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Grade 11Mechanics

ii) State the approximation that has to be made in this equation to reduce it to an equation of simpleharmonic motion.

abhishek shekhar

7 Years agoGrade 11

1 Answer

Samyak Jain

Approved Tutor Answer7 Years ago

Displace the pendulum slightly from mean (vertical) position making angle $\theta$ with vertical and resolve mg along the string and perpendicular to it.

You will find that the restoring force of simple pendulum is mgsin $\theta$ .

$\therefore$ restoring force is F = – mgsin $\theta$ . Also, F = ma.

a = – gsin $\theta$ . $\because$ a = d²x / dt² , d²x / dt² = – gsin $\theta$ ….(1)

Now, x is slight displacement, considering the circular part of moion, we get

x = $l$ $\theta$ $\Rightarrow$ $\theta$ = x / $l$

Differentiate both sides wrt t i.e. d $\theta$ /dt = (1/ $l$ )dx/dt. Here 1/ $l$ is constant, $l$ being length of string.

Again differentiate both sides wrt t to get d² $\theta$ /dt² = (1/ $l$ )d²x/dt² or d²x/dt² = $l$ d² $\theta$ /dt² .

Put in (1).

$l$ d² $\theta$ /dt² = – gsin $\theta$ , which is the required equation to be obtained.
sin $\theta$ = $\theta$ is the approximation to be made to reduce above equation to an equation of simple harmonic motion.

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