Question

The period T of a simple pendulum is the amount of time required for it to undergo one complete oscillation. If the length of the pendulum is L and the acceleration of gravity is g, then T is given by
`T = 2\pi L^pg^q`

Find the powers p,q required for dimensional consistency.

Answer 1

1/2, - 1/2

Step-by-step explanation:

`T=2\pi L^(p)* g^(q)`

Write the dimensions on both the sides

The dimensions of T are [T]

The dimensions of L are [L]

The dimensions of g are
`[LT^(-2)]`

So,
`[T]=[L]^(p)][LT^(-2)]^(q)`

`[T]=[L^(p+q)T^(-2q)]`

Now compare the dimensions on both the sides for the corresponding terms

p + q = 0 ...... (1)

-2q = 1 ......... (2)

By solving equation (1) and (2), we get

q = - 1/2

p = 1 / 2

Thus, the value of p and q are 1/2 and - 1/2 respectively.