Question

Jason knows that the equation to calculate the period of a simple pendulum is , where T is the period, L is the length of the rod, and g is the acceleration due to gravity. He also knows that the frequency (f) of the pendulum is the reciprocal of its period. How can he express L in terms of g and f?

Answer 1

Answer-

$\boxed{\boxed{L=\dfrac{g}{4\pi^2 f^2}}}$

Solution-

The equation for time period of a simple pendulum is given by,

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

Where,

T = Time period,

L = Length of the rod,

g = Acceleration due to gravity.

Frequency (f) of the pendulum is the reciprocal of its period, i.e

$f=\dfrac{1}{T}\ \Rightarrow T=\dfrac{1}{f}$

Putting the values,

$\Rightarrow \dfrac{1}{f}=2\pi \sqrt{\dfrac{L}{g}}$

$\Rightarrow (\dfrac{1}{f})^2=(2\pi \sqrt{\dfrac{L}{g}})^2$

$\Rightarrow \dfrac{1}{f^2}=4\pi^2 \dfrac{L}{g}$

$\Rightarrow L=\dfrac{g}{4\pi^2 f^2}$

Answer 2

1/f = 2π√(L/g)
1/(2πf) = √(L/g) . . . . . divide by 2π
1/(2πf)^2 = L/g . . . . . .square both sides
g/(2πf)^2 = L . . . . . . .multiply by g

L = g/(4π^2f^2) . . . . . . . matches the 1st selection