Question

The period (T) of a simple pendulum depends on the length (I) of the simple pendulum and the acceleration due to gravity (g) at a place obtains an expression for (T) by the method of dimensions​

Answer 1

EXPRESSION- T∝√L/√g⇒ T∝ k √L/√g, Where K is the proportionality constant and K= 2π

Let's discuss that how this expression was obtained.

As we are given with the information that the time period is a dependent quantity and depends on the length of the pendulum and the acceleration due to the gravity, So, the expression obtained from this is:

        T  ∝$L^{a}$$g^{b}$ ____eq1

    Then now by converting the g(acceleration due to gravity) to its dimensional value we get:

                                     g= {L$T^{-2}$]____eq2

Now, By combining eq1 and eq2 we get;

                          T∝ $[L]^{a}$$[LT^{-2}]^{b}$

                      =>$L^{0}$$T^{1}$∝$L^{a+b}$$T^{-2b}$

Now comparing the power of T and L we get:

                     a=1/2 and b=  - 1/2

Now by putting it to the eqn1 we can easily get the expression:

                     T  ∝$L^{1/2}$$g^{1/2}$