Question

derive an expression t=2π√l/g using dimentional analysis​

Answer 1

Answer:

Hint: Here, we will proceed by determining the dimensions of the quantities involved in the given equation (i.e., time, length and acceleration due to gravity) and then, we will take dimensions of the quantities on both the sides of the given equation.

Complete Step-by-Step solution:

Given, Time period of a simple pendulum, T=2πlg−−√ →(1) where l is length of the pendulum and g is acceleration due to gravity.

As we know that

Dimension of time period T = T

Dimension of length of the pendulum l = Dimension of length = L

Since, Acceleration = VelocityTime=(DisplacementTime)Time

⇒ Acceleration = Displacement(Time)2 →(2)

As, dimension of displacement (length) is L and that of time is T

By applying dimensional analysis on equation (2), we get

Dimension of acceleration = Dimension of displacement(Dimension of time)2=[L][T2]=[L][T−2]=[LT−2]

Dimension of acceleration due to gravity (acceleration) is [LT−2]

When we will be applying dimensional analysis on equation (1), 2π is a constant which is getting multiplied so it will be neglected.

By applying dimensional analysis on equation (1), we get

Dimension of T = Dimension of lDimension of g−−−−−−−−−−−√

⇒[T]=[L][LT−2]−−−−−−⎷⇒[T]=[[L][L−1T2]]12⇒[T]=[[L1−1T2]]12⇒[T]=[[L0T2]]12⇒[T]=⎡⎣⎢T2×12⎤⎦⎥⇒[T]=[T]