Question

0.23 The frequency (n) of oscillation of a simple plndulum depends on length(i), mass of the bob(m) and acceleration due to gravity (g Using dimensional analysis derive relation between n, 1 m and g.​

Answer 1

Answer:

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−21L21

      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

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