Question

Derive the equation a pentulam T.It depends on mass of the bob,length of the pentulam and acceleration due gravity?

Answer 1

Period $T=$ time to complete one oscillation

At Equilibrium $T_0=mg$

Restoring force on the pendulum mass displaced $F_e=-mgsin\theta$

For small displacements $sin\theta=\theta$

$\theta=\frac{displacement x}{Length l}=\frac{ x}{ l}$

Thus:

$Fe=-mg(\frac{x}{ l})$

From Newton's second law $a=\frac{F_e}{m}=- \frac{g x}{ l}$

Also for simple harmonic motion $a=-x\omega^2$

$-x\omega^2=-\frac{gx}{l}$

$\omega=\sqrt{g/l}$

Since period $T=2\pi/\omega$

$T=2\pi\sqrt{l/g}$