Question

obtain the formula for time period of the simple pendulum using dimensional analysis depends on mass M and length L and acceleration due to gravity G and K=2π​

Answer 1

Answer:

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Answer 2

DIMENSIONAL ANALYSIS AND OBTAINING FORMULA:

LET time period be denoted as :

$T \propto {M}^{x} {L}^{y} {g}^{z}$

• Expressing which of the quantities in terms of basic physical quantities like mass, length and time.

$\implies T \propto {M}^{x} {L}^{y} {(L{T}^{ - 2} )}^{z}$

$\implies T \propto {M}^{x} {L}^{y + z} {T}^{ - 2z}$

• Now, compare the exponentials on both sides and we obtain the following equations:

$1) \: x = 0$

$2) \: y + z = 0$

$3) \: - 2z = 1$

• After solving the equations, we get:

$x = 0,y = \dfrac{1}{2} ,z = - \dfrac{1}{2}$

So, our final equation is:

$\boxed{T = 2\pi \sqrt{ \dfrac{L}{g} } }$