Question

obtain the expression for time period of oscillating bob of simple pendulum.



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Answer 1

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.?
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}TαmagbLc 
      [T] \alpha [m] ^{a}[g] ^{b}[L] ^{c}[T]α[m]a[g]b[L]c 
      M^{0}L^{0}T^{1}=M^{a}L^{b}T^{-2b}L^{c}M0L0T1=MaLbT−2bLc 
      M^{0}L^{0}T^{1}=M^{a}L^{b+c}T^{-2b}M0L0T1=MaLb+cT−2b 
      ⇒a=0 ⇒ Time period of oscillation is independent of mass of the bob
      
      -2b=1
      ⇒b=-\frac{1}{2}21​ 
      
      b+c = 0
      -\frac{1}{2}21​ + c =0
      c=\frac{1}{2}21​ 
      
Giving values to a,b and c in first equation
      T \alpha m ^{0}g ^{- \frac{1}{2} }L ^{ \frac{1}{2} }Tαm0g−21​L21​ 
      T \alpha \sqrt{ \frac{L}{g} }TαgL​​ 

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

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