Q: Derive an expression for the time period of a simple pendulum of mass (m), length (l) at a place where acceleration due to gravity is (g).

Sol. Let the time period of a simple pendulum depend upon the mass of bob m. length of pendulum l, and acceleration due to gravity g, then

t ∝ m^{a} l^{b} g^{c}

⇒ t = k m^{a} l^{b} g^{c}

M^{0} L^{0} T^{1} = M^{a} L^{b} [LT^{-2}]^{c}

⇒ M^{0} L^{0} T^{1} = M^{a} L^{b+c} T^{-2c}

Comparing the powers of M, L, and T on both sides, we get a = 0, b =1/2 and c = -1/2. Putting these values,

⇒ a = 0,b = 1/2 and c = -1/2.Putting these values,

$\large T = k\sqrt{\frac{l}{g}}$

which is the required relation.