Question

The time period of oscillation of a simple pendulum depends on the following quantities. length of the pendulum, mass of the bob, and acceleration due to gravity. Derive the expression for time period using dimensional method.?

Answer 1

Let Time period =T
      Mass of the bob = m
      Acceleration due to gravity = g
     Length of string = L

Let $T \alpha m ^{a}g ^{b}L ^{c}$
      $[T] \alpha [m] ^{a}[g] ^{b}[L] ^{c}$
      $M^{0}L^{0}T^{1}=M^{a}L^{b}T^{-2b}L^{c}$
      $M^{0}L^{0}T^{1}=M^{a}L^{b+c}T^{-2b}$
      ⇒a=0 ⇒ Time period of oscillation is independent of mass of the bob
     
      -2b=1
      ⇒b=-$\frac{1}{2}$
      
      b+c = 0
      -$\frac{1}{2}$ + c =0
      c=$\frac{1}{2}$
      
Giving values to a,b and c in first equation
      $T \alpha m ^{0}g ^{- \frac{1}{2} }L ^{ \frac{1}{2} }$
      $T \alpha \sqrt{ \frac{L}{g} }$

The real expression for Time period is
      $T =2 \pi \sqrt{ \frac{L}{g} }$

Therefore time period of oscillation depends only on gravity and length of the string.
Not on mass of the bob.

Answer 2

Answer:

Hope this answer helps you!!