Balancing a pencil!

Consider a pencil that stands upright on its tip and then falls over. Let’s idealize the pencil as a mass $m$ sitting at the end of a mass less rod of length $l$

Assume that the pencil makes an initial (small) angle $\theta_{0}$ with the vertical, and that its initial angular speed is $\omega_{0}$ . The angle will eventually become large, but while it is small (so that $\sin \theta \approx \theta$ ), what is $\theta$ as a function of time?

#Physics #Mechanics

Easy Math Editor

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Comments

First tell whether slipping is taking place or not, and if it is taking place , is the surface frictionless.

If slipping is not taking place , then , take moment about lowest point.

$mglsin\theta$ = $ml^2\alpha$ [Using : $sin\theta = \theta$ ]

$\Rightarrow \frac{d^2\theta}{dt^2} = \frac{g}{l}\theta$

Solve this 2nd order differential equation to get :

$\Rightarrow \theta = \frac{\theta_{0} + w_{0}\sqrt{\frac{l}{g}}}{2} e^{\sqrt{\frac{g}{l}}t} + \frac{\theta_{0} - w_{0}\sqrt{\frac{l}{g}}}{2} e^{-\sqrt{\frac{g}{l}}t}$

jatin yadav - 7 years, 7 months ago

How did you solve the differential equation?

Krishna Jha - 7 years, 7 months ago

Something known as wolfram :)

Actually, 2nd order diff. equations aren't taught in class 12th.

jatin yadav - 7 years, 7 months ago

@Jatin Yadav – This is a second order equation with constant coefficients. So we simply need to find roots of auxiliary equation. Solution that I wrote below is equivalent to your solution after we rearrange the terms.

Snehal Shekatkar - 7 years, 7 months ago

I did not recheck my calculation so answer that I am posting might be wrong. My answer is $\theta(t)=\theta_{0} \cosh(\omega t)+\frac{\omega_{0}}{\omega}\sinh(\omega t)$ for small $\theta$ . Here $\omega=\sqrt{\frac{g}{l}}$ and $\theta_{0}$ is an initial angular displacement.

Snehal Shekatkar - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$