Derive an expression for the time period of simple pendulum using dimensional analysis?

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Answer

Derivation of Time Period of Simple Pendulum using Dimensional Analysis

**Identifying Variables:** Let's consider the variables involved in the time period of a simple pendulum: length (L), acceleration due to gravity (g), and time period (T).
**Formulating the Equation:** The time period (T) of a simple pendulum can be expressed as a function of length (L) and acceleration due to gravity (g) as T = f(L, g).
**Analyzing Dimensions:** The dimensions of length, acceleration, and time are [L], [LT^-2], and [T], respectively.
**Using Dimensional Analysis:** By examining the dimensions of the variables involved, we can write the equation as [T] = [L]^a [g]^b, where 'a' and 'b' are exponents to be determined.
**Determining Exponents:** Equating dimensions on both sides, we get [T] = [L]^a [g]^b. This leads to:
[T] = L^a (LT^-2)^b
[T] = L^a L^b T^-2b
[T] = L^(a+b) T^(-2b)
**Equating Dimensions:** Equating the dimensions of time on both sides gives:
1 = -2b
b = -1/2
**Substitute Exponents:** Substituting the value of 'b' back into the equation, we get:
[T] = L^(a-1/2) T
**Equating Dimensions (again):** Equating dimensions of time on both sides gives:
1 = a - 1/2
a = 3/2
**Final Expression:** Therefore, the expression for the time period of a simple pendulum using dimensional analysis is: T = k√(L/g), where 'k' is a constant.