Question

Show that the expression of the time period $T$ of a simple pendulum of length $l$ given by $T = 2\pi \sqrt {\dfrac{l}{g}} $ is dimensionally correct.
As in the above equation, the dimension of both sides is the same. The given formula is
A. dimensionally correct.
B. The dimension of L.H.S and R.H.S are not same
C. It cannot be solved
D. None of these

Answer 1

Hint: To solve this problem we should understand about the Units and Measurement also the Dimensions and Dimensional Analysis and the concept behind them. Then with our knowledge and the given information, we will approach our answer.

Complete answer:
Unit: A unit is an internationally accepted standard for measurements of quantities. As a physical quantity it is an arbitrary chosen standard which is widely accepted by the society and by which other quantities of similar nature may be measured.
Measurement consists of a numeric quantity along with a relevant unit. Units for fundamental or base quantities are called fundamental units. Units including a combination of fundamental units are called Derived units. Fundamental units are derived units together form a system of units.
Measurement: The process of measurement is primarily a comparison process. To measure a physical quantity, we have to find out how many times a standard amount of that quantity is present in the quantity being measured.
Dimension: The dimension of a physical quantity states the powers to which the fundamental units of mass, length and time must be raised to represent the given physical quantity.
$T = 2\pi \sqrt {\dfrac{l}{g}} $

Dimensionally $\left[ T \right]\, = \sqrt {\dfrac{{\left[ L \right]}}{{\left[ {L{T^{ - 2}}} \right]}}} = \left[ T \right]$

As we can see (in the above equation), the dimensions of both sides are the same. The given formula is dimensionally correct.

Hence, the correct answer is option A - As in the above equation, the dimension of both sides is the same. The given formula is dimensionally correct.

Note: The concept of dimension and dimensional formulae are put to the following uses, checking the result obtained, Conversion from one system of unit to another, Deriving relationships between physical quantities.