Answer:
The expression is shown in the explanation below:
Explanation:
Thinking process:
Let the time period of a simple pendulum be given by the expression:
[tex]T = \pi \sqrt{\frac{l}{g} }[/tex]
Let the fundamental units be mass= M, time = t, length = L
Then the equation will be in the form
[tex]T = M^{a}l^{b}g^{c}[/tex]
[tex]T = KM^{a}l^{b}g^{c}[/tex]
where k is the constant of proportionality.
Now putting the dimensional formula:
[tex]T = KM^{a}L^{b} [LT^{-} ^{2}]^{c}[/tex]
[tex]M^{0}L^{0}T^{1} = KM^{a}L^{b+c}[/tex]
Equating the powers gives:
a = 0
b + c = 0
2c = 1, c = -1/2
b = 1/2
so;
a = 0 , b = 1/2 , c = -1/2
Therefore:
[tex]T = KM^{0}l^{\frac{1}{2} } g^{\frac{1}{2} }[/tex]
T = [tex]2\pi \sqrt{\frac{l}{g} }[/tex]
where k = [tex]2\pi[/tex]
