Question

Time period of simple pendulum depens upon 1-mass (m) of bob, 2-effective length of pendulum(l), 3-acceleration due to gravity at that place(g).Obtain an expression for the time period of a simple pendulum.

Answer 1

Howdy!!

your answer is -----

Given, time period of simple pendulum(T) depends upon , mass (m) ,effective length (l), and acceleration due to gravity (g),

so ,

T = $k \: {l}^{x} \: {m}^{y} \: {g}^{z}$
......... (1)
k = some constant

now, write the dimensions of each,

[M°L°T^1] = [L]^x [M]^y [LT^-2]^z
= [L^x+z M^y T^-2z ]

comparing both side , we get

y = 0 , x + z = 0, -2z = 1

=> y = 0 , x = 1/2 , z = -1/2

now, put this value in equation (1),

T = k l^1/2 m ^0 g^-1/2

=> T = k √l/√g

and , by experimentally the value of
k = 2π,

so, final formula is

T = $2\pi \frac{ \sqrt{l} }{ \sqrt{g} } = 2\pi \sqrt{ \frac{l}{g} }$
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hope it help you