Question

The dimensions of $\left(\dfrac{force\ constant}{mass}\right)^{\dfrac{1}{2}}$ are the same as that
a) acceleration
b) angular acceleration
c) angular velocity
d) none of the above.

Answer 1

The dimensions of a physical term are the exponents through which the primary units of mass, length, time, etc., must be raised to express it. M, L, and T represent the dimensions of the three mechanical terms mass, length, and time. They can also express by using the brackets [M], [L], and [T].

Complete answer:
Formula for force constant is-
$\dfrac{Force}{Length}$
So, dimensions for force constant is-
$\dfrac{\left[MLT^{-2}\right]}{[L]} = \left[MT^{-2}\right]$
$\left(\dfrac{force\ constant}{mass}\right)^{\dfrac{1}{2}}$ having dimension-
$\left[\dfrac{\left[MT^{-2}\right]}{[M]}\right]^\dfrac{1}{2} = \left[T^{-1}\right]$
Now we will check options.
a) acceleration
Formula for acceleration is-
$\dfrac{Force}{Mass}$ . So, Dimensions for acceleration will be
$\dfrac{\left[MLT^{-2}\right]}{[M]} = \left[LT^{-2}\right]$
b) angular acceleration
Formula for angular acceleration is-
$\dfrac{angular\ velocity}{time}$ . So, Dimensions for angular acceleration will be
$\left[T^{-2}\right]$.
c) angular velocity
Formula for angular acceleration is-
$\dfrac{angular\ rotation}{time}$. So, Dimensions for angular velocity will be
$\left[T^{-1}\right]$.

Option (c) is correct.

Additional Information:
Other dimensions are represented by mol(for the amount of substance), K(for temperature), I(for electric current), cd(for luminous intensity). The dimensions of a physical term and the dimensions of its unit are equal.

Note:
Dimensions help in determining the accuracy of a given relation, obtain a connection between the several physical quantities, and change the physical quantity value from one system to a different one. Also noted that there are various physical quantities with the same dimensions, it is challenging to know them by knowing dimensions alone.