Question

The time period of oscillation of a body is given by \[T = 2\pi \sqrt {\dfrac{{mgA}}{K}} \]. $K$: Represents the kinetic energy, in mass, g acceleration due to gravity and A is unknown.
If $\left[ A \right] = {M^x}{L^y}{T^z}$; then what is the value of $x + y + z.?$
A. $3$
B. $2$
C. $1$
D. $5$

Answer 1

Hint: We can solve this problem with the concept of unit and dimensions. If we measure a quantity, the measurement has two parts: unit and dimension. There are some fundamental quantities which are independent of the others and the units which measure these quantities are called fundamental units. The fundamental quantities are length, mass and time. Fundamental quantities are used for derivation of some other quantities known as derived quantities.

Complete answer:
A unit is a way to present a number or the measurement of the dimension while the dimension is a physical variable without number. In the other word, dimensions are the power to which the basic units are raised to obtain one unit of that quantity and the mathematical representation is called the dimensional formula of that quantity. For example, if p is the unit of a derived quantity represented as $P = \left[ {{M^a}{L^b}{T^c}} \right]$, the exponents $a,b$ and $c$ are called the dimension.
By the law of homogeneity of dimension, dimensions of all the terms must be the same on both sides and the same dimensional quantities can be added or subtracted. For the given question, the time period of oscillation of a body is given by \[T = 2\pi \sqrt {\dfrac{{mgA}}{K}} \]. $K$: Represents the kinetic energy, in mass, g acceleration due to gravity and A is unknown.
Dimension of time is $\left[ T \right]$, $K = \left[ {M{L^2}{T^{ - 2}}} \right]$,$g = \left[ {L{T^{ - 2}}} \right]$ and $A = {M^x}{L^y}{T^z}$
 $\left[ T \right]$= ${\left[ {\dfrac{{\left[ M \right]\left[ {L{T^{ - 2}}} \right]\left[ {{M^x}{L^y}{T^z}} \right]}}{{M{L^2}{T^{ - 2}}}}} \right]^{\dfrac{1}{2}}}$
${\left[ {\dfrac{{{M^x}{L^y}{T^z}}}{L}} \right]^{\dfrac{1}{2}}} = T$
$
  {M^x}{L^y}{T^z} = {M^0}L{T^2} \\
  x + y + z = 0 + 1 + 2 = 3 \\
$
$\left[ A \right] = {M^x}{L^y}{T^z}$; the value of $x + y + z = 3$.

So, the correct answer is “Option A”.

Note:
Dimensional analysis has some limitations such as that dimensional laws are not applicable for trigonometric, logarithmic and exponential functions. The quantities which have no dimension can not be determined by this method.