Question

Time period of oscillation of pendulum depends on the mass of the bog, length of the pendulum and Acceleration due to gravity. Obtain the relation between them using dimensional analysis.
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Answer 1

Answer:

Explanation:

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

check the attachment :)