the same as the force applied so from here we can say that the term muscular strength used in the question is actually the force applied on a body to get the desired speed.